1284:
2680:
782:
333:, and for studying mixed-state dynamics in one-dimensional quantum lattice systems,. Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states.
2285:
27:
219:
fully characterize a quantum many-body system is seriously impeded by the lavishly exponential buildup with the system size of the amount of variables needed for simulation, which leads, in the best cases, to unreasonably long computational times and extended use of memory. To get around this problem a number of various methods have been developed and put into practice in the course of time, one of the most successful ones being the
9600:
16423:
6260:
5491:
324:, or in systems far from equilibrium in quantum transport. From this point of view, TEBD had a certain ascendance over DMRG, a very powerful technique, but until recently not very well suited for simulating time-evolutions. With the Matrix Product States formalism being at the mathematical heart of DMRG, the TEBD scheme was adopted by the DMRG community, thus giving birth to the time dependent DMRG
15886:
8266:
4468:
1279:{\displaystyle c_{i_{1}i_{2}..i_{N}}=\sum \limits _{\alpha _{1},..,\alpha _{N-1}=0}^{\chi }\Gamma _{\alpha _{1}}^{i_{1}}\lambda _{\alpha _{1}}^{}\Gamma _{\alpha _{1}\alpha _{2}}^{i_{2}}\lambda _{\alpha _{2}}^{}\Gamma _{\alpha _{2}\alpha _{3}}^{i_{3}}\lambda _{\alpha _{3}}^{}\cdot ..\cdot \Gamma _{\alpha _{N-2}\alpha _{N-1}}^{i_{N-1}}\lambda _{\alpha _{N-1}}^{}\Gamma _{\alpha _{N-1}}^{i_{N}}}
162:
2992:
5116:
5970:
14670:
16112:
5123:
2675:{\displaystyle |{\phi _{1}^{A}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{A}}\rangle +|{1_{A}}\rangle ),\ \ |{\phi _{1}^{B}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{B}}\rangle +|{1_{B}}\rangle ),\ \ |{\phi _{2}^{A}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{A}}\rangle -|{1_{A}}\rangle ),\ \ |{\phi _{2}^{B}}\rangle ={\frac {1}{\sqrt {2}}}(|{1_{B}}\rangle -|{0_{B}}\rangle )}
17041:
Schmidt dimension is just two. Hence, at the first bond, instead of futilely diagonalizing, let us say, 10 by 10 or 20 by 20 matrices, we can just restrict ourselves to ordinary 2 by 2 ones, thus making the algorithm generally faster. What we can do instead is set a threshold for the eigenvalues of the SD, keeping only those that are above the threshold.
2280:
14447:
9073:
9315:
7924:
7629:
8582:
15579:
8029:
4169:
17027:
2829:
8759:
3456:
2824:
4836:
15239:
4157:
12921:
16104:
14452:
282:-like Hamiltonians. The method exhibits a low-degree polynomial behavior in the increase of computational time with respect to the amount of entanglement present in the system. The algorithm is based on a scheme that exploits the fact that in these one-dimensional systems the eigenvalues of the reduced
17040:
One thing that can save a lot of computational time without loss of accuracy is to use a different
Schmidt dimension for each bond instead of a fixed one for all bonds, keeping only the necessary amount of relevant coefficients, as usual. For example, taking the first bond, in the case of qubits, the
17031:
When using the
Trotter expansion, we do not move from bond to bond, but between bonds of same parity; moreover, for the ST2, we make a sweep of the even ones and two for the odd. But nevertheless, the calculation presented above still holds. The error is evaluated by successively multiplying with the
12074:
2107:
239:
theory (QIT), for genuine quantum algorithms, appropriate for problems that would perform badly when trying to be solved on a classical computer, but pretty fast and successful on a quantum one. The search for such algorithms is still going, the best-known (and almost the only ones found) being the
15025:
14228:
255:
In the field of QIT one has to identify the primary resources necessary for genuine quantum computation. Such a resource may be responsible for the speedup gain in quantum versus classical, identifying them means also identifying systems that can be simulated in a reasonably efficient manner on a
218:
in parameters with the size of the system, and correspondingly, the high computational costs, one solution would be to look for numerical methods that deal with special cases, where one can profit from the physics of the system. The raw approach, by directly dealing with all the parameters used to
293:
One can also estimate the amount of computational resources required for the simulation of a quantum system on a classical computer, knowing how the entanglement contained in the system scales with the size of the system. The classically (and quantum, as well) feasible simulations are those that
234:
is plugged in and functioning, the perspectives for the field of computational physics will look rather promising, but until that day one has to restrict oneself to the mundane tools offered by classical computers. While experimental physicists are putting a lot of effort in trying to build the
7642:
7367:
16640:
2102:
17044:
TEBD also offers the possibility of straightforward parallelization due to the factorization of the exponential time-evolution operator using the Suzuki–Trotter expansion. A parallel-TEBD has the same mathematics as its non-parallelized counterpart, the only difference is in the numerical
16800:
4827:
14873:
13575:
6255:{\displaystyle |{\Psi }\rangle ={\frac {1}{\sqrt {\sum \limits _{{\alpha _{l}}=1}^{{\chi }_{c}}{|\lambda _{{\alpha }_{l}}^{}|}^{2}}}}\cdot \sum \limits _{{{\alpha }_{l}}=1}^{{\chi }_{c}}\lambda _{{\alpha }_{l}}^{}|{\Phi _{\alpha _{l}}^{}}\rangle |{\Phi _{\alpha _{l}}^{}}\rangle ,}
8430:
16812:
6989:
1510:
16418:{\displaystyle n_{2}=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}\cdot ({\lambda '}_{\alpha _{l}}^{})^{2}=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}{\frac {(\lambda _{\alpha _{l}}^{})^{2}}{R}}={\frac {S_{1}}{R}}}
1964:. The Schmidt rank of a bipartite split is given by the number of non-zero Schmidt coefficients. If the Schmidt rank is one, the split is characterized by a product state. The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique.
11226:, conserving the norm (unlike power series expansions), and there's where the Trotter-Suzuki expansion comes in. In problems of quantum dynamics the unitarity of the operators in the ST expansion proves quite practical, since the error tends to concentrate in the overall
9815:
5486:{\displaystyle |{\Phi _{\alpha _{k}}^{}}\rangle =\sum _{\alpha _{k+1},\alpha _{k+2}..\alpha _{N}}\Gamma _{\alpha _{k}\alpha _{k+1}}^{i_{k+1}}\lambda _{\alpha _{k+1}}^{}\cdot \cdot \lambda _{\alpha _{N-1}}^{N-1}\Gamma _{\alpha _{N-1}}^{i_{N}}|{i_{k+1}i_{k+2}..i_{N}}\rangle }
14223:
9308:
8587:
3252:
9595:{\displaystyle \lambda _{\beta }^{'}|{\Phi _{\beta }^{'}}\rangle =\langle {\Phi _{\beta }^{'}}|{\psi '}\rangle =\sum _{i,j,\alpha ,\gamma }(\Gamma _{\beta \gamma }^{'j})^{*}\Theta _{\alpha \gamma }^{ij}(\lambda _{\gamma })^{2}\lambda _{\alpha }|{{\alpha }i}\rangle }
12443:
and the appropriate
Schmidt eigenvectors. Mind this would imply diagonalizing somewhat generous reduced density matrices, which, depending on the system one has to simulate, might be a task beyond our reach and patience. Instead, one can try to do the following:
13826:
2687:
15474:
3671:
3957:
12760:
710:
15881:{\displaystyle n_{1}=1=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}(\lambda _{\alpha _{l}}^{})^{2}+\sum \limits _{\alpha _{l}=\chi _{c}}^{\chi }(c_{\alpha _{l-1}\alpha _{l}})^{2}(\lambda _{\alpha _{l}}^{})^{2}=S_{1}+S_{2},}
15030:
8863:
8261:{\displaystyle |{\psi }\rangle =\sum \limits _{\alpha ,\beta ,\gamma =1}^{\chi }\sum \limits _{i,j=1}^{M}\lambda _{\alpha }^{}\Gamma _{\alpha \beta }^{i}\lambda _{\beta }^{}\Gamma _{\beta \gamma }^{j}\lambda _{\gamma }^{}|{{\alpha }ij{\gamma }}\rangle }
1623:
4463:{\displaystyle |{\Psi }\rangle =\sum _{i_{1},i_{2},\alpha _{1},\alpha _{2}}\Gamma _{\alpha _{1}}^{i_{1}}\lambda _{\alpha _{1}}^{}\Gamma _{\alpha _{1}\alpha _{2}}^{i_{2}}\lambda _{{\alpha }_{2}}^{}|{i_{1}i_{2}}\rangle |{\Phi _{\alpha _{2}}^{}}\rangle }
3190:
11917:
10708:
2987:{\displaystyle \left|\Phi \right\rangle =\left({\frac {1}{\sqrt {3}}}\left|0_{A}\right\rangle -{\frac {i}{\sqrt {3}}}\left|1_{A}\right\rangle \right)\otimes \left(\left|0_{B}\right\rangle +{\frac {1}{\sqrt {2}}}\left|1_{B}\right\rangle \right)}
10833:
14030:
15894:
11653:
139:
many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block
Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original
11084:
16495:
1970:
16651:
14880:
5111:{\displaystyle |{\Phi _{\alpha _{k}}^{}}\rangle =\sum _{\alpha _{1},\alpha _{2}..\alpha _{k-1}}\Gamma _{\alpha _{1}}^{i_{1}}\lambda _{\alpha _{1}}^{}\cdot \cdot \Gamma _{\alpha _{k-1}\alpha _{k}}^{i_{k}}|{i_{1}i_{2}..i_{k}}\rangle }
4664:
11190:
10512:
14734:
13938:
13447:
14665:{\displaystyle |{\psi _{D}^{\bot }}\rangle ={\frac {1}{\sqrt {\epsilon _{n}}}}\sum \limits _{{{\alpha }_{n}}={\chi }_{c}}^{\chi }\lambda _{{\alpha }_{n}}^{}|{\Phi _{\alpha _{n}}^{}}\rangle |{\Phi _{\alpha _{n}}^{}}\rangle }
5961:
1398:
10210:
5850:
is useful when dealing with systems that exhibit a low degree of entanglement, which fortunately is the case with many 1D systems, where the
Schmidt coefficients of the ground state decay in an exponential manner with
14113:
13389:
6817:
2275:{\displaystyle \left|{\Psi }\right\rangle ={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}\left|{\phi _{1}^{A}\phi _{1}^{B}}\right\rangle +{\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\left|{\phi _{2}^{A}\phi _{2}^{B}}\right\rangle }
11730:
14442:{\displaystyle |{\psi _{D}}\rangle ={\frac {1}{\sqrt {1-\epsilon _{n}}}}\sum \limits _{{{\alpha }_{n}}=1}^{{\chi }_{c}}\lambda _{{\alpha }_{n}}^{}|{\Phi _{\alpha _{n}}^{}}\rangle |{\Phi _{\alpha _{n}}^{}}\rangle }
13205:
7280:
7919:{\displaystyle \rho ^{}=\sum _{\gamma }{(\lambda _{\gamma }^{})^{2}}|{\Phi _{\gamma }^{}}\rangle \langle {\Phi _{\gamma }^{}}|=\sum _{\gamma }{(\lambda _{\gamma }^{})^{2}}|{\gamma }\rangle \langle {\gamma }|.}
7624:{\displaystyle \rho ^{}=\sum _{\alpha }{(\lambda _{\alpha }^{})}^{2}|{\Phi _{\alpha }^{}}\rangle \langle {\Phi _{\alpha }^{}}|=\sum _{\alpha }{(\lambda _{\alpha }^{})^{2}}|{\alpha }\rangle \langle {\alpha }|.}
9679:
9174:
13690:
7360:
5616:
15345:
14729:
3527:
3245:
12683:
1363:
10352:, hence its role is not very relevant for the order of magnitude of the number of basic operations, but in the case when the on-site dimension is higher than two it has a rather decisive contribution.
1962:
546:
8425:
10129:
330:
Around the same time, other groups have developed similar approaches in which quantum information plays a predominant role as, for example, in DMRG implementations for periodic boundary conditions
12416:
for an arbitrary n-particle state, since this would mean one has to compute the
Schmidt decomposition at each bond, to arrange the Schmidt eigenvalues in decreasing order and to choose the first
3821:
3741:
1864:
541:
10324:
10267:
8577:{\displaystyle |{\psi '}\rangle =\sum \limits _{\alpha ,\gamma =1}^{\chi }\sum \limits _{i,j=1}^{M}\lambda _{\alpha }\Theta _{\alpha \gamma }^{ij}\lambda _{\gamma }|{{\alpha }ij\gamma }\rangle }
17022:{\displaystyle \epsilon ({D})=1-\prod \limits _{n=1}^{N-1}(1-\epsilon _{n})\prod \limits _{n=1}^{N-1}{\frac {1-2\epsilon _{n}}{1-\epsilon _{n}}}=1-\prod \limits _{n=1}^{N-1}(1-2\epsilon _{n})}
3905:
3522:
397:
13440:
11433:
15574:
12259:
12152:
16490:
13662:
10081:
9916:
10587:
10026:
8754:{\displaystyle \Theta _{\alpha \gamma }^{ij}=\sum \limits _{\beta =1}^{\chi }\sum \limits _{m,n=1}^{M}V_{mn}^{ij}\Gamma _{\alpha \beta }^{m}\lambda _{\beta }\Gamma _{\beta \gamma }^{n}.}
3952:
3094:
10712:
11531:
9878:
6810:
6294:
Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition. Consider for instance the case of 50
3451:{\displaystyle |{\Psi }\rangle =\sum \limits _{i_{1},{\alpha _{1}=1}}^{M,\chi }\Gamma _{\alpha _{1}}^{i_{1}}\lambda _{\alpha _{1}}^{}|{i_{1}}\rangle |{\Phi _{\alpha _{1}}^{}}\rangle }
1734:
1665:
9639:
9134:
7173:
13324:
11466:
13617:
12754:
12588:
11570:
6551:
5676:
1515:
777:
13092:
13055:
12986:
12548:
12508:
12360:
12323:
10990:
5788:
12203:
11230:, thus allowing us to faithfully compute expectation values and conserved quantities. Because the ST conserves the phase-space volume, it is also called a symplectic integrator.
5965:
Therefore, it is possible to take into account only some of the
Schmidt coefficients (namely the largest ones), dropping the others and consequently normalizing again the state:
2819:{\displaystyle |{\Phi }\rangle ={\frac {1}{\sqrt {3}}}|{00}\rangle +{\frac {1}{\sqrt {6}}}|{01}\rangle -{\frac {i}{\sqrt {3}}}|{10}\rangle -{\frac {i}{\sqrt {6}}}|{11}\rangle }
14084:
8022:
7206:
6620:
1393:
13291:
11907:
3099:
13256:
11563:
11089:
8854:
425:
10385:
4152:{\displaystyle |\tau _{\alpha _{1}i_{2}}^{}\rangle =\sum _{\alpha _{2}}\Gamma _{\alpha _{1}\alpha _{2}}^{i_{2}}\lambda _{{\alpha }_{2}}^{}|{\Phi _{\alpha _{2}}^{}}\rangle }
13015:
12916:{\displaystyle |{\psi _{gr}}\rangle =\lim _{\tau \rightarrow \infty }{\frac {e^{-{\tilde {H}}\tau }|{\psi _{P}}\rangle }{\|e^{-{\tilde {H}}\tau }|{\psi _{P}}\rangle \|}},}
12714:
12617:
11218:
15506:
9169:
8369:
5817:
9674:
8313:
7127:
7088:
6759:
6706:
6667:
4536:
272:"any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted"
16099:{\displaystyle (c_{\alpha _{l-1}\alpha _{l}})^{2}=\sum \limits _{i_{l}=1}^{d}(\Gamma _{\alpha _{l-1}\alpha _{l}}^{i_{l}})^{*}\Gamma _{\alpha _{l-1}\alpha _{l}}^{i_{l}}}
15234:{\displaystyle \epsilon =1-|\langle {\psi }|\psi '_{D}\rangle |^{2}=1-(1-\epsilon _{n+1})|\langle {\psi }|\psi _{D}\rangle |^{2}=1-(1-\epsilon _{n+1})(1-\epsilon _{n})}
14108:
14054:
13943:
13852:
12471:
12441:
12414:
12283:
10973:
10904:
10580:
9984:
9960:
8807:
6327:
6289:
11267:
9095:
9068:{\displaystyle \rho ^{'}=Tr_{JC}|{\psi '}\rangle \langle {\psi '}|=\sum _{j,j',\gamma ,\gamma '}\rho _{\gamma \gamma '}^{jj'}|{j\gamma }\rangle \langle {j'\gamma '}|.}
8781:
8333:
7040:
12096:
7020:
6726:
5869:
5735:
4494:
1763:
9936:
6401:
6374:
6347:
4596:
4566:
13119:
12949:
12387:
10541:
7980:
7953:
6571:
6491:
6448:
6428:
5844:
5700:
5529:
3847:
1790:
1692:
452:
11874:
11836:
11803:
11765:
5876:
15340:
10350:
172:
14731:
because they are spanned by vectors corresponding to orthogonal spaces. Using the same argument as for the
Trotter expansion, the error after the truncation is:
15285:
15262:
13685:
11486:
10378:
4646:
3047:
13329:
12069:{\displaystyle |{{\tilde {\psi }}_{t+\delta }}\rangle =e^{-i{\frac {\delta }{2}}F}e^{{-i\delta }G}e^{{\frac {-i\delta }{2}}F}|{{\tilde {\psi }}_{t}}\rangle .}
11657:
286:
on a bipartite split of the system are exponentially decaying, thus allowing us to work in a re-sized space spanned by the eigenvectors corresponding to the
13124:
227:(DMRG) method, next to QMC, is a very reliable method, with an expanding community of users and an increasing number of applications to physical systems.
17191:
M. Zwolak; G. Vidal (2004). "Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm".
7213:
16635:{\displaystyle \epsilon ={\frac {S_{1}}{R}}-1\leq {\frac {1-R}{R}}={\frac {\epsilon _{l}}{1-\epsilon _{l}}}{\to }0\ \ as\ \ {\epsilon _{l}{\to }{0}}}
2097:{\displaystyle |{\Psi }\rangle ={\frac {1}{2{\sqrt {2}}}}\left(|{00}\rangle +{\sqrt {3}}|{01}\rangle +{\sqrt {3}}|{10}\rangle +|{11}\rangle \right)}
294:
involve systems only lightly entangled—the strongly entangled ones being, on the other hand, good candidates only for genuine quantum computations.
16795:{\displaystyle |\langle {\psi _{D}}|\psi _{D}\rangle |^{2}=1-{\frac {\epsilon _{l}}{1-\epsilon _{l}}}={\frac {1-2\epsilon _{l}}{1-\epsilon _{l}}}}
15020:{\displaystyle |{\psi _{D}}\rangle ={\sqrt {1-\epsilon _{n+1}}}|{{{\psi }'}_{D}}\rangle +{\sqrt {\epsilon _{n+1}}}|{{\psi '}_{D}^{\bot }}\rangle }
4822:{\displaystyle |{\Psi }\rangle =\sum _{\alpha _{k}}\lambda _{{\alpha }_{k}}^{}|{\Phi _{\alpha _{k}}^{}}\rangle |{\Phi _{\alpha _{k}}^{}}\rangle }
44:
183:
13849:
First, as we have seen above, the smallest contributions to the
Schmidt spectrum are left away, the state being faithfully represented up to:
91:
17128:
F. Verstraete; J. J. Garcia-Ripoll; J. I. Cirac (2004). "Matrix
Product Density Operators: Simulation of finite-T and dissipative systems".
14868:{\displaystyle \epsilon _{n}=1-|\langle {\psi }|\psi _{D}\rangle |^{2}=\sum \limits _{\alpha =\chi _{c}}^{\chi }(\lambda _{\alpha }^{})^{2}}
63:
13570:{\displaystyle |{{\tilde {\psi }}_{Tr}}\rangle ={\sqrt {1-{\epsilon }^{2}}}|{\psi _{Tr}}\rangle +{\epsilon }|{\psi _{Tr}^{\bot }}\rangle }
10134:
11272:
1505:{\displaystyle \left\vert \Psi \right\rangle =\sum \limits _{i=1}^{M_{A|B}}a_{i}\left\vert {\Phi _{i}^{A}\Phi _{i}^{B}}\right\rangle }
70:
13216:
The errors in the simulation are resulting from the Suzuki–Trotter approximation and the involved truncation of the Hilbert space.
14672:
is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected. Now,
12924:
or, alternatively, simulate an isentropic evolution using a time-dependent Hamiltonian, which interpolates between the Hamiltonian
11909:
and an ST expansion of the first order keeps only the product of the exponentials, the approximation becoming, in this case, exact.
7294:
5534:
14675:
14218:{\displaystyle |{\psi }\rangle ={\sqrt {1-\epsilon _{n}}}|{\psi _{D}}\rangle +{\sqrt {\epsilon _{n}}}|{\psi _{D}^{\bot }}\rangle }
3195:
12622:
6984:{\displaystyle \Gamma _{\alpha _{k-1}\alpha _{k}}^{'i_{k}}=\sum _{j}U_{j_{k}}^{i_{k}}\Gamma _{\alpha _{k-1}\alpha _{k}}^{j_{k}}.}
1314:
305:
interpolations between a target Hamiltonian and a Hamiltonian with an already-known ground state. The computational time scales
77:
1869:
287:
12261:
again to the odd ones; this is basically a sequence of TQG's, and it has been explained above how to update the decomposition
17340:
13017:; the evolution must be done slow enough, such that the system is always in the ground state or, at least, very close to it.
3758:
3678:
260:; hence, it is possible to establish a distinct lower bound for the entanglement needed for quantum computational speedups.
457:
224:
59:
9810:{\displaystyle |{\Phi ^{'}}\rangle =\sum _{i,\alpha }\Gamma _{\alpha \beta }^{'i}\lambda _{\alpha }|{{\alpha }i}\rangle .}
1795:
10272:
10215:
8376:
6406:
Even if the Schmidt eigenvalues don't have this exponential decay, but they show an algebraic decrease, we can still use
9303:{\displaystyle |{\Phi ^{'}}\rangle =\sum _{j,\gamma }\Gamma _{\beta \gamma }^{'j}\lambda _{\gamma }|{j\gamma }\rangle .}
13821:{\displaystyle \epsilon (T)=1-|\langle {\tilde {\psi _{Tr}}}|{\psi _{Tr}}\rangle |^{2}=1-1+\epsilon ^{2}=\epsilon ^{2}}
3852:
3469:
356:
267:
15469:{\displaystyle R={\sum \limits _{{\alpha _{l}}=1}^{{\chi }_{c}}{|\lambda _{{\alpha }_{l}}^{}|}^{2}}={1-\epsilon _{l}}}
13396:
3666:{\displaystyle |{\Phi _{\alpha _{1}}^{}}\rangle =\sum _{i_{2}}|{i_{2}}\rangle |{\tau _{\alpha _{1}i_{2}}^{}}\rangle }
201:
110:
15511:
12208:
12101:
16430:
13622:
10086:
10031:
9883:
10975:
This is done to make the Suzuki–Trotter expansion (ST) of the exponential operator, named after Masuo Suzuki and
705:{\displaystyle |\Psi \rangle =\sum \limits _{i=1}^{M}c_{i_{1}i_{2}..i_{N}}|{i_{1},i_{2},..,i_{N-1},i_{N}}\rangle }
148:
in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.
10987:
The Suzuki–Trotter expansion of the first order (ST1) represents a general way of writing exponential operators:
9989:
3910:
3052:
275:
17382:
11499:
9835:
6764:
48:
3460:
The process can be decomposed in three steps, iterated for each bond (and, correspondingly, SD) in the chain:
1697:
1628:
1618:{\displaystyle |{\Phi _{i}^{A}\Phi _{i}^{B}}\rangle =|{\Phi _{i}^{A}}\rangle \otimes |{\Phi _{i}^{B}}\rangle }
325:
270:, has recently proposed a scheme useful for simulating a certain category of quantum systems. He asserts that
9605:
9100:
7132:
13296:
11438:
84:
17032:
normalization constant, each time we build the reduced density matrix and select its relevant eigenvalues.
13582:
12719:
12553:
6496:
5621:
1300:
722:
13060:
13023:
12954:
12516:
12476:
12328:
12291:
5740:
12157:
3185:{\displaystyle \left|{\Phi _{\alpha _{1}}^{}}\right\rangle \left|{\Phi _{\alpha _{1}}^{}}\right\rangle }
179:
17377:
17313:
Hatano, Naomichi; Suzuki, Masuo (2005-11-16). "Finding Exponential Product Formulas of Higher Orders".
13842:
Considering the errors arising from the truncation of the Hilbert space comprised in the decomposition
12619:
by a sufficiently local transformation Q (one that can be expressed as a product of TQGs, for example)
14059:
7997:
7181:
6595:
1368:
144:. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of
13261:
10703:{\displaystyle F\equiv \sum _{{\text{even }}l}(K_{1}^{l}+K_{2}^{l,l+1})=\sum _{{\text{even }}l}F^{},}
3744:
13227:
11536:
3001:
At this point we know enough to try to see how we explicitly build the decomposition (let's call it
402:
10828:{\displaystyle G\equiv \sum _{{\text{odd }}l}(K_{1}^{l}+K_{2}^{l,l+1})=\sum _{{\text{odd }}l}G^{}.}
14025:{\displaystyle \epsilon _{n}=\sum \limits _{\alpha =\chi _{c}}^{\chi }(\lambda _{\alpha }^{})^{2}}
12991:
12690:
12593:
11197:
176:
that states a Knowledge editor's personal feelings or presents an original argument about a topic.
17252:
Vidal, Guifré (2004-07-19). "Efficient Simulation of One-Dimensional Quantum Many-Body Systems".
15481:
9139:
8338:
5793:
37:
11879:
9644:
8285:
7093:
7060:
6731:
6672:
6633:
4499:
1866:. This is called the Schmidt decomposition (SD) of a state. In general the summation goes up to
16645:
14089:
14035:
12452:
12419:
12395:
12264:
10909:
10840:
10546:
9965:
9941:
8816:
8790:
6578:
6305:
6267:
11648:{\displaystyle e^{{\frac {\delta }{2}}F}=\prod _{{\text{even }}l}e^{{\frac {\delta }{2}}F^{}}}
11236:
9080:
8766:
8318:
7025:
12081:
11079:{\displaystyle e^{A+B}=\lim _{n\to \infty }\left(e^{\frac {A}{n}}e^{\frac {B}{n}}\right)^{n}}
8857:
7005:
6711:
5854:
5705:
4496:'s are a special case, like the first ones, expressing the right-hand Schmidt vectors at the
4479:
1739:
1296:
249:
245:
10360:
The numerical simulation is targeting (possibly time-dependent) Hamiltonians of a system of
9921:
8787:
and their corresponding Schmidt eigenvectors must be computed and expressed in terms of the
6379:
6352:
6332:
4571:
4541:
17210:
17147:
17067:(2003-10-01). "Efficient Classical Simulation of Slightly Entangled Quantum Computations".
13097:
12927:
12365:
10519:
7958:
7931:
6556:
6469:
6466:(OQG) and two-qubit gates (TQG) acting on neighbouring qubits. Instead of updating all the
6433:
6413:
5822:
5685:
5507:
3832:
1768:
1670:
1289:
430:
257:
214:
Considering the inherent difficulties of simulating general quantum many-body systems, the
145:
11841:
11808:
11770:
11737:
8:
15289:
10329:
4568:
lattice place. As shown in, it is straightforward to obtain the Schmidt decomposition at
316:
simulations of time-dependent Hamiltonians, describing systems that can be realized with
241:
236:
220:
17214:
17151:
15267:
As we calculated before, the normalization constant after making the truncation at bond
14032:
is the sum of all the discarded eigenvalues of the reduced density matrix, at the bond
17354:
17318:
17295:
17261:
17234:
17200:
17171:
17137:
17110:
17076:
15270:
15247:
13670:
11471:
11185:{\displaystyle e^{{\delta }(A+B)}=e^{{\delta }A}e^{{\delta }B}+{\it {O}}(\delta ^{2}).}
10363:
4601:
3011:
302:
215:
10507:{\displaystyle H_{N}=\sum \limits _{l=1}^{N}K_{1}^{}+\sum \limits _{l=1}^{N}K_{2}^{}.}
297:
The numerical method is efficient in simulating real-time dynamics or calculations of
17358:
17346:
17336:
17287:
17279:
17226:
17163:
17102:
17094:
17299:
17238:
17175:
17114:
6592:
The OQGs are affecting only the qubit they are acting upon, the update of the state
17328:
17271:
17218:
17155:
17086:
11223:
298:
231:
17275:
17222:
17159:
17090:
13933:{\displaystyle \epsilon ({\it {D}})=1-\prod \limits _{n=1}^{N-1}(1-\epsilon _{n})}
6450:
may be sensibly larger, but still within reach of eventual numerical simulations.
7285:
6630:
does not modify the Schmidt eigenvalues or vectors on the left, consequently the
321:
17064:
263:
7043:
6623:
6299:
4472:
Repeating the steps 1 to 3, one can construct the whole decomposition of state
313:
312:
A useful feature of the TEBD algorithm is that it can be reliably employed for
283:
309:
with the system size, hence many-particles systems in 1D can be investigated.
235:
first quantum computer, theoretical physicists are searching, in the field of
17371:
17350:
17283:
17098:
12687:
make an imaginary-time evolution towards the ground state of the Hamiltonian
12392:
It is rather troublesome, if at all possible, to construct the decomposition
11227:
10380:
particles arranged in a line, which are composed of arbitrary OQGs and TQGs:
5956:{\displaystyle \lambda _{{\alpha }_{l}}^{}{\sim }e^{-K\alpha _{l}},\ K>0.}
317:
279:
141:
17291:
17230:
17167:
17127:
17106:
16809:
The total truncation error, considering both sources, is upper bounded by:
11222:
For simulations of quantum dynamics it is useful to use operators that are
10976:
6463:
5702:
for a bipartite cut in the middle of the chain can have a maximal value of
17323:
6302:
chain, for the sake of simplicity. A dimension of 12, let's say, for the
17266:
17205:
17180:
17142:
17081:
13837:
9097:'s. Expressing the eigenvectors of the diagonalized matrix in the basis:
331:
13384:{\displaystyle \epsilon ={\frac {T}{\delta }}\delta ^{p+1}=T\delta ^{p}}
6553:, we will restrict ourselves to a number of operations that increase in
17332:
11725:{\displaystyle e^{{\delta }G}=\prod _{{\text{odd }}l}e^{{\delta }G^{}}}
6574:
6458:
One can proceed now to investigate the behaviour of the decomposition
6430:. The number of coefficients to account for a faithful description of
13200:{\displaystyle |{\psi _{T}}\rangle =e^{-iH_{n}T}|{\psi _{0}}\rangle }
132:
5618:. Apparently this is just a fancy way of rewriting the coefficients
26:
7275:{\displaystyle {{H}=J{\otimes }H_{C}{\otimes }H_{D}{\otimes }K}.\,}
6329:
would be a reasonable choice, keeping the discarded eigenvalues at
17317:. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 37–68.
13664:
accounts for the part that is neglected when doing the expansion.
13219:
10205:{\displaystyle \lambda _{\beta }^{'}|{\Phi _{\beta }^{'}}\rangle }
6295:
136:
15576:; taking into account the full Schmidt dimension, the norm is:
9825:
6349:% of the total, as shown by numerical studies, meaning roughly
306:
8270:
Using the same reasoning as for the OQG, the applying the TQG
7637:
is spanned by the eigenvectors of the reduced density matrix:
7355:{\displaystyle \rho ^{J}=Tr_{CDK}|\psi \rangle \langle \psi |}
5611:{\displaystyle {\chi }^{2}{\cdot }M(N-2)+2{\chi }M+(N-1)\chi }
14724:{\displaystyle \langle \psi _{D}^{\bot }|\psi _{D}\rangle =0}
10083:
operations. The same holds for the formation of the elements
7291:
is spanned by the eigenvectors of the reduced density matrix
350:
3240:{\displaystyle \left|{\Phi _{\alpha _{1}}^{}}\right\rangle }
12678:{\displaystyle |{\psi _{0}}\rangle =Q|{\psi _{gr}}\rangle }
12473:
for a simple initial state, let us say, some product state
5678:, but in fact there is more to it than that. Assuming that
1358:{\displaystyle \vert \Psi \rangle \in {H_{A}\otimes H_{B}}}
173:
personal reflection, personal essay, or argumentative essay
6993:
1303:
to express a state with limited entanglement more simply.
17035:
16644:
Hence, when constructing the reduced density matrix, the
15508:
and calculate the norm of the right-hand Schmidt vectors
15264:
is more subtle and requires a little bit of calculation.
13241:
10188:
10013:
10010:
1957:{\displaystyle M_{A|B}=\min(\dim({H_{A}}),\dim({H_{B}}))}
341:
14877:
After moving to the next bond, the state is, similarly:
1395:
can be represented in an appropriately chosen basis as:
11233:
The trick of the ST2 is to write the unitary operators
135:
is a numerical scheme used to simulate one-dimensional
16109:
Taking into account the truncated space, the norm is:
15244:
The second error source enfolded in the decomposition
13838:
Errors coming from the truncation of the Hilbert space
7208:
can be regarded as belonging to only four subsystems:
3816:{\displaystyle |{\tau _{\alpha _{1}i_{2}}^{}}\rangle }
3736:{\displaystyle |{\tau _{\alpha _{1}i_{2}}^{}}\rangle }
2996:
1798:
16815:
16654:
16498:
16433:
16115:
15897:
15582:
15514:
15484:
15348:
15292:
15273:
15250:
15033:
14883:
14737:
14678:
14455:
14231:
14116:
14092:
14062:
14038:
13946:
13855:
13693:
13673:
13625:
13585:
13450:
13399:
13332:
13299:
13264:
13230:
13127:
13100:
13063:
13026:
12994:
12957:
12930:
12763:
12722:
12693:
12625:
12596:
12556:
12519:
12479:
12455:
12422:
12398:
12368:
12331:
12294:
12267:
12211:
12160:
12104:
12084:
11920:
11882:
11844:
11811:
11773:
11740:
11660:
11573:
11539:
11502:
11474:
11441:
11275:
11239:
11200:
11092:
10993:
10912:
10843:
10715:
10590:
10549:
10522:
10388:
10366:
10332:
10275:
10218:
10137:
10089:
10034:
9992:
9968:
9944:
9924:
9886:
9838:
9682:
9647:
9608:
9318:
9177:
9142:
9103:
9083:
8866:
8819:
8793:
8769:
8590:
8433:
8379:
8341:
8321:
8288:
8032:
8000:
7961:
7934:
7645:
7370:
7297:
7216:
7184:
7135:
7096:
7063:
7028:
7008:
6820:
6767:
6734:
6714:
6675:
6636:
6598:
6559:
6499:
6472:
6436:
6416:
6382:
6355:
6335:
6308:
6270:
5973:
5879:
5857:
5825:
5796:
5743:
5708:
5688:
5624:
5537:
5510:
5126:
4839:
4667:
4604:
4574:
4544:
4502:
4482:
4172:
3960:
3913:
3855:
3835:
3761:
3681:
3530:
3472:
3255:
3198:
3102:
3055:
3014:
2832:
2690:
2288:
2110:
1973:
1872:
1859:{\textstyle \sum \limits _{i=1}^{M_{A|B}}a_{i}^{2}=1}
1771:
1742:
1700:
1673:
1631:
1518:
1401:
1371:
1317:
785:
725:
549:
536:{\displaystyle |i_{1},i_{2},..,i_{N-1},i_{N}\rangle }
460:
433:
405:
359:
6584:
11491:
10319:{\displaystyle {\it {O}}(M^{2}{\cdot }{\chi }^{3})}
10262:{\displaystyle {\it {O}}(M^{3}{\cdot }{\chi }^{3})}
8420:{\displaystyle |{\psi '}\rangle =V|{\psi }\rangle }
51:. Unsourced material may be challenged and removed.
17021:
16794:
16634:
16484:
16417:
16098:
15880:
15568:
15500:
15468:
15334:
15279:
15256:
15233:
15019:
14867:
14723:
14664:
14441:
14217:
14102:
14078:
14048:
14024:
13932:
13820:
13679:
13656:
13619:is the state kept after the Trotter expansion and
13611:
13569:
13434:
13383:
13318:
13285:
13250:
13199:
13113:
13086:
13049:
13009:
12980:
12943:
12915:
12748:
12708:
12677:
12611:
12582:
12542:
12502:
12465:
12435:
12408:
12381:
12354:
12317:
12288:Our goal is to make the time evolution of a state
12277:
12253:
12197:
12146:
12090:
12068:
11901:
11868:
11830:
11797:
11759:
11724:
11647:
11557:
11525:
11480:
11460:
11427:
11261:
11212:
11184:
11078:
10967:
10898:
10827:
10702:
10574:
10535:
10506:
10372:
10344:
10318:
10261:
10204:
10123:
10075:
10020:
9978:
9954:
9930:
9910:
9872:
9809:
9668:
9633:
9594:
9302:
9163:
9128:
9089:
9067:
8848:
8801:
8775:
8753:
8576:
8419:
8363:
8327:
8307:
8260:
8016:
7974:
7947:
7918:
7623:
7354:
7274:
7200:
7167:
7121:
7082:
7034:
7014:
6983:
6804:
6753:
6720:
6700:
6661:
6614:
6565:
6545:
6485:
6453:
6442:
6422:
6395:
6368:
6341:
6321:
6283:
6254:
5955:
5863:
5838:
5811:
5782:
5729:
5694:
5670:
5610:
5523:
5485:
5110:
4821:
4640:
4590:
4560:
4530:
4488:
4462:
4151:
3946:
3899:
3841:
3815:
3735:
3665:
3516:
3450:
3239:
3184:
3088:
3041:
2986:
2818:
2674:
2274:
2096:
1956:
1858:
1784:
1757:
1728:
1686:
1659:
1617:
1504:
1387:
1357:
1278:
771:
719:The trick of TEBD is to re-write the coefficients
704:
535:
446:
419:
391:
266:, then at the Institute for Quantum Information,
17190:
12510:, for which the decomposition is straightforward.
4655:The Schmidt eigenvalues, are given explicitly in
3900:{\displaystyle |{\Phi _{\alpha _{2}}^{}}\rangle }
3517:{\displaystyle |{\Phi _{\alpha _{1}}^{}}\rangle }
392:{\displaystyle |\Psi \rangle \in H^{{\otimes }N}}
17369:
17315:Quantum Annealing and Other Optimization Methods
13435:{\displaystyle |{{\tilde {\psi }}_{Tr}}\rangle }
12791:
12154:are applied successively to all odd sites, then
11428:{\displaystyle e^{-iH_{N}T}=^{T/{\delta }}=^{n}}
11014:
9077:The square roots of its eigenvalues are the new
1894:
278:displaying local interactions, as for example,
17184:
17121:
13220:Errors coming from the Suzuki–Trotter expansion
10982:
15569:{\displaystyle \|{\Phi _{\alpha _{l-1}}^{}}\|}
13020:finally, make the time-evolution of the state
12254:{\displaystyle e^{-i{\frac {\delta }{2}}F^{}}}
12147:{\displaystyle e^{-i{\frac {\delta }{2}}F^{}}}
10543:as a sum of two possibly non-commuting terms,
10131:, or for computing the left-hand eigenvectors
16804:
16485:{\displaystyle \epsilon =n_{2}-n_{1}=n_{2}-1}
13657:{\displaystyle |{\psi _{Tr}^{\bot }}\rangle }
10124:{\displaystyle \rho _{\gamma \gamma '}^{jj'}}
10076:{\displaystyle {O}(M^{4}{\cdot }{\chi }^{3})}
9911:{\displaystyle \Theta _{\alpha \gamma }^{ij}}
336:
17312:
16690:
16660:
15563:
15515:
15157:
15134:
15077:
15051:
15027:The error, after the second truncation, is:
15014:
14959:
14901:
14785:
14762:
14712:
14679:
14659:
14612:
14478:
14436:
14389:
14249:
14212:
14170:
14127:
14073:
13765:
13720:
13651:
13606:
13564:
13528:
13480:
13429:
13194:
13145:
13081:
13044:
12975:
12904:
12901:
12856:
12851:
12784:
12743:
12672:
12643:
12577:
12537:
12497:
12349:
12312:
12060:
11953:
11912:The time-evolution can be made according to
10199:
9801:
9718:
9628:
9625:
9609:
9589:
9437:
9384:
9378:
9294:
9213:
9123:
9120:
9104:
9036:
9033:
8930:
8927:
8571:
8449:
8414:
8395:
8255:
8043:
8011:
7990:+ 1. Using this basis and the decomposition
7900:
7897:
7784:
7781:
7605:
7602:
7495:
7492:
7341:
7338:
7195:
6609:
6376:coefficients, as compared to the originally
6291:is the number of kept Schmidt coefficients.
6246:
6199:
5984:
5480:
5171:
5105:
4878:
4816:
4769:
4678:
4457:
4416:
4183:
4146:
4007:
3894:
3810:
3730:
3660:
3609:
3569:
3511:
3445:
3404:
3266:
2813:
2785:
2757:
2729:
2701:
2666:
2643:
2605:
2568:
2545:
2507:
2470:
2447:
2409:
2372:
2349:
2311:
2086:
2070:
2047:
2024:
1984:
1723:
1654:
1612:
1584:
1556:
1382:
1324:
1318:
1306:
699:
558:
530:
414:
368:
16648:of the matrix is multiplied by the factor:
14449:is the state kept after the truncation and
10355:
10021:{\displaystyle \gamma ,\alpha ,{\it {i,j}}}
8763:To find out the new decomposition, the new
3947:{\displaystyle \lambda _{{\alpha }_{2}}^{}}
3089:{\displaystyle \lambda _{{\alpha }_{1}}^{}}
274:. This happens to be the case with generic
14110:, described by the Schmidt decomposition:
13224:In the case of a Trotter approximation of
12988:as its ground state, and the Hamiltonian
11194:The correction term vanishes in the limit
17322:
17265:
17204:
17141:
17080:
15241:and so on, as we move from bond to bond.
11526:{\displaystyle e^{{\frac {\delta }{2}}F}}
10326:basic operations. In the case of qubits,
9873:{\displaystyle {O}(M{\cdot }{\chi }^{2})}
9819:
7271:
6805:{\displaystyle {O}(M^{2}\cdot \chi ^{2})}
1311:Consider the state of a bipartite system
202:Learn how and when to remove this message
111:Learn how and when to remove this message
4538:bond in terms of the local basis at the
1729:{\displaystyle |{\Phi _{i}^{B}}\rangle }
1660:{\displaystyle |{\Phi _{i}^{A}}\rangle }
17059:
17057:
9634:{\displaystyle \{|{i\alpha }\rangle \}}
9129:{\displaystyle \{|{j\gamma }\rangle \}}
7168:{\displaystyle {O}({M\cdot \chi }^{3})}
5737:; in this case we end up with at least
1295:To understand why, one can look at the
1292:, simplifies the calculations greatly.
256:classical computer. Such a resource is
17370:
13326:steps, the error after the time T is:
13319:{\displaystyle n={\frac {T}{\delta }}}
11461:{\displaystyle n={\frac {T}{\delta }}}
5846:! The truth is that the decomposition
3247:'s in the local basis, one can write:
342:Introducing the decomposition of State
18:Quantum many-body simulation algorithm
17306:
17251:
17245:
17063:
13612:{\displaystyle |{\psi _{Tr}}\rangle }
12749:{\displaystyle |{\psi _{gr}}\rangle }
12583:{\displaystyle |{\psi _{gr}}\rangle }
7178:Following Vidal's original approach,
6546:{\displaystyle c_{i_{1}i_{2}..i_{N}}}
5819:ones, slightly more than the initial
5671:{\displaystyle c_{i_{1}i_{2}..i_{N}}}
4831:The Schmidt eigenvectors are simply:
4166:: make the substitutions and obtain:
1736:, which form an orthonormal basis in
772:{\displaystyle c_{i_{1}i_{2}..i_{N}}}
399:. The most natural way of describing
17054:
13087:{\displaystyle |{\psi _{T}}\rangle }
13050:{\displaystyle |{\psi _{0}}\rangle }
12981:{\displaystyle |{\psi _{P}}\rangle }
12543:{\displaystyle |{\psi _{0}}\rangle }
12503:{\displaystyle |{\psi _{P}}\rangle }
12355:{\displaystyle |{\psi _{T}}\rangle }
12318:{\displaystyle |{\psi _{0}}\rangle }
7002:The changes required to update the
5783:{\displaystyle M^{N+1}{\cdot }(N-2)}
225:density matrix renormalization group
155:
49:adding citations to reliable sources
20:
16968:
16889:
16840:
16265:
16130:
15948:
15728:
15603:
15357:
14804:
14504:
14281:
13961:
13882:
12198:{\displaystyle e^{{-i\delta }G^{}}}
10448:
10403:
9602:after expressing them in the basis
8637:
8616:
8483:
8456:
8083:
8050:
6091:
5997:
5790:coefficients, considering only the
3907:and, correspondingly, coefficients
3273:
2997:Building the decomposition of state
1800:
1417:
833:
565:
13:
16043:
15979:
15520:
15487:
15008:
14692:
14622:
14581:
14472:
14399:
14358:
14206:
14095:
14041:
13864:
13645:
13558:
13238:
13234:
12801:
12458:
12401:
12270:
11158:
11024:
10278:
10221:
10185:
10164:
10007:
9971:
9947:
9888:
9741:
9690:
9649:
9521:
9475:
9389:
9345:
9236:
9185:
9144:
8795:
8722:
8685:
8592:
8520:
8343:
8290:
8185:
8137:
7789:
7743:
7500:
7460:
7098:
7065:
7009:
6925:
6822:
6736:
6715:
6677:
6638:
6209:
6168:
5980:
5382:
5236:
5134:
5009:
4937:
4847:
4779:
4738:
4674:
4483:
4426:
4312:
4246:
4179:
4115:
4031:
3863:
3538:
3480:
3414:
3322:
3262:
3205:
3147:
3109:
2837:
2697:
2116:
1980:
1967:For example, the two-qubit state:
1708:
1667:that make an orthonormal basis in
1639:
1597:
1569:
1541:
1526:
1483:
1468:
1406:
1378:
1321:
1233:
1119:
1031:
955:
889:
555:
411:
365:
301:using imaginary-time evolution or
14:
17394:
13667:The total error scales with time
12362:using the n-particle Hamiltonian
9312:From the left-hand eigenvectors,
6994:Two-qubit gates acting on qubits
3524:'s in a local basis for qubit 2:
3008:Consider the bipartite splitting
14079:{\displaystyle |{\psi }\rangle }
13211:
12325:for a time T, towards the state
11492:Simulation of the time-evolution
8017:{\displaystyle |{\psi }\rangle }
7201:{\displaystyle |{\psi }\rangle }
6728:'s that will be updated are the
6615:{\displaystyle |{\psi }\rangle }
6585:One-qubit gates acting on qubit
1388:{\displaystyle |{\Psi }\rangle }
160:
60:"Time-evolving block decimation"
25:
13834:of the dimension of the chain.
13286:{\displaystyle {\delta }^{p+1}}
7633:In a similar way, the subspace
6669:'s, or on the right, hence the
6462:when acted upon with one-qubit
6454:The update of the decomposition
151:
36:needs additional citations for
17016:
16994:
16885:
16866:
16827:
16819:
16695:
16676:
16656:
16622:
16585:
16380:
16374:
16368:
16348:
16336:
16299:
16252:
16246:
16240:
16213:
16201:
16164:
16081:
16075:
16033:
16017:
16011:
15975:
15935:
15898:
15840:
15834:
15828:
15808:
15799:
15762:
15715:
15709:
15703:
15683:
15674:
15637:
15557:
15542:
15433:
15427:
15421:
15398:
15329:
15326:
15311:
15305:
15296:
15293:
15228:
15209:
15206:
15181:
15162:
15143:
15130:
15126:
15101:
15082:
15060:
15047:
14986:
14934:
14885:
14856:
14850:
14844:
14831:
14790:
14771:
14758:
14698:
14653:
14638:
14616:
14606:
14597:
14575:
14569:
14563:
14457:
14430:
14415:
14393:
14383:
14374:
14352:
14346:
14340:
14233:
14191:
14154:
14118:
14064:
14013:
14007:
14001:
13988:
13927:
13908:
13869:
13859:
13770:
13746:
13739:
13716:
13703:
13697:
13627:
13587:
13540:
13509:
13464:
13452:
13413:
13401:
13251:{\displaystyle {\it {p^{th}}}}
13178:
13129:
13065:
13028:
13001:
12959:
12951:, which has the product state
12885:
12873:
12835:
12823:
12798:
12765:
12724:
12700:
12653:
12627:
12603:
12558:
12521:
12481:
12333:
12296:
12244:
12238:
12188:
12182:
12137:
12131:
12047:
12035:
11934:
11922:
11861:
11850:
11823:
11817:
11790:
11779:
11752:
11746:
11715:
11709:
11638:
11632:
11558:{\displaystyle e^{{\delta }G}}
11488:is called the Trotter number.
11416:
11357:
11335:
11305:
11204:
11176:
11163:
11115:
11103:
11021:
10956:
10951:
10940:
10927:
10921:
10913:
10887:
10882:
10871:
10858:
10852:
10844:
10817:
10811:
10785:
10737:
10692:
10686:
10660:
10612:
10496:
10478:
10439:
10433:
10313:
10283:
10256:
10226:
10193:
10180:
10158:
10070:
10040:
9867:
9844:
9787:
9768:
9760:
9712:
9701:
9684:
9661:
9653:
9613:
9575:
9555:
9541:
9511:
9502:
9494:
9471:
9423:
9416:
9405:
9372:
9361:
9339:
9282:
9263:
9255:
9207:
9196:
9179:
9155:
9148:
9108:
9058:
9021:
8944:
8913:
8888:
8879:
8841:
8832:
8740:
8734:
8703:
8697:
8551:
8435:
8405:
8381:
8353:
8347:
8300:
8294:
8233:
8227:
8221:
8203:
8197:
8179:
8173:
8155:
8149:
8131:
8119:
8034:
8002:
7909:
7888:
7877:
7871:
7859:
7846:
7828:
7821:
7798:
7775:
7752:
7737:
7726:
7720:
7708:
7695:
7676:
7651:
7614:
7593:
7582:
7576:
7564:
7551:
7533:
7526:
7509:
7486:
7469:
7454:
7443:
7438:
7426:
7413:
7393:
7376:
7348:
7331:
7186:
7162:
7141:
7129:. They consist of a number of
7114:
7102:
7075:
7069:
6963:
6957:
6867:
6861:
6799:
6773:
6746:
6740:
6693:
6681:
6654:
6642:
6600:
6240:
6225:
6203:
6193:
6184:
6162:
6156:
6150:
6073:
6067:
6061:
6038:
5975:
5905:
5899:
5777:
5765:
5602:
5590:
5570:
5558:
5426:
5410:
5404:
5336:
5324:
5280:
5268:
5165:
5150:
5128:
5063:
5047:
5041:
4997:
4991:
4959:
4953:
4872:
4863:
4841:
4810:
4795:
4773:
4763:
4754:
4732:
4726:
4720:
4669:
4635:
4620:
4614:
4605:
4516:
4503:
4451:
4442:
4420:
4390:
4384:
4378:
4344:
4338:
4306:
4300:
4268:
4262:
4174:
4140:
4131:
4109:
4103:
4097:
4063:
4057:
4002:
3993:
3962:
3939:
3933:
3888:
3879:
3857:
3804:
3795:
3763:
3724:
3715:
3683:
3654:
3645:
3613:
3593:
3563:
3554:
3532:
3505:
3496:
3474:
3439:
3430:
3408:
3388:
3382:
3376:
3344:
3338:
3257:
3227:
3221:
3172:
3163:
3131:
3125:
3081:
3075:
3049:. The SD has the coefficients
3036:
3027:
3021:
3015:
2804:
2776:
2748:
2720:
2692:
2684:On the other hand, the state:
2669:
2650:
2627:
2623:
2584:
2571:
2552:
2529:
2525:
2486:
2473:
2454:
2431:
2427:
2388:
2375:
2356:
2333:
2329:
2290:
2077:
2061:
2038:
2015:
1975:
1951:
1948:
1933:
1921:
1906:
1897:
1882:
1824:
1702:
1694:and, correspondingly, vectors
1633:
1591:
1563:
1520:
1441:
1373:
1261:
1255:
1227:
1215:
1171:
1157:
1101:
1095:
1063:
1057:
1025:
1019:
987:
981:
949:
943:
911:
905:
629:
551:
462:
420:{\displaystyle |\Psi \rangle }
407:
361:
125:time-evolving block decimation
1:
17276:10.1103/physrevlett.93.040502
17223:10.1103/PhysRevLett.93.207205
17160:10.1103/PhysRevLett.93.207204
17091:10.1103/physrevlett.91.147902
17048:
13258:order, the error is of order
9918:one makes the summation over
9824:The dimension of the largest
8813:. The reduced density matrix
8282:+ 1 one needs only to update
17036:"Adaptive" Schmidt dimension
13010:{\displaystyle {\tilde {H}}}
12709:{\displaystyle {\tilde {H}}}
12612:{\displaystyle {\tilde {H}}}
12449:construct the decomposition
11213:{\displaystyle \delta \to 0}
10983:The Suzuki–Trotter expansion
10837:Any two-body terms commute:
5495:
1301:singular value decomposition
353:, described by the function
7:
15501:{\displaystyle {\it {l}}-1}
9164:{\displaystyle \Gamma ^{}}}
8364:{\displaystyle \Gamma ^{}.}
6761:'s (requiring only at most
6577:of low degree, thus saving
5812:{\displaystyle {\chi }^{2}}
10:
17399:
16805:The total truncation error
15478:Now let us go to the bond
11902:{\displaystyle l{\neq }l'}
10516:It is useful to decompose
10028:, adding up to a total of
9669:{\displaystyle \Gamma ^{}}
8308:{\displaystyle \Gamma ^{}}
7122:{\displaystyle \Gamma ^{}}
7083:{\displaystyle \Gamma ^{}}
6754:{\displaystyle \Gamma ^{}}
6701:{\displaystyle \Gamma ^{}}
6662:{\displaystyle \Gamma ^{}}
5682:is even, the Schmidt rank
4531:{\displaystyle (N-1)^{th}}
1792:being real and positive,
716:is the on-site dimension.
337:The decomposition of state
221:quantum Monte Carlo method
14103:{\displaystyle {\it {n}}}
14049:{\displaystyle {\it {n}}}
13393:The unapproximated state
12466:{\displaystyle {\it {D}}}
12436:{\displaystyle \chi _{c}}
12409:{\displaystyle {\it {D}}}
12278:{\displaystyle {\it {D}}}
11565:are easy to express, as:
10968:{\displaystyle },G^{}]=0}
10899:{\displaystyle },F^{}]=0}
10575:{\displaystyle H_{N}=F+G}
9979:{\displaystyle {\it {n}}}
9955:{\displaystyle {\it {m}}}
9171:'s are obtained as well:
8849:{\displaystyle \rho ^{'}}
8802:{\displaystyle {\Gamma }}
6322:{\displaystyle \chi _{c}}
6284:{\displaystyle \chi _{c}}
5531:initial terms, there are
1307:The Schmidt decomposition
427:would be using the local
250:Grover's search algorithm
11734:since any two operators
11262:{\displaystyle e^{-iHt}}
10356:The numerical simulation
9880:; when constructing the
9090:{\displaystyle \lambda }
8809:'s of the decomposition
8776:{\displaystyle \lambda }
8328:{\displaystyle \lambda }
7035:{\displaystyle \lambda }
1765:, with the coefficients
1625:are formed with vectors
17254:Physical Review Letters
17069:Physical Review Letters
16427:Taking the difference,
12091:{\displaystyle \delta }
7015:{\displaystyle \Gamma }
6721:{\displaystyle \Gamma }
5864:{\displaystyle \alpha }
5730:{\displaystyle M^{N/2}}
4489:{\displaystyle \Gamma }
1758:{\displaystyle {H_{B}}}
1299:of a state, which uses
17023:
16993:
16914:
16865:
16796:
16636:
16486:
16419:
16298:
16163:
16100:
15974:
15882:
15761:
15636:
15570:
15502:
15470:
15394:
15336:
15281:
15258:
15235:
15021:
14869:
14830:
14725:
14666:
14543:
14443:
14320:
14219:
14104:
14080:
14050:
14026:
13987:
13934:
13907:
13822:
13681:
13658:
13613:
13571:
13436:
13385:
13320:
13293:. Taking into account
13287:
13252:
13201:
13115:
13094:using the Hamiltonian
13088:
13051:
13011:
12982:
12945:
12917:
12750:
12710:
12679:
12613:
12584:
12544:
12504:
12467:
12437:
12410:
12383:
12356:
12319:
12279:
12255:
12205:to the even ones, and
12199:
12148:
12092:
12070:
11903:
11870:
11832:
11799:
11761:
11726:
11649:
11559:
11527:
11482:
11462:
11429:
11263:
11214:
11186:
11080:
10969:
10900:
10829:
10704:
10576:
10537:
10508:
10467:
10422:
10374:
10346:
10320:
10263:
10206:
10125:
10077:
10022:
9980:
9956:
9932:
9931:{\displaystyle \beta }
9912:
9874:
9820:The computational cost
9811:
9670:
9635:
9596:
9304:
9165:
9130:
9091:
9069:
8850:
8803:
8777:
8755:
8662:
8635:
8578:
8508:
8481:
8421:
8365:
8329:
8309:
8262:
8108:
8081:
8018:
7976:
7949:
7920:
7625:
7356:
7276:
7202:
7169:
7123:
7084:
7036:
7016:
6985:
6806:
6755:
6722:
6702:
6663:
6616:
6567:
6547:
6487:
6444:
6424:
6410:to describe our state
6397:
6396:{\displaystyle 2^{50}}
6370:
6369:{\displaystyle 2^{14}}
6343:
6342:{\displaystyle 0.0001}
6323:
6285:
6256:
6130:
6034:
5957:
5865:
5840:
5813:
5784:
5731:
5696:
5672:
5612:
5525:
5487:
5112:
4823:
4642:
4592:
4591:{\displaystyle k^{th}}
4562:
4561:{\displaystyle N^{th}}
4532:
4490:
4464:
4153:
3948:
3901:
3843:
3817:
3737:
3667:
3518:
3452:
3320:
3241:
3186:
3090:
3043:
2988:
2820:
2676:
2276:
2104:has the following SD:
2098:
1958:
1860:
1834:
1786:
1759:
1730:
1688:
1661:
1619:
1506:
1451:
1389:
1359:
1288:This form, known as a
1280:
887:
773:
706:
584:
537:
448:
421:
393:
182:by rewriting it in an
17383:Computational physics
17024:
16967:
16888:
16839:
16797:
16637:
16487:
16420:
16264:
16129:
16101:
15947:
15883:
15727:
15602:
15571:
15503:
15471:
15356:
15337:
15282:
15259:
15236:
15022:
14870:
14803:
14726:
14667:
14503:
14444:
14280:
14220:
14105:
14081:
14051:
14027:
13960:
13935:
13881:
13830:The Trotter error is
13823:
13682:
13659:
13614:
13572:
13437:
13386:
13321:
13288:
13253:
13202:
13116:
13114:{\displaystyle H_{n}}
13089:
13052:
13012:
12983:
12946:
12944:{\displaystyle H_{1}}
12918:
12751:
12711:
12680:
12614:
12585:
12545:
12505:
12468:
12438:
12411:
12384:
12382:{\displaystyle H_{n}}
12357:
12320:
12280:
12256:
12200:
12149:
12093:
12078:For each "time-step"
12071:
11904:
11871:
11833:
11800:
11762:
11727:
11650:
11560:
11528:
11483:
11463:
11430:
11264:
11215:
11187:
11081:
10970:
10901:
10830:
10705:
10577:
10538:
10536:{\displaystyle H_{N}}
10509:
10447:
10402:
10375:
10347:
10321:
10264:
10207:
10126:
10078:
10023:
9981:
9957:
9933:
9913:
9875:
9812:
9671:
9636:
9597:
9305:
9166:
9131:
9092:
9070:
8851:
8804:
8778:
8756:
8636:
8615:
8579:
8482:
8455:
8422:
8366:
8330:
8310:
8263:
8082:
8049:
8019:
7982:belong to the qubits
7977:
7975:{\displaystyle H_{D}}
7950:
7948:{\displaystyle H_{C}}
7921:
7626:
7357:
7277:
7203:
7170:
7124:
7085:
7037:
7017:
6986:
6807:
6756:
6723:
6703:
6664:
6617:
6568:
6566:{\displaystyle \chi }
6548:
6488:
6486:{\displaystyle M^{N}}
6445:
6443:{\displaystyle \psi }
6425:
6423:{\displaystyle \psi }
6398:
6371:
6344:
6324:
6286:
6257:
6090:
5996:
5958:
5866:
5841:
5839:{\displaystyle M^{N}}
5814:
5785:
5732:
5697:
5695:{\displaystyle \chi }
5673:
5613:
5526:
5524:{\displaystyle M^{N}}
5488:
5113:
4824:
4643:
4593:
4563:
4533:
4491:
4465:
4154:
3949:
3902:
3844:
3842:{\displaystyle \chi }
3818:
3738:
3668:
3519:
3453:
3272:
3242:
3187:
3091:
3044:
2989:
2821:
2677:
2277:
2099:
1959:
1861:
1799:
1787:
1785:{\displaystyle a_{i}}
1760:
1731:
1689:
1687:{\displaystyle H_{A}}
1662:
1620:
1507:
1416:
1390:
1360:
1297:Schmidt decomposition
1281:
832:
774:
707:
564:
538:
449:
447:{\displaystyle M^{N}}
422:
394:
16813:
16652:
16496:
16431:
16113:
15895:
15580:
15512:
15482:
15346:
15290:
15271:
15248:
15031:
14881:
14735:
14676:
14453:
14229:
14114:
14090:
14086:is, at a given bond
14060:
14036:
13944:
13853:
13846:, they are twofold.
13691:
13671:
13623:
13583:
13448:
13397:
13330:
13297:
13262:
13228:
13125:
13098:
13061:
13024:
12992:
12955:
12928:
12761:
12720:
12691:
12623:
12594:
12554:
12550:to the ground state
12517:
12477:
12453:
12420:
12396:
12366:
12329:
12292:
12285:when applying them.
12265:
12209:
12158:
12102:
12082:
11918:
11880:
11869:{\displaystyle G^{}}
11842:
11831:{\displaystyle G^{}}
11809:
11798:{\displaystyle F^{}}
11771:
11760:{\displaystyle F^{}}
11738:
11658:
11571:
11537:
11500:
11472:
11439:
11273:
11237:
11198:
11090:
10991:
10910:
10841:
10713:
10588:
10547:
10520:
10386:
10364:
10330:
10273:
10216:
10135:
10087:
10032:
9990:
9966:
9942:
9922:
9884:
9836:
9680:
9645:
9606:
9316:
9175:
9140:
9101:
9081:
8864:
8817:
8791:
8767:
8588:
8431:
8377:
8339:
8319:
8286:
8030:
7998:
7959:
7932:
7643:
7368:
7295:
7214:
7182:
7133:
7094:
7061:
7026:
7006:
6818:
6765:
6732:
6712:
6673:
6634:
6596:
6557:
6497:
6470:
6434:
6414:
6380:
6353:
6333:
6306:
6268:
5971:
5877:
5855:
5823:
5794:
5741:
5706:
5686:
5622:
5535:
5508:
5124:
4837:
4665:
4602:
4572:
4542:
4500:
4480:
4170:
3958:
3911:
3853:
3833:
3759:
3755:: write each vector
3743:are not necessarily
3679:
3528:
3470:
3253:
3196:
3100:
3053:
3012:
2830:
2826:is a product state:
2688:
2286:
2108:
1971:
1870:
1796:
1769:
1740:
1698:
1671:
1629:
1516:
1399:
1369:
1315:
1290:Matrix product state
783:
723:
547:
458:
431:
403:
357:
346:Consider a chain of
327:, t-DMRG for short.
258:quantum entanglement
216:exponential increase
45:improve this article
17215:2004PhRvL..93t7205Z
17152:2004PhRvL..93t7204V
16378:
16250:
16095:
16031:
15838:
15713:
15561:
15431:
15335:{\displaystyle (:)}
15076:
15012:
14854:
14696:
14657:
14610:
14573:
14476:
14434:
14387:
14350:
14210:
14011:
13649:
13562:
10784:
10754:
10659:
10629:
10500:
10443:
10345:{\displaystyle M=2}
10197:
10156:
10120:
9907:
9775:
9540:
9509:
9420:
9376:
9337:
9270:
9019:
8747:
8710:
8683:
8611:
8539:
8231:
8210:
8183:
8162:
8135:
8024:can be written as:
7875:
7825:
7779:
7724:
7580:
7530:
7490:
7442:
6977:
6923:
6881:
6244:
6197:
6160:
6071:
5909:
5424:
5380:
5340:
5300:
5169:
5061:
5001:
4973:
4876:
4814:
4767:
4730:
4455:
4388:
4358:
4310:
4282:
4144:
4107:
4077:
4006:
3943:
3892:
3829:(Vidal's emphasis)
3808:
3728:
3658:
3567:
3509:
3443:
3386:
3358:
3231:
3192:. By expanding the
3176:
3135:
3085:
2603:
2505:
2407:
2309:
2266:
2251:
2192:
2177:
1849:
1721:
1652:
1610:
1582:
1554:
1539:
1496:
1481:
1365:. Every such state
1275:
1231:
1191:
1105:
1077:
1029:
1001:
953:
925:
454:-dimensional basis
248:large numbers, and
237:quantum information
17333:10.1007/11526216_2
17019:
16792:
16632:
16482:
16415:
16351:
16216:
16096:
16042:
15978:
15878:
15811:
15686:
15566:
15519:
15498:
15466:
15402:
15332:
15277:
15254:
15231:
15064:
15017:
14991:
14865:
14834:
14721:
14682:
14662:
14621:
14580:
14544:
14462:
14439:
14398:
14357:
14321:
14215:
14196:
14100:
14076:
14046:
14022:
13991:
13930:
13818:
13677:
13654:
13632:
13609:
13567:
13545:
13432:
13381:
13316:
13283:
13248:
13197:
13111:
13084:
13047:
13007:
12978:
12941:
12913:
12805:
12746:
12706:
12675:
12609:
12580:
12540:
12500:
12463:
12433:
12406:
12379:
12352:
12315:
12275:
12251:
12195:
12144:
12088:
12066:
11899:
11866:
11828:
11795:
11757:
11722:
11693:
11645:
11611:
11555:
11523:
11478:
11458:
11425:
11259:
11210:
11182:
11076:
11028:
10965:
10896:
10825:
10805:
10758:
10740:
10736:
10700:
10680:
10633:
10615:
10611:
10572:
10533:
10504:
10468:
10423:
10370:
10342:
10316:
10259:
10202:
10163:
10138:
10121:
10090:
10073:
10018:
9976:
9952:
9928:
9908:
9887:
9870:
9807:
9740:
9739:
9666:
9631:
9592:
9520:
9474:
9470:
9388:
9344:
9319:
9300:
9235:
9234:
9161:
9126:
9087:
9065:
8989:
8988:
8846:
8799:
8773:
8751:
8721:
8684:
8663:
8591:
8574:
8519:
8417:
8361:
8325:
8305:
8258:
8211:
8184:
8163:
8136:
8109:
8014:
7972:
7945:
7916:
7849:
7844:
7788:
7742:
7698:
7693:
7621:
7554:
7549:
7499:
7459:
7416:
7410:
7352:
7272:
7198:
7175:basic operations.
7165:
7119:
7080:
7032:
7012:
6981:
6924:
6895:
6894:
6821:
6802:
6751:
6718:
6698:
6659:
6612:
6579:computational time
6563:
6543:
6483:
6440:
6420:
6393:
6366:
6339:
6319:
6281:
6252:
6208:
6167:
6131:
6042:
5953:
5880:
5861:
5836:
5809:
5780:
5727:
5692:
5668:
5608:
5521:
5483:
5381:
5347:
5301:
5235:
5234:
5133:
5108:
5008:
4974:
4936:
4935:
4846:
4819:
4778:
4737:
4701:
4700:
4638:
4588:
4558:
4528:
4486:
4460:
4425:
4359:
4311:
4283:
4245:
4244:
4149:
4114:
4078:
4030:
4029:
3966:
3944:
3914:
3897:
3862:
3839:
3813:
3768:
3733:
3688:
3663:
3618:
3591:
3537:
3514:
3479:
3448:
3413:
3359:
3321:
3237:
3204:
3182:
3146:
3108:
3096:and eigenvectors
3086:
3056:
3039:
2984:
2816:
2672:
2589:
2491:
2393:
2295:
2272:
2252:
2237:
2178:
2163:
2094:
1954:
1856:
1835:
1782:
1755:
1726:
1707:
1684:
1657:
1638:
1615:
1596:
1568:
1540:
1525:
1502:
1482:
1467:
1385:
1355:
1276:
1232:
1192:
1118:
1078:
1030:
1002:
954:
926:
888:
769:
702:
533:
444:
417:
389:
184:encyclopedic style
171:is written like a
17378:Quantum mechanics
17342:978-3-540-27987-7
17324:math-ph/0506007v1
16956:
16790:
16744:
16609:
16606:
16597:
16594:
16582:
16547:
16520:
16413:
16393:
15280:{\displaystyle l}
15257:{\displaystyle D}
14983:
14931:
14501:
14500:
14278:
14277:
14188:
14151:
13742:
13680:{\displaystyle T}
13506:
13467:
13416:
13347:
13314:
13004:
12908:
12876:
12826:
12790:
12703:
12606:
12590:of a Hamiltonian
12231:
12124:
12050:
12027:
11978:
11937:
11687:
11679:
11625:
11605:
11597:
11587:
11516:
11481:{\displaystyle n}
11456:
11408:
11373:
11086:or, equivalently
11062:
11047:
11013:
10799:
10791:
10730:
10722:
10674:
10666:
10605:
10597:
10373:{\displaystyle N}
9724:
9443:
9219:
8951:
7835:
7684:
7540:
7401:
7057:+1, concern only
7044:unitary operation
6885:
6085:
6084:
5943:
5177:
4884:
4684:
4641:{\displaystyle :}
4189:
4013:
3575:
3042:{\displaystyle :}
2959:
2958:
2895:
2894:
2862:
2861:
2801:
2800:
2773:
2772:
2745:
2744:
2717:
2716:
2621:
2620:
2582:
2579:
2523:
2522:
2484:
2481:
2425:
2424:
2386:
2383:
2327:
2326:
2230:
2227:
2209:
2156:
2153:
2135:
2058:
2035:
2007:
2004:
212:
211:
204:
121:
120:
113:
95:
17390:
17363:
17362:
17326:
17310:
17304:
17303:
17269:
17267:quant-ph/0310089
17249:
17243:
17242:
17208:
17206:cond-mat/0406440
17188:
17182:
17179:
17145:
17143:cond-mat/0406426
17125:
17119:
17118:
17084:
17082:quant-ph/0301063
17061:
17045:implementation.
17028:
17026:
17025:
17020:
17015:
17014:
16992:
16981:
16957:
16955:
16954:
16953:
16937:
16936:
16935:
16916:
16913:
16902:
16884:
16883:
16864:
16853:
16826:
16801:
16799:
16798:
16793:
16791:
16789:
16788:
16787:
16771:
16770:
16769:
16750:
16745:
16743:
16742:
16741:
16725:
16724:
16715:
16704:
16703:
16698:
16689:
16688:
16679:
16674:
16673:
16672:
16659:
16641:
16639:
16638:
16633:
16631:
16630:
16625:
16620:
16619:
16607:
16604:
16595:
16592:
16588:
16583:
16581:
16580:
16579:
16563:
16562:
16553:
16548:
16543:
16532:
16521:
16516:
16515:
16506:
16491:
16489:
16488:
16483:
16475:
16474:
16462:
16461:
16449:
16448:
16424:
16422:
16421:
16416:
16414:
16409:
16408:
16399:
16394:
16389:
16388:
16387:
16377:
16366:
16365:
16364:
16346:
16344:
16343:
16334:
16333:
16332:
16331:
16322:
16321:
16297:
16296:
16295:
16285:
16278:
16277:
16260:
16259:
16249:
16238:
16237:
16236:
16226:
16225:
16209:
16208:
16199:
16198:
16197:
16196:
16187:
16186:
16162:
16161:
16160:
16150:
16143:
16142:
16125:
16124:
16105:
16103:
16102:
16097:
16094:
16093:
16092:
16073:
16072:
16071:
16062:
16061:
16041:
16040:
16030:
16029:
16028:
16009:
16008:
16007:
15998:
15997:
15973:
15968:
15961:
15960:
15943:
15942:
15933:
15932:
15931:
15930:
15921:
15920:
15887:
15885:
15884:
15879:
15874:
15873:
15861:
15860:
15848:
15847:
15837:
15826:
15825:
15824:
15807:
15806:
15797:
15796:
15795:
15794:
15785:
15784:
15760:
15755:
15754:
15753:
15741:
15740:
15723:
15722:
15712:
15701:
15700:
15699:
15682:
15681:
15672:
15671:
15670:
15669:
15660:
15659:
15635:
15634:
15633:
15623:
15616:
15615:
15592:
15591:
15575:
15573:
15572:
15567:
15562:
15560:
15540:
15539:
15538:
15507:
15505:
15504:
15499:
15491:
15490:
15475:
15473:
15472:
15467:
15465:
15464:
15463:
15444:
15443:
15442:
15437:
15436:
15430:
15419:
15418:
15417:
15412:
15401:
15393:
15392:
15391:
15386:
15379:
15372:
15371:
15370:
15341:
15339:
15338:
15333:
15286:
15284:
15283:
15278:
15263:
15261:
15260:
15255:
15240:
15238:
15237:
15232:
15227:
15226:
15205:
15204:
15171:
15170:
15165:
15156:
15155:
15146:
15141:
15133:
15125:
15124:
15091:
15090:
15085:
15072:
15063:
15058:
15050:
15026:
15024:
15023:
15018:
15013:
15011:
15006:
15001:
15000:
14989:
14984:
14982:
14981:
14966:
14958:
14957:
14956:
14951:
14950:
14946:
14937:
14932:
14930:
14929:
14908:
14900:
14899:
14898:
14888:
14874:
14872:
14871:
14866:
14864:
14863:
14853:
14842:
14829:
14824:
14823:
14822:
14799:
14798:
14793:
14784:
14783:
14774:
14769:
14761:
14747:
14746:
14730:
14728:
14727:
14722:
14711:
14710:
14701:
14695:
14690:
14671:
14669:
14668:
14663:
14658:
14656:
14636:
14635:
14634:
14619:
14611:
14609:
14595:
14594:
14593:
14578:
14572:
14561:
14560:
14559:
14554:
14542:
14537:
14536:
14535:
14530:
14521:
14520:
14519:
14514:
14502:
14499:
14498:
14489:
14485:
14477:
14475:
14470:
14460:
14448:
14446:
14445:
14440:
14435:
14433:
14413:
14412:
14411:
14396:
14388:
14386:
14372:
14371:
14370:
14355:
14349:
14338:
14337:
14336:
14331:
14319:
14318:
14317:
14312:
14305:
14298:
14297:
14296:
14291:
14279:
14276:
14275:
14260:
14256:
14248:
14247:
14246:
14236:
14224:
14222:
14221:
14216:
14211:
14209:
14204:
14194:
14189:
14187:
14186:
14177:
14169:
14168:
14167:
14157:
14152:
14150:
14149:
14134:
14126:
14121:
14109:
14107:
14106:
14101:
14099:
14098:
14085:
14083:
14082:
14077:
14072:
14067:
14055:
14053:
14052:
14047:
14045:
14044:
14031:
14029:
14028:
14023:
14021:
14020:
14010:
13999:
13986:
13981:
13980:
13979:
13956:
13955:
13939:
13937:
13936:
13931:
13926:
13925:
13906:
13895:
13868:
13867:
13827:
13825:
13824:
13819:
13817:
13816:
13804:
13803:
13779:
13778:
13773:
13764:
13763:
13762:
13749:
13744:
13743:
13738:
13737:
13725:
13719:
13686:
13684:
13683:
13678:
13663:
13661:
13660:
13655:
13650:
13648:
13643:
13630:
13618:
13616:
13615:
13610:
13605:
13604:
13603:
13590:
13576:
13574:
13573:
13568:
13563:
13561:
13556:
13543:
13538:
13527:
13526:
13525:
13512:
13507:
13505:
13504:
13499:
13487:
13479:
13478:
13477:
13469:
13468:
13460:
13455:
13441:
13439:
13438:
13433:
13428:
13427:
13426:
13418:
13417:
13409:
13404:
13390:
13388:
13387:
13382:
13380:
13379:
13364:
13363:
13348:
13340:
13325:
13323:
13322:
13317:
13315:
13307:
13292:
13290:
13289:
13284:
13282:
13281:
13270:
13257:
13255:
13254:
13249:
13247:
13246:
13245:
13244:
13206:
13204:
13203:
13198:
13193:
13192:
13191:
13181:
13176:
13175:
13171:
13170:
13144:
13143:
13142:
13132:
13120:
13118:
13117:
13112:
13110:
13109:
13093:
13091:
13090:
13085:
13080:
13079:
13078:
13068:
13056:
13054:
13053:
13048:
13043:
13042:
13041:
13031:
13016:
13014:
13013:
13008:
13006:
13005:
12997:
12987:
12985:
12984:
12979:
12974:
12973:
12972:
12962:
12950:
12948:
12947:
12942:
12940:
12939:
12922:
12920:
12919:
12914:
12909:
12907:
12900:
12899:
12898:
12888:
12883:
12882:
12878:
12877:
12869:
12854:
12850:
12849:
12848:
12838:
12833:
12832:
12828:
12827:
12819:
12807:
12804:
12783:
12782:
12781:
12768:
12755:
12753:
12752:
12747:
12742:
12741:
12740:
12727:
12715:
12713:
12712:
12707:
12705:
12704:
12696:
12684:
12682:
12681:
12676:
12671:
12670:
12669:
12656:
12642:
12641:
12640:
12630:
12618:
12616:
12615:
12610:
12608:
12607:
12599:
12589:
12587:
12586:
12581:
12576:
12575:
12574:
12561:
12549:
12547:
12546:
12541:
12536:
12535:
12534:
12524:
12509:
12507:
12506:
12501:
12496:
12495:
12494:
12484:
12472:
12470:
12469:
12464:
12462:
12461:
12442:
12440:
12439:
12434:
12432:
12431:
12415:
12413:
12412:
12407:
12405:
12404:
12388:
12386:
12385:
12380:
12378:
12377:
12361:
12359:
12358:
12353:
12348:
12347:
12346:
12336:
12324:
12322:
12321:
12316:
12311:
12310:
12309:
12299:
12284:
12282:
12281:
12276:
12274:
12273:
12260:
12258:
12257:
12252:
12250:
12249:
12248:
12247:
12232:
12224:
12204:
12202:
12201:
12196:
12194:
12193:
12192:
12191:
12176:
12153:
12151:
12150:
12145:
12143:
12142:
12141:
12140:
12125:
12117:
12097:
12095:
12094:
12089:
12075:
12073:
12072:
12067:
12059:
12058:
12057:
12052:
12051:
12043:
12038:
12033:
12032:
12028:
12023:
12012:
12005:
12004:
12000:
11984:
11983:
11979:
11971:
11952:
11951:
11950:
11939:
11938:
11930:
11925:
11908:
11906:
11905:
11900:
11898:
11890:
11875:
11873:
11872:
11867:
11865:
11864:
11860:
11837:
11835:
11834:
11829:
11827:
11826:
11804:
11802:
11801:
11796:
11794:
11793:
11789:
11766:
11764:
11763:
11758:
11756:
11755:
11731:
11729:
11728:
11723:
11721:
11720:
11719:
11718:
11703:
11692:
11688:
11685:
11675:
11674:
11670:
11654:
11652:
11651:
11646:
11644:
11643:
11642:
11641:
11626:
11618:
11610:
11606:
11603:
11593:
11592:
11588:
11580:
11564:
11562:
11561:
11556:
11554:
11553:
11549:
11532:
11530:
11529:
11524:
11522:
11521:
11517:
11509:
11487:
11485:
11484:
11479:
11467:
11465:
11464:
11459:
11457:
11449:
11434:
11432:
11431:
11426:
11424:
11423:
11414:
11413:
11409:
11401:
11394:
11393:
11389:
11379:
11378:
11374:
11366:
11353:
11352:
11351:
11346:
11333:
11332:
11328:
11327:
11301:
11300:
11296:
11295:
11268:
11266:
11265:
11260:
11258:
11257:
11219:
11217:
11216:
11211:
11191:
11189:
11188:
11183:
11175:
11174:
11162:
11161:
11152:
11151:
11147:
11137:
11136:
11132:
11119:
11118:
11102:
11085:
11083:
11082:
11077:
11075:
11074:
11069:
11065:
11064:
11063:
11055:
11049:
11048:
11040:
11027:
11009:
11008:
10974:
10972:
10971:
10966:
10955:
10954:
10950:
10931:
10930:
10905:
10903:
10902:
10897:
10886:
10885:
10881:
10862:
10861:
10834:
10832:
10831:
10826:
10821:
10820:
10804:
10800:
10797:
10783:
10766:
10753:
10748:
10735:
10731:
10728:
10709:
10707:
10706:
10701:
10696:
10695:
10679:
10675:
10672:
10658:
10641:
10628:
10623:
10610:
10606:
10603:
10581:
10579:
10578:
10573:
10559:
10558:
10542:
10540:
10539:
10534:
10532:
10531:
10513:
10511:
10510:
10505:
10499:
10476:
10466:
10461:
10442:
10431:
10421:
10416:
10398:
10397:
10379:
10377:
10376:
10371:
10351:
10349:
10348:
10343:
10325:
10323:
10322:
10317:
10312:
10311:
10306:
10300:
10295:
10294:
10282:
10281:
10268:
10266:
10265:
10260:
10255:
10254:
10249:
10243:
10238:
10237:
10225:
10224:
10211:
10209:
10208:
10203:
10198:
10196:
10192:
10191:
10179:
10171:
10161:
10155:
10154:
10146:
10130:
10128:
10127:
10122:
10119:
10118:
10106:
10105:
10082:
10080:
10079:
10074:
10069:
10068:
10063:
10057:
10052:
10051:
10039:
10027:
10025:
10024:
10019:
10017:
10016:
9985:
9983:
9982:
9977:
9975:
9974:
9961:
9959:
9958:
9953:
9951:
9950:
9937:
9935:
9934:
9929:
9917:
9915:
9914:
9909:
9906:
9898:
9879:
9877:
9876:
9871:
9866:
9865:
9860:
9854:
9843:
9832:is of the order
9816:
9814:
9813:
9808:
9800:
9796:
9790:
9785:
9784:
9774:
9767:
9759:
9751:
9738:
9717:
9716:
9715:
9711:
9700:
9687:
9675:
9673:
9672:
9667:
9665:
9664:
9660:
9640:
9638:
9637:
9632:
9624:
9616:
9601:
9599:
9598:
9593:
9588:
9584:
9578:
9573:
9572:
9563:
9562:
9553:
9552:
9539:
9531:
9519:
9518:
9508:
9501:
9493:
9485:
9469:
9436:
9435:
9426:
9421:
9419:
9415:
9404:
9396:
9377:
9375:
9371:
9360:
9352:
9342:
9336:
9335:
9327:
9309:
9307:
9306:
9301:
9293:
9285:
9280:
9279:
9269:
9262:
9254:
9246:
9233:
9212:
9211:
9210:
9206:
9195:
9182:
9170:
9168:
9167:
9162:
9160:
9159:
9158:
9135:
9133:
9132:
9127:
9119:
9111:
9096:
9094:
9093:
9088:
9074:
9072:
9071:
9066:
9061:
9056:
9055:
9047:
9032:
9024:
9018:
9017:
9005:
9004:
8987:
8986:
8969:
8947:
8942:
8941:
8926:
8925:
8916:
8911:
8910:
8892:
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8878:
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8844:
8831:
8808:
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8805:
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8798:
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8774:
8760:
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8709:
8695:
8682:
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8634:
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8554:
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8548:
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8530:
8518:
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8507:
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8480:
8475:
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8438:
8426:
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8423:
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8413:
8408:
8394:
8393:
8384:
8370:
8368:
8367:
8362:
8357:
8356:
8334:
8332:
8331:
8326:
8314:
8312:
8311:
8306:
8304:
8303:
8267:
8265:
8264:
8259:
8254:
8253:
8242:
8236:
8230:
8219:
8209:
8195:
8182:
8171:
8161:
8147:
8134:
8117:
8107:
8102:
8080:
8075:
8042:
8037:
8023:
8021:
8020:
8015:
8010:
8005:
7981:
7979:
7978:
7973:
7971:
7970:
7954:
7952:
7951:
7946:
7944:
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7925:
7923:
7922:
7917:
7912:
7907:
7896:
7891:
7886:
7885:
7884:
7874:
7857:
7843:
7831:
7826:
7824:
7811:
7796:
7780:
7778:
7765:
7750:
7740:
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7734:
7733:
7723:
7706:
7692:
7680:
7679:
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7617:
7612:
7601:
7596:
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7579:
7562:
7548:
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7531:
7529:
7525:
7507:
7491:
7489:
7485:
7467:
7457:
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7451:
7446:
7441:
7424:
7409:
7397:
7396:
7392:
7361:
7359:
7358:
7353:
7351:
7334:
7329:
7328:
7307:
7306:
7281:
7279:
7278:
7273:
7267:
7263:
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7257:
7248:
7243:
7242:
7233:
7222:
7207:
7205:
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7199:
7194:
7189:
7174:
7172:
7171:
7166:
7161:
7160:
7155:
7140:
7128:
7126:
7125:
7120:
7118:
7117:
7089:
7087:
7086:
7081:
7079:
7078:
7042:'s, following a
7041:
7039:
7038:
7033:
7021:
7019:
7018:
7013:
6990:
6988:
6987:
6982:
6976:
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6974:
6955:
6954:
6953:
6944:
6943:
6922:
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6910:
6909:
6908:
6893:
6880:
6879:
6878:
6860:
6852:
6851:
6850:
6841:
6840:
6812:operations), as
6811:
6809:
6808:
6803:
6798:
6797:
6785:
6784:
6772:
6760:
6758:
6757:
6752:
6750:
6749:
6727:
6725:
6724:
6719:
6707:
6705:
6704:
6699:
6697:
6696:
6668:
6666:
6665:
6660:
6658:
6657:
6624:unitary operator
6621:
6619:
6618:
6613:
6608:
6603:
6572:
6570:
6569:
6564:
6552:
6550:
6549:
6544:
6542:
6541:
6540:
6539:
6524:
6523:
6514:
6513:
6492:
6490:
6489:
6484:
6482:
6481:
6449:
6447:
6446:
6441:
6429:
6427:
6426:
6421:
6402:
6400:
6399:
6394:
6392:
6391:
6375:
6373:
6372:
6367:
6365:
6364:
6348:
6346:
6345:
6340:
6328:
6326:
6325:
6320:
6318:
6317:
6290:
6288:
6287:
6282:
6280:
6279:
6261:
6259:
6258:
6253:
6245:
6243:
6223:
6222:
6221:
6206:
6198:
6196:
6182:
6181:
6180:
6165:
6159:
6148:
6147:
6146:
6141:
6129:
6128:
6127:
6122:
6115:
6108:
6107:
6106:
6101:
6086:
6083:
6082:
6077:
6076:
6070:
6059:
6058:
6057:
6052:
6041:
6033:
6032:
6031:
6026:
6019:
6012:
6011:
6010:
5995:
5991:
5983:
5978:
5962:
5960:
5959:
5954:
5941:
5937:
5936:
5935:
5934:
5914:
5908:
5897:
5896:
5895:
5890:
5870:
5868:
5867:
5862:
5845:
5843:
5842:
5837:
5835:
5834:
5818:
5816:
5815:
5810:
5808:
5807:
5802:
5789:
5787:
5786:
5781:
5764:
5759:
5758:
5736:
5734:
5733:
5728:
5726:
5725:
5721:
5701:
5699:
5698:
5693:
5677:
5675:
5674:
5669:
5667:
5666:
5665:
5664:
5649:
5648:
5639:
5638:
5617:
5615:
5614:
5609:
5583:
5554:
5549:
5548:
5543:
5530:
5528:
5527:
5522:
5520:
5519:
5500:Now, looking at
5492:
5490:
5489:
5484:
5479:
5478:
5477:
5462:
5461:
5446:
5445:
5429:
5423:
5422:
5421:
5402:
5401:
5400:
5379:
5368:
5367:
5366:
5339:
5322:
5321:
5320:
5299:
5298:
5297:
5266:
5265:
5264:
5249:
5248:
5233:
5232:
5231:
5216:
5215:
5197:
5196:
5170:
5168:
5148:
5147:
5146:
5131:
5117:
5115:
5114:
5109:
5104:
5103:
5102:
5087:
5086:
5077:
5076:
5066:
5060:
5059:
5058:
5039:
5038:
5037:
5028:
5027:
5000:
4989:
4988:
4987:
4972:
4971:
4970:
4951:
4950:
4949:
4934:
4933:
4932:
4911:
4910:
4898:
4897:
4877:
4875:
4861:
4860:
4859:
4844:
4828:
4826:
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4820:
4815:
4813:
4793:
4792:
4791:
4776:
4768:
4766:
4752:
4751:
4750:
4735:
4729:
4718:
4717:
4716:
4711:
4699:
4698:
4697:
4677:
4672:
4647:
4645:
4644:
4639:
4597:
4595:
4594:
4589:
4587:
4586:
4567:
4565:
4564:
4559:
4557:
4556:
4537:
4535:
4534:
4529:
4527:
4526:
4495:
4493:
4492:
4487:
4469:
4467:
4466:
4461:
4456:
4454:
4440:
4439:
4438:
4423:
4415:
4414:
4413:
4404:
4403:
4393:
4387:
4376:
4375:
4374:
4369:
4357:
4356:
4355:
4336:
4335:
4334:
4325:
4324:
4309:
4298:
4297:
4296:
4281:
4280:
4279:
4260:
4259:
4258:
4243:
4242:
4241:
4229:
4228:
4216:
4215:
4203:
4202:
4182:
4177:
4158:
4156:
4155:
4150:
4145:
4143:
4129:
4128:
4127:
4112:
4106:
4095:
4094:
4093:
4088:
4076:
4075:
4074:
4055:
4054:
4053:
4044:
4043:
4028:
4027:
4026:
4005:
3991:
3990:
3989:
3980:
3979:
3965:
3953:
3951:
3950:
3945:
3942:
3931:
3930:
3929:
3924:
3906:
3904:
3903:
3898:
3893:
3891:
3877:
3876:
3875:
3860:
3849:Schmidt vectors
3848:
3846:
3845:
3840:
3823:in terms of the
3822:
3820:
3819:
3814:
3809:
3807:
3793:
3792:
3791:
3782:
3781:
3766:
3742:
3740:
3739:
3734:
3729:
3727:
3713:
3712:
3711:
3702:
3701:
3686:
3672:
3670:
3669:
3664:
3659:
3657:
3643:
3642:
3641:
3632:
3631:
3616:
3608:
3607:
3606:
3596:
3590:
3589:
3588:
3568:
3566:
3552:
3551:
3550:
3535:
3523:
3521:
3520:
3515:
3510:
3508:
3494:
3493:
3492:
3477:
3457:
3455:
3454:
3449:
3444:
3442:
3428:
3427:
3426:
3411:
3403:
3402:
3401:
3391:
3385:
3374:
3373:
3372:
3357:
3356:
3355:
3336:
3335:
3334:
3319:
3308:
3307:
3300:
3299:
3286:
3285:
3265:
3260:
3246:
3244:
3243:
3238:
3236:
3232:
3230:
3219:
3218:
3217:
3191:
3189:
3188:
3183:
3181:
3177:
3175:
3161:
3160:
3159:
3140:
3136:
3134:
3123:
3122:
3121:
3095:
3093:
3092:
3087:
3084:
3073:
3072:
3071:
3066:
3048:
3046:
3045:
3040:
2993:
2991:
2990:
2985:
2983:
2979:
2978:
2974:
2973:
2960:
2954:
2950:
2945:
2941:
2940:
2919:
2915:
2914:
2910:
2909:
2896:
2890:
2886:
2881:
2877:
2876:
2863:
2857:
2853:
2843:
2825:
2823:
2822:
2817:
2812:
2807:
2802:
2796:
2792:
2784:
2779:
2774:
2768:
2764:
2756:
2751:
2746:
2740:
2736:
2728:
2723:
2718:
2712:
2708:
2700:
2695:
2681:
2679:
2678:
2673:
2665:
2664:
2663:
2653:
2642:
2641:
2640:
2630:
2622:
2616:
2612:
2604:
2602:
2597:
2587:
2580:
2577:
2567:
2566:
2565:
2555:
2544:
2543:
2542:
2532:
2524:
2518:
2514:
2506:
2504:
2499:
2489:
2482:
2479:
2469:
2468:
2467:
2457:
2446:
2445:
2444:
2434:
2426:
2420:
2416:
2408:
2406:
2401:
2391:
2384:
2381:
2371:
2370:
2369:
2359:
2348:
2347:
2346:
2336:
2328:
2322:
2318:
2310:
2308:
2303:
2293:
2281:
2279:
2278:
2273:
2271:
2267:
2265:
2260:
2250:
2245:
2231:
2229:
2228:
2223:
2217:
2210:
2205:
2202:
2197:
2193:
2191:
2186:
2176:
2171:
2157:
2155:
2154:
2149:
2143:
2136:
2131:
2128:
2123:
2119:
2103:
2101:
2100:
2095:
2093:
2089:
2085:
2080:
2069:
2064:
2059:
2054:
2046:
2041:
2036:
2031:
2023:
2018:
2008:
2006:
2005:
2000:
1991:
1983:
1978:
1963:
1961:
1960:
1955:
1947:
1946:
1945:
1920:
1919:
1918:
1890:
1889:
1885:
1865:
1863:
1862:
1857:
1848:
1843:
1833:
1832:
1831:
1827:
1813:
1791:
1789:
1788:
1783:
1781:
1780:
1764:
1762:
1761:
1756:
1754:
1753:
1752:
1735:
1733:
1732:
1727:
1722:
1720:
1715:
1705:
1693:
1691:
1690:
1685:
1683:
1682:
1666:
1664:
1663:
1658:
1653:
1651:
1646:
1636:
1624:
1622:
1621:
1616:
1611:
1609:
1604:
1594:
1583:
1581:
1576:
1566:
1555:
1553:
1548:
1538:
1533:
1523:
1511:
1509:
1508:
1503:
1501:
1497:
1495:
1490:
1480:
1475:
1461:
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1412:
1394:
1392:
1391:
1386:
1381:
1376:
1364:
1362:
1361:
1356:
1354:
1353:
1352:
1340:
1339:
1285:
1283:
1282:
1277:
1274:
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1272:
1253:
1252:
1251:
1230:
1213:
1212:
1211:
1190:
1189:
1188:
1170:
1155:
1154:
1153:
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1137:
1104:
1093:
1092:
1091:
1076:
1075:
1074:
1055:
1054:
1053:
1044:
1043:
1028:
1017:
1016:
1015:
1000:
999:
998:
979:
978:
977:
968:
967:
952:
941:
940:
939:
924:
923:
922:
903:
902:
901:
886:
881:
874:
873:
846:
845:
828:
827:
826:
825:
810:
809:
800:
799:
778:
776:
775:
770:
768:
767:
766:
765:
750:
749:
740:
739:
711:
709:
708:
703:
698:
697:
696:
684:
683:
656:
655:
643:
642:
632:
627:
626:
625:
624:
609:
608:
599:
598:
583:
578:
554:
542:
540:
539:
534:
529:
528:
516:
515:
488:
487:
475:
474:
465:
453:
451:
450:
445:
443:
442:
426:
424:
423:
418:
410:
398:
396:
395:
390:
388:
387:
383:
364:
322:optical lattices
242:Shor's algorithm
232:quantum computer
223:(QMC). Also the
207:
200:
196:
193:
187:
164:
163:
156:
116:
109:
105:
102:
96:
94:
53:
29:
21:
17398:
17397:
17393:
17392:
17391:
17389:
17388:
17387:
17368:
17367:
17366:
17343:
17311:
17307:
17250:
17246:
17193:Phys. Rev. Lett
17189:
17185:
17130:Phys. Rev. Lett
17126:
17122:
17062:
17055:
17051:
17038:
17010:
17006:
16982:
16971:
16949:
16945:
16938:
16931:
16927:
16917:
16915:
16903:
16892:
16879:
16875:
16854:
16843:
16822:
16814:
16811:
16810:
16807:
16783:
16779:
16772:
16765:
16761:
16751:
16749:
16737:
16733:
16726:
16720:
16716:
16714:
16699:
16694:
16693:
16684:
16680:
16675:
16668:
16664:
16663:
16655:
16653:
16650:
16649:
16626:
16621:
16615:
16611:
16610:
16584:
16575:
16571:
16564:
16558:
16554:
16552:
16533:
16531:
16511:
16507:
16505:
16497:
16494:
16493:
16470:
16466:
16457:
16453:
16444:
16440:
16432:
16429:
16428:
16404:
16400:
16398:
16383:
16379:
16367:
16360:
16356:
16355:
16347:
16345:
16339:
16335:
16327:
16323:
16311:
16307:
16306:
16302:
16291:
16287:
16286:
16273:
16269:
16268:
16255:
16251:
16239:
16232:
16228:
16227:
16218:
16217:
16204:
16200:
16192:
16188:
16176:
16172:
16171:
16167:
16156:
16152:
16151:
16138:
16134:
16133:
16120:
16116:
16114:
16111:
16110:
16088:
16084:
16074:
16067:
16063:
16051:
16047:
16046:
16036:
16032:
16024:
16020:
16010:
16003:
15999:
15987:
15983:
15982:
15969:
15956:
15952:
15951:
15938:
15934:
15926:
15922:
15910:
15906:
15905:
15901:
15896:
15893:
15892:
15890:
15869:
15865:
15856:
15852:
15843:
15839:
15827:
15820:
15816:
15815:
15802:
15798:
15790:
15786:
15774:
15770:
15769:
15765:
15756:
15749:
15745:
15736:
15732:
15731:
15718:
15714:
15702:
15695:
15691:
15690:
15677:
15673:
15665:
15661:
15649:
15645:
15644:
15640:
15629:
15625:
15624:
15611:
15607:
15606:
15587:
15583:
15581:
15578:
15577:
15541:
15528:
15524:
15523:
15518:
15513:
15510:
15509:
15486:
15485:
15483:
15480:
15479:
15459:
15455:
15448:
15438:
15432:
15420:
15413:
15408:
15407:
15406:
15397:
15396:
15395:
15387:
15382:
15381:
15380:
15366:
15362:
15361:
15360:
15355:
15347:
15344:
15343:
15291:
15288:
15287:
15272:
15269:
15268:
15249:
15246:
15245:
15222:
15218:
15194:
15190:
15166:
15161:
15160:
15151:
15147:
15142:
15137:
15129:
15114:
15110:
15086:
15081:
15080:
15068:
15059:
15054:
15046:
15032:
15029:
15028:
15007:
15002:
14993:
14992:
14990:
14985:
14971:
14967:
14965:
14952:
14942:
14941:
14940:
14939:
14938:
14933:
14919:
14915:
14907:
14894:
14890:
14889:
14884:
14882:
14879:
14878:
14859:
14855:
14843:
14838:
14825:
14818:
14814:
14807:
14794:
14789:
14788:
14779:
14775:
14770:
14765:
14757:
14742:
14738:
14736:
14733:
14732:
14706:
14702:
14697:
14691:
14686:
14677:
14674:
14673:
14637:
14630:
14626:
14625:
14620:
14615:
14596:
14589:
14585:
14584:
14579:
14574:
14562:
14555:
14550:
14549:
14548:
14538:
14531:
14526:
14525:
14515:
14510:
14509:
14508:
14507:
14494:
14490:
14484:
14471:
14466:
14461:
14456:
14454:
14451:
14450:
14414:
14407:
14403:
14402:
14397:
14392:
14373:
14366:
14362:
14361:
14356:
14351:
14339:
14332:
14327:
14326:
14325:
14313:
14308:
14307:
14306:
14292:
14287:
14286:
14285:
14284:
14271:
14267:
14255:
14242:
14238:
14237:
14232:
14230:
14227:
14226:
14205:
14200:
14195:
14190:
14182:
14178:
14176:
14163:
14159:
14158:
14153:
14145:
14141:
14133:
14122:
14117:
14115:
14112:
14111:
14094:
14093:
14091:
14088:
14087:
14068:
14063:
14061:
14058:
14057:
14040:
14039:
14037:
14034:
14033:
14016:
14012:
14000:
13995:
13982:
13975:
13971:
13964:
13951:
13947:
13945:
13942:
13941:
13921:
13917:
13896:
13885:
13863:
13862:
13854:
13851:
13850:
13840:
13812:
13808:
13799:
13795:
13774:
13769:
13768:
13755:
13751:
13750:
13745:
13730:
13726:
13724:
13723:
13715:
13692:
13689:
13688:
13672:
13669:
13668:
13644:
13636:
13631:
13626:
13624:
13621:
13620:
13596:
13592:
13591:
13586:
13584:
13581:
13580:
13557:
13549:
13544:
13539:
13534:
13518:
13514:
13513:
13508:
13500:
13495:
13494:
13486:
13470:
13459:
13458:
13457:
13456:
13451:
13449:
13446:
13445:
13419:
13408:
13407:
13406:
13405:
13400:
13398:
13395:
13394:
13375:
13371:
13353:
13349:
13339:
13331:
13328:
13327:
13306:
13298:
13295:
13294:
13271:
13266:
13265:
13263:
13260:
13259:
13237:
13233:
13232:
13231:
13229:
13226:
13225:
13222:
13214:
13209:
13187:
13183:
13182:
13177:
13166:
13162:
13155:
13151:
13138:
13134:
13133:
13128:
13126:
13123:
13122:
13105:
13101:
13099:
13096:
13095:
13074:
13070:
13069:
13064:
13062:
13059:
13058:
13037:
13033:
13032:
13027:
13025:
13022:
13021:
12996:
12995:
12993:
12990:
12989:
12968:
12964:
12963:
12958:
12956:
12953:
12952:
12935:
12931:
12929:
12926:
12925:
12894:
12890:
12889:
12884:
12868:
12867:
12863:
12859:
12855:
12844:
12840:
12839:
12834:
12818:
12817:
12813:
12809:
12808:
12806:
12794:
12774:
12770:
12769:
12764:
12762:
12759:
12758:
12733:
12729:
12728:
12723:
12721:
12718:
12717:
12695:
12694:
12692:
12689:
12688:
12662:
12658:
12657:
12652:
12636:
12632:
12631:
12626:
12624:
12621:
12620:
12598:
12597:
12595:
12592:
12591:
12567:
12563:
12562:
12557:
12555:
12552:
12551:
12530:
12526:
12525:
12520:
12518:
12515:
12514:
12490:
12486:
12485:
12480:
12478:
12475:
12474:
12457:
12456:
12454:
12451:
12450:
12427:
12423:
12421:
12418:
12417:
12400:
12399:
12397:
12394:
12393:
12373:
12369:
12367:
12364:
12363:
12342:
12338:
12337:
12332:
12330:
12327:
12326:
12305:
12301:
12300:
12295:
12293:
12290:
12289:
12269:
12268:
12266:
12263:
12262:
12237:
12233:
12223:
12216:
12212:
12210:
12207:
12206:
12181:
12177:
12166:
12165:
12161:
12159:
12156:
12155:
12130:
12126:
12116:
12109:
12105:
12103:
12100:
12099:
12083:
12080:
12079:
12053:
12042:
12041:
12040:
12039:
12034:
12013:
12011:
12010:
12006:
11990:
11989:
11985:
11970:
11963:
11959:
11940:
11929:
11928:
11927:
11926:
11921:
11919:
11916:
11915:
11891:
11886:
11881:
11878:
11877:
11853:
11849:
11845:
11843:
11840:
11839:
11816:
11812:
11810:
11807:
11806:
11805:(respectively,
11782:
11778:
11774:
11772:
11769:
11768:
11745:
11741:
11739:
11736:
11735:
11708:
11704:
11699:
11698:
11694:
11684:
11683:
11666:
11665:
11661:
11659:
11656:
11655:
11631:
11627:
11617:
11616:
11612:
11602:
11601:
11579:
11578:
11574:
11572:
11569:
11568:
11545:
11544:
11540:
11538:
11535:
11534:
11508:
11507:
11503:
11501:
11498:
11497:
11494:
11473:
11470:
11469:
11448:
11440:
11437:
11436:
11419:
11415:
11400:
11399:
11395:
11385:
11384:
11380:
11365:
11364:
11360:
11347:
11342:
11338:
11334:
11323:
11319:
11312:
11308:
11291:
11287:
11280:
11276:
11274:
11271:
11270:
11244:
11240:
11238:
11235:
11234:
11199:
11196:
11195:
11170:
11166:
11157:
11156:
11143:
11142:
11138:
11128:
11127:
11123:
11098:
11097:
11093:
11091:
11088:
11087:
11070:
11054:
11050:
11039:
11035:
11034:
11030:
11029:
11017:
10998:
10994:
10992:
10989:
10988:
10985:
10943:
10939:
10935:
10920:
10916:
10911:
10908:
10907:
10874:
10870:
10866:
10851:
10847:
10842:
10839:
10838:
10810:
10806:
10796:
10795:
10767:
10762:
10749:
10744:
10727:
10726:
10714:
10711:
10710:
10685:
10681:
10671:
10670:
10642:
10637:
10624:
10619:
10602:
10601:
10589:
10586:
10585:
10554:
10550:
10548:
10545:
10544:
10527:
10523:
10521:
10518:
10517:
10477:
10472:
10462:
10451:
10432:
10427:
10417:
10406:
10393:
10389:
10387:
10384:
10383:
10365:
10362:
10361:
10358:
10331:
10328:
10327:
10307:
10302:
10301:
10296:
10290:
10286:
10277:
10276:
10274:
10271:
10270:
10269:, respectively
10250:
10245:
10244:
10239:
10233:
10229:
10220:
10219:
10217:
10214:
10213:
10212:, a maximum of
10184:
10183:
10173:
10172:
10167:
10162:
10157:
10148:
10147:
10142:
10136:
10133:
10132:
10111:
10107:
10098:
10094:
10088:
10085:
10084:
10064:
10059:
10058:
10053:
10047:
10043:
10035:
10033:
10030:
10029:
10006:
10005:
9991:
9988:
9987:
9970:
9969:
9967:
9964:
9963:
9946:
9945:
9943:
9940:
9939:
9923:
9920:
9919:
9899:
9891:
9885:
9882:
9881:
9861:
9856:
9855:
9850:
9839:
9837:
9834:
9833:
9822:
9792:
9791:
9786:
9780:
9776:
9763:
9753:
9752:
9744:
9728:
9704:
9694:
9693:
9689:
9688:
9683:
9681:
9678:
9677:
9656:
9652:
9648:
9646:
9643:
9642:
9617:
9612:
9607:
9604:
9603:
9580:
9579:
9574:
9568:
9564:
9558:
9554:
9548:
9544:
9532:
9524:
9514:
9510:
9497:
9487:
9486:
9478:
9447:
9428:
9427:
9422:
9408:
9398:
9397:
9392:
9387:
9364:
9354:
9353:
9348:
9343:
9338:
9329:
9328:
9323:
9317:
9314:
9313:
9286:
9281:
9275:
9271:
9258:
9248:
9247:
9239:
9223:
9199:
9189:
9188:
9184:
9183:
9178:
9176:
9173:
9172:
9151:
9147:
9143:
9141:
9138:
9137:
9112:
9107:
9102:
9099:
9098:
9082:
9079:
9078:
9057:
9048:
9040:
9039:
9025:
9020:
9010:
9006:
8997:
8993:
8979:
8962:
8955:
8943:
8934:
8933:
8918:
8917:
8912:
8903:
8899:
8872:
8871:
8867:
8865:
8862:
8861:
8825:
8824:
8820:
8818:
8815:
8814:
8794:
8792:
8789:
8788:
8783:'s at the bond
8768:
8765:
8764:
8733:
8725:
8715:
8711:
8696:
8688:
8675:
8667:
8657:
8640:
8630:
8619:
8603:
8595:
8589:
8586:
8585:
8556:
8555:
8550:
8544:
8540:
8531:
8523:
8513:
8509:
8503:
8486:
8476:
8459:
8440:
8439:
8434:
8432:
8429:
8428:
8409:
8404:
8386:
8385:
8380:
8378:
8375:
8374:
8346:
8342:
8340:
8337:
8336:
8320:
8317:
8316:
8293:
8289:
8287:
8284:
8283:
8249:
8238:
8237:
8232:
8220:
8215:
8196:
8188:
8172:
8167:
8148:
8140:
8118:
8113:
8103:
8086:
8076:
8053:
8038:
8033:
8031:
8028:
8027:
8006:
8001:
7999:
7996:
7995:
7966:
7962:
7960:
7957:
7956:
7939:
7935:
7933:
7930:
7929:
7908:
7903:
7892:
7887:
7880:
7876:
7858:
7853:
7845:
7839:
7827:
7801:
7797:
7792:
7787:
7755:
7751:
7746:
7741:
7736:
7729:
7725:
7707:
7702:
7694:
7688:
7671:
7654:
7650:
7646:
7644:
7641:
7640:
7613:
7608:
7597:
7592:
7585:
7581:
7563:
7558:
7550:
7544:
7532:
7515:
7508:
7503:
7498:
7475:
7468:
7463:
7458:
7453:
7447:
7425:
7420:
7412:
7411:
7405:
7382:
7375:
7371:
7369:
7366:
7365:
7347:
7330:
7318:
7314:
7302:
7298:
7296:
7293:
7292:
7259:
7253:
7249:
7244:
7238:
7234:
7229:
7218:
7217:
7215:
7212:
7211:
7190:
7185:
7183:
7180:
7179:
7156:
7145:
7144:
7136:
7134:
7131:
7130:
7101:
7097:
7095:
7092:
7091:
7068:
7064:
7062:
7059:
7058:
7027:
7024:
7023:
7007:
7004:
7003:
7000:
6970:
6966:
6956:
6949:
6945:
6933:
6929:
6928:
6916:
6912:
6911:
6904:
6900:
6899:
6889:
6874:
6870:
6854:
6853:
6846:
6842:
6830:
6826:
6825:
6819:
6816:
6815:
6793:
6789:
6780:
6776:
6768:
6766:
6763:
6762:
6739:
6735:
6733:
6730:
6729:
6713:
6710:
6709:
6680:
6676:
6674:
6671:
6670:
6641:
6637:
6635:
6632:
6631:
6604:
6599:
6597:
6594:
6593:
6590:
6558:
6555:
6554:
6535:
6531:
6519:
6515:
6509:
6505:
6504:
6500:
6498:
6495:
6494:
6477:
6473:
6471:
6468:
6467:
6456:
6435:
6432:
6431:
6415:
6412:
6411:
6387:
6383:
6381:
6378:
6377:
6360:
6356:
6354:
6351:
6350:
6334:
6331:
6330:
6313:
6309:
6307:
6304:
6303:
6275:
6271:
6269:
6266:
6265:
6224:
6217:
6213:
6212:
6207:
6202:
6183:
6176:
6172:
6171:
6166:
6161:
6149:
6142:
6137:
6136:
6135:
6123:
6118:
6117:
6116:
6102:
6097:
6096:
6095:
6094:
6078:
6072:
6060:
6053:
6048:
6047:
6046:
6037:
6036:
6035:
6027:
6022:
6021:
6020:
6006:
6002:
6001:
6000:
5990:
5979:
5974:
5972:
5969:
5968:
5930:
5926:
5919:
5915:
5910:
5898:
5891:
5886:
5885:
5884:
5878:
5875:
5874:
5856:
5853:
5852:
5830:
5826:
5824:
5821:
5820:
5803:
5798:
5797:
5795:
5792:
5791:
5760:
5748:
5744:
5742:
5739:
5738:
5717:
5713:
5709:
5707:
5704:
5703:
5687:
5684:
5683:
5660:
5656:
5644:
5640:
5634:
5630:
5629:
5625:
5623:
5620:
5619:
5579:
5550:
5544:
5539:
5538:
5536:
5533:
5532:
5515:
5511:
5509:
5506:
5505:
5498:
5473:
5469:
5451:
5447:
5435:
5431:
5430:
5425:
5417:
5413:
5403:
5390:
5386:
5385:
5369:
5356:
5352:
5351:
5323:
5310:
5306:
5305:
5287:
5283:
5267:
5254:
5250:
5244:
5240:
5239:
5227:
5223:
5205:
5201:
5186:
5182:
5181:
5149:
5142:
5138:
5137:
5132:
5127:
5125:
5122:
5121:
5098:
5094:
5082:
5078:
5072:
5068:
5067:
5062:
5054:
5050:
5040:
5033:
5029:
5017:
5013:
5012:
4990:
4983:
4979:
4978:
4966:
4962:
4952:
4945:
4941:
4940:
4922:
4918:
4906:
4902:
4893:
4889:
4888:
4862:
4855:
4851:
4850:
4845:
4840:
4838:
4835:
4834:
4794:
4787:
4783:
4782:
4777:
4772:
4753:
4746:
4742:
4741:
4736:
4731:
4719:
4712:
4707:
4706:
4705:
4693:
4689:
4688:
4673:
4668:
4666:
4663:
4662:
4603:
4600:
4599:
4579:
4575:
4573:
4570:
4569:
4549:
4545:
4543:
4540:
4539:
4519:
4515:
4501:
4498:
4497:
4481:
4478:
4477:
4441:
4434:
4430:
4429:
4424:
4419:
4409:
4405:
4399:
4395:
4394:
4389:
4377:
4370:
4365:
4364:
4363:
4351:
4347:
4337:
4330:
4326:
4320:
4316:
4315:
4299:
4292:
4288:
4287:
4275:
4271:
4261:
4254:
4250:
4249:
4237:
4233:
4224:
4220:
4211:
4207:
4198:
4194:
4193:
4178:
4173:
4171:
4168:
4167:
4130:
4123:
4119:
4118:
4113:
4108:
4096:
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10816:
10813:
10809:
10803:
10794:
10790:
10787:
10782:
10779:
10776:
10773:
10770:
10765:
10761:
10757:
10752:
10747:
10743:
10739:
10734:
10725:
10721:
10718:
10699:
10694:
10691:
10688:
10684:
10678:
10669:
10665:
10662:
10657:
10654:
10651:
10648:
10645:
10640:
10636:
10632:
10627:
10622:
10618:
10614:
10609:
10600:
10596:
10593:
10571:
10568:
10565:
10562:
10557:
10553:
10530:
10526:
10503:
10498:
10495:
10492:
10489:
10486:
10483:
10480:
10475:
10471:
10465:
10460:
10457:
10454:
10450:
10446:
10441:
10438:
10435:
10430:
10426:
10420:
10415:
10412:
10409:
10405:
10401:
10396:
10392:
10369:
10357:
10354:
10341:
10338:
10335:
10315:
10310:
10305:
10299:
10293:
10289:
10285:
10280:
10258:
10253:
10248:
10242:
10236:
10232:
10228:
10223:
10201:
10195:
10190:
10187:
10182:
10178:
10175:
10170:
10166:
10160:
10153:
10150:
10145:
10141:
10117:
10114:
10110:
10104:
10101:
10097:
10093:
10072:
10067:
10062:
10056:
10050:
10046:
10042:
10038:
10015:
10012:
10009:
10004:
10001:
9998:
9995:
9973:
9949:
9927:
9905:
9902:
9897:
9894:
9890:
9869:
9864:
9859:
9853:
9849:
9846:
9842:
9821:
9818:
9806:
9803:
9799:
9795:
9789:
9783:
9779:
9773:
9770:
9766:
9762:
9758:
9755:
9750:
9747:
9743:
9737:
9734:
9731:
9727:
9723:
9720:
9714:
9710:
9707:
9703:
9699:
9696:
9692:
9686:
9663:
9659:
9655:
9651:
9630:
9627:
9623:
9620:
9615:
9611:
9591:
9587:
9583:
9577:
9571:
9567:
9561:
9557:
9551:
9547:
9543:
9538:
9535:
9530:
9527:
9523:
9517:
9513:
9507:
9504:
9500:
9496:
9492:
9489:
9484:
9481:
9477:
9473:
9468:
9465:
9462:
9459:
9456:
9453:
9450:
9446:
9442:
9439:
9434:
9431:
9425:
9418:
9414:
9411:
9407:
9403:
9400:
9395:
9391:
9386:
9383:
9380:
9374:
9370:
9367:
9363:
9359:
9356:
9351:
9347:
9341:
9334:
9331:
9326:
9322:
9299:
9296:
9292:
9289:
9284:
9278:
9274:
9268:
9265:
9261:
9257:
9253:
9250:
9245:
9242:
9238:
9232:
9229:
9226:
9222:
9218:
9215:
9209:
9205:
9202:
9198:
9194:
9191:
9187:
9181:
9157:
9154:
9150:
9146:
9125:
9122:
9118:
9115:
9110:
9106:
9086:
9064:
9060:
9054:
9051:
9046:
9043:
9038:
9035:
9031:
9028:
9023:
9016:
9013:
9009:
9003:
9000:
8996:
8992:
8985:
8982:
8978:
8975:
8972:
8968:
8965:
8961:
8958:
8954:
8950:
8946:
8940:
8937:
8932:
8929:
8924:
8921:
8915:
8909:
8906:
8902:
8898:
8895:
8890:
8887:
8884:
8881:
8877:
8874:
8870:
8843:
8840:
8837:
8834:
8830:
8827:
8823:
8797:
8772:
8750:
8745:
8742:
8739:
8736:
8731:
8728:
8724:
8718:
8714:
8708:
8705:
8702:
8699:
8694:
8691:
8687:
8681:
8678:
8673:
8670:
8666:
8660:
8655:
8652:
8649:
8646:
8643:
8639:
8633:
8628:
8625:
8622:
8618:
8614:
8609:
8606:
8601:
8598:
8594:
8573:
8569:
8566:
8563:
8559:
8553:
8547:
8543:
8537:
8534:
8529:
8526:
8522:
8516:
8512:
8506:
8501:
8498:
8495:
8492:
8489:
8485:
8479:
8474:
8471:
8468:
8465:
8462:
8458:
8454:
8451:
8446:
8443:
8437:
8416:
8412:
8407:
8403:
8400:
8397:
8392:
8389:
8383:
8360:
8355:
8352:
8349:
8345:
8324:
8302:
8299:
8296:
8292:
8257:
8252:
8248:
8245:
8241:
8235:
8229:
8226:
8223:
8218:
8214:
8208:
8205:
8202:
8199:
8194:
8191:
8187:
8181:
8178:
8175:
8170:
8166:
8160:
8157:
8154:
8151:
8146:
8143:
8139:
8133:
8130:
8127:
8124:
8121:
8116:
8112:
8106:
8101:
8098:
8095:
8092:
8089:
8085:
8079:
8074:
8071:
8068:
8065:
8062:
8059:
8056:
8052:
8048:
8045:
8041:
8036:
8013:
8009:
8004:
7969:
7965:
7942:
7938:
7928:The subspaces
7915:
7911:
7906:
7902:
7899:
7895:
7890:
7883:
7879:
7873:
7870:
7867:
7864:
7861:
7856:
7852:
7848:
7842:
7838:
7834:
7830:
7823:
7820:
7817:
7814:
7810:
7807:
7804:
7800:
7795:
7791:
7786:
7783:
7777:
7774:
7771:
7768:
7764:
7761:
7758:
7754:
7749:
7745:
7739:
7732:
7728:
7722:
7719:
7716:
7713:
7710:
7705:
7701:
7697:
7691:
7687:
7683:
7678:
7674:
7670:
7667:
7663:
7660:
7657:
7653:
7649:
7620:
7616:
7611:
7607:
7604:
7600:
7595:
7588:
7584:
7578:
7575:
7572:
7569:
7566:
7561:
7557:
7553:
7547:
7543:
7539:
7535:
7528:
7524:
7521:
7518:
7514:
7511:
7506:
7502:
7497:
7494:
7488:
7484:
7481:
7478:
7474:
7471:
7466:
7462:
7456:
7450:
7445:
7440:
7437:
7434:
7431:
7428:
7423:
7419:
7415:
7408:
7404:
7400:
7395:
7391:
7388:
7385:
7381:
7378:
7374:
7350:
7346:
7343:
7340:
7337:
7333:
7327:
7324:
7321:
7317:
7313:
7310:
7305:
7301:
7270:
7266:
7262:
7256:
7252:
7247:
7241:
7237:
7232:
7228:
7225:
7221:
7197:
7193:
7188:
7164:
7159:
7154:
7151:
7148:
7143:
7139:
7116:
7113:
7110:
7107:
7104:
7100:
7077:
7074:
7071:
7067:
7031:
7011:
6999:
6992:
6980:
6973:
6969:
6965:
6962:
6959:
6952:
6948:
6942:
6939:
6936:
6932:
6927:
6919:
6915:
6907:
6903:
6898:
6892:
6888:
6884:
6877:
6873:
6869:
6866:
6863:
6859:
6856:
6849:
6845:
6839:
6836:
6833:
6829:
6824:
6801:
6796:
6792:
6788:
6783:
6779:
6775:
6771:
6748:
6745:
6742:
6738:
6717:
6695:
6692:
6689:
6686:
6683:
6679:
6656:
6653:
6650:
6647:
6644:
6640:
6611:
6607:
6602:
6589:
6583:
6562:
6538:
6534:
6530:
6527:
6522:
6518:
6512:
6508:
6503:
6480:
6476:
6455:
6452:
6439:
6419:
6390:
6386:
6363:
6359:
6338:
6316:
6312:
6278:
6274:
6251:
6248:
6242:
6239:
6236:
6233:
6230:
6227:
6220:
6216:
6211:
6205:
6201:
6195:
6192:
6189:
6186:
6179:
6175:
6170:
6164:
6158:
6155:
6152:
6145:
6140:
6134:
6126:
6121:
6114:
6111:
6105:
6100:
6093:
6089:
6081:
6075:
6069:
6066:
6063:
6056:
6051:
6045:
6040:
6030:
6025:
6018:
6015:
6009:
6005:
5999:
5994:
5989:
5986:
5982:
5977:
5952:
5949:
5946:
5940:
5933:
5929:
5925:
5922:
5918:
5913:
5907:
5904:
5901:
5894:
5889:
5883:
5860:
5833:
5829:
5806:
5801:
5779:
5776:
5773:
5770:
5767:
5763:
5757:
5754:
5751:
5747:
5724:
5720:
5716:
5712:
5691:
5663:
5659:
5655:
5652:
5647:
5643:
5637:
5633:
5628:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5582:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5553:
5547:
5542:
5518:
5514:
5497:
5494:
5482:
5476:
5472:
5468:
5465:
5460:
5457:
5454:
5450:
5444:
5441:
5438:
5434:
5428:
5420:
5416:
5412:
5409:
5406:
5399:
5396:
5393:
5389:
5384:
5378:
5375:
5372:
5365:
5362:
5359:
5355:
5350:
5346:
5343:
5338:
5335:
5332:
5329:
5326:
5319:
5316:
5313:
5309:
5304:
5296:
5293:
5290:
5286:
5282:
5279:
5276:
5273:
5270:
5263:
5260:
5257:
5253:
5247:
5243:
5238:
5230:
5226:
5222:
5219:
5214:
5211:
5208:
5204:
5200:
5195:
5192:
5189:
5185:
5180:
5176:
5173:
5167:
5164:
5161:
5158:
5155:
5152:
5145:
5141:
5136:
5130:
5107:
5101:
5097:
5093:
5090:
5085:
5081:
5075:
5071:
5065:
5057:
5053:
5049:
5046:
5043:
5036:
5032:
5026:
5023:
5020:
5016:
5011:
5007:
5004:
4999:
4996:
4993:
4986:
4982:
4977:
4969:
4965:
4961:
4958:
4955:
4948:
4944:
4939:
4931:
4928:
4925:
4921:
4917:
4914:
4909:
4905:
4901:
4896:
4892:
4887:
4883:
4880:
4874:
4871:
4868:
4865:
4858:
4854:
4849:
4843:
4818:
4812:
4809:
4806:
4803:
4800:
4797:
4790:
4786:
4781:
4775:
4771:
4765:
4762:
4759:
4756:
4749:
4745:
4740:
4734:
4728:
4725:
4722:
4715:
4710:
4704:
4696:
4692:
4687:
4683:
4680:
4676:
4671:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4585:
4582:
4578:
4555:
4552:
4548:
4525:
4522:
4518:
4514:
4511:
4508:
4505:
4485:
4459:
4453:
4450:
4447:
4444:
4437:
4433:
4428:
4422:
4418:
4412:
4408:
4402:
4398:
4392:
4386:
4383:
4380:
4373:
4368:
4362:
4354:
4350:
4346:
4343:
4340:
4333:
4329:
4323:
4319:
4314:
4308:
4305:
4302:
4295:
4291:
4286:
4278:
4274:
4270:
4267:
4264:
4257:
4253:
4248:
4240:
4236:
4232:
4227:
4223:
4219:
4214:
4210:
4206:
4201:
4197:
4192:
4188:
4185:
4181:
4176:
4148:
4142:
4139:
4136:
4133:
4126:
4122:
4117:
4111:
4105:
4102:
4099:
4092:
4087:
4081:
4073:
4069:
4065:
4062:
4059:
4052:
4048:
4042:
4038:
4033:
4025:
4021:
4016:
4012:
4009:
4004:
4001:
3998:
3995:
3988:
3984:
3978:
3974:
3969:
3964:
3941:
3938:
3935:
3928:
3923:
3917:
3896:
3890:
3887:
3884:
3881:
3874:
3870:
3865:
3859:
3838:
3812:
3806:
3803:
3800:
3797:
3790:
3786:
3780:
3776:
3771:
3765:
3732:
3726:
3723:
3720:
3717:
3710:
3706:
3700:
3696:
3691:
3685:
3662:
3656:
3653:
3650:
3647:
3640:
3636:
3630:
3626:
3621:
3615:
3611:
3605:
3601:
3595:
3587:
3583:
3578:
3574:
3571:
3565:
3562:
3559:
3556:
3549:
3545:
3540:
3534:
3513:
3507:
3504:
3501:
3498:
3491:
3487:
3482:
3476:
3466:: express the
3447:
3441:
3438:
3435:
3432:
3425:
3421:
3416:
3410:
3406:
3400:
3396:
3390:
3384:
3381:
3378:
3371:
3367:
3362:
3354:
3350:
3346:
3343:
3340:
3333:
3329:
3324:
3318:
3315:
3312:
3306:
3303:
3298:
3294:
3289:
3284:
3280:
3275:
3271:
3268:
3264:
3259:
3235:
3229:
3226:
3223:
3216:
3212:
3207:
3202:
3180:
3174:
3171:
3168:
3165:
3158:
3154:
3149:
3144:
3139:
3133:
3130:
3127:
3120:
3116:
3111:
3106:
3083:
3080:
3077:
3070:
3065:
3059:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
2998:
2995:
2982:
2977:
2972:
2968:
2964:
2957:
2953:
2948:
2944:
2939:
2935:
2931:
2926:
2922:
2918:
2913:
2908:
2904:
2900:
2893:
2889:
2884:
2880:
2875:
2871:
2867:
2860:
2856:
2850:
2846:
2842:
2839:
2836:
2815:
2811:
2806:
2799:
2795:
2790:
2787:
2783:
2778:
2771:
2767:
2762:
2759:
2755:
2750:
2743:
2739:
2734:
2731:
2727:
2722:
2715:
2711:
2706:
2703:
2699:
2694:
2671:
2668:
2662:
2658:
2652:
2648:
2645:
2639:
2635:
2629:
2625:
2619:
2615:
2610:
2607:
2601:
2596:
2592:
2586:
2576:
2573:
2570:
2564:
2560:
2554:
2550:
2547:
2541:
2537:
2531:
2527:
2521:
2517:
2512:
2509:
2503:
2498:
2494:
2488:
2478:
2475:
2472:
2466:
2462:
2456:
2452:
2449:
2443:
2439:
2433:
2429:
2423:
2419:
2414:
2411:
2405:
2400:
2396:
2390:
2380:
2377:
2374:
2368:
2364:
2358:
2354:
2351:
2345:
2341:
2335:
2331:
2325:
2321:
2316:
2313:
2307:
2302:
2298:
2292:
2270:
2264:
2259:
2255:
2249:
2244:
2240:
2235:
2226:
2221:
2216:
2213:
2208:
2200:
2196:
2190:
2185:
2181:
2175:
2170:
2166:
2161:
2152:
2147:
2142:
2139:
2134:
2126:
2122:
2118:
2114:
2092:
2088:
2084:
2079:
2075:
2072:
2068:
2063:
2057:
2052:
2049:
2045:
2040:
2034:
2029:
2026:
2022:
2017:
2012:
2003:
1998:
1994:
1989:
1986:
1982:
1977:
1953:
1950:
1944:
1940:
1935:
1932:
1929:
1926:
1923:
1917:
1913:
1908:
1905:
1902:
1899:
1896:
1893:
1888:
1884:
1880:
1876:
1855:
1852:
1847:
1842:
1838:
1830:
1826:
1822:
1818:
1812:
1809:
1806:
1802:
1779:
1775:
1751:
1747:
1725:
1719:
1714:
1710:
1704:
1681:
1677:
1656:
1650:
1645:
1641:
1635:
1614:
1608:
1603:
1599:
1593:
1589:
1586:
1580:
1575:
1571:
1565:
1561:
1558:
1552:
1547:
1543:
1537:
1532:
1528:
1522:
1500:
1494:
1489:
1485:
1479:
1474:
1470:
1465:
1459:
1455:
1447:
1443:
1439:
1435:
1429:
1426:
1423:
1419:
1415:
1411:
1408:
1405:
1384:
1380:
1375:
1351:
1347:
1343:
1338:
1334:
1329:
1326:
1323:
1320:
1308:
1305:
1271:
1267:
1263:
1260:
1257:
1250:
1247:
1244:
1240:
1235:
1229:
1226:
1223:
1220:
1217:
1210:
1207:
1204:
1200:
1195:
1187:
1184:
1181:
1177:
1173:
1169:
1166:
1163:
1159:
1152:
1149:
1146:
1142:
1136:
1133:
1130:
1126:
1121:
1117:
1114:
1111:
1108:
1103:
1100:
1097:
1090:
1086:
1081:
1073:
1069:
1065:
1062:
1059:
1052:
1048:
1042:
1038:
1033:
1027:
1024:
1021:
1014:
1010:
1005:
997:
993:
989:
986:
983:
976:
972:
966:
962:
957:
951:
948:
945:
938:
934:
929:
921:
917:
913:
910:
907:
900:
896:
891:
885:
880:
877:
872:
869:
866:
862:
858:
855:
852:
849:
844:
840:
835:
831:
824:
820:
816:
813:
808:
804:
798:
794:
789:
764:
760:
756:
753:
748:
744:
738:
734:
729:
701:
695:
691:
687:
682:
679:
676:
672:
668:
665:
662:
659:
654:
650:
646:
641:
637:
631:
623:
619:
615:
612:
607:
603:
597:
593:
588:
582:
577:
574:
571:
567:
563:
560:
557:
553:
532:
527:
523:
519:
514:
511:
508:
504:
500:
497:
494:
491:
486:
482:
478:
473:
469:
464:
441:
437:
416:
413:
409:
386:
382:
377:
373:
370:
367:
363:
343:
340:
338:
335:
314:time evolution
284:density matrix
210:
209:
168:
166:
159:
153:
150:
119:
118:
33:
31:
24:
17:
9:
6:
4:
3:
2:
17395:
17384:
17381:
17379:
17376:
17375:
17373:
17360:
17356:
17352:
17348:
17344:
17338:
17334:
17330:
17325:
17320:
17316:
17309:
17301:
17297:
17293:
17289:
17285:
17281:
17277:
17273:
17268:
17263:
17260:(4): 040502.
17259:
17255:
17248:
17240:
17236:
17232:
17228:
17224:
17220:
17216:
17212:
17207:
17202:
17198:
17194:
17187:
17181:
17177:
17173:
17169:
17165:
17161:
17157:
17153:
17149:
17144:
17139:
17135:
17131:
17124:
17116:
17112:
17108:
17104:
17100:
17096:
17092:
17088:
17083:
17078:
17074:
17070:
17066:
17065:Vidal, Guifré
17060:
17058:
17053:
17046:
17042:
17033:
17029:
17011:
17007:
17003:
17000:
16997:
16989:
16986:
16983:
16978:
16975:
16972:
16964:
16961:
16958:
16950:
16946:
16942:
16939:
16932:
16928:
16924:
16921:
16918:
16910:
16907:
16904:
16899:
16896:
16893:
16880:
16876:
16872:
16869:
16861:
16858:
16855:
16850:
16847:
16844:
16836:
16833:
16830:
16823:
16816:
16802:
16784:
16780:
16776:
16773:
16766:
16762:
16758:
16755:
16752:
16746:
16738:
16734:
16730:
16727:
16721:
16717:
16711:
16708:
16705:
16700:
16685:
16681:
16669:
16665:
16647:
16642:
16627:
16616:
16612:
16601:
16598:
16589:
16576:
16572:
16568:
16565:
16559:
16555:
16549:
16544:
16540:
16537:
16534:
16528:
16525:
16522:
16517:
16512:
16508:
16502:
16499:
16479:
16476:
16471:
16467:
16463:
16458:
16454:
16450:
16445:
16441:
16437:
16434:
16425:
16410:
16405:
16401:
16395:
16390:
16384:
16371:
16361:
16357:
16352:
16340:
16328:
16324:
16318:
16315:
16312:
16308:
16303:
16292:
16288:
16282:
16279:
16274:
16270:
16261:
16256:
16243:
16233:
16229:
16222:
16219:
16210:
16205:
16193:
16189:
16183:
16180:
16177:
16173:
16168:
16157:
16153:
16147:
16144:
16139:
16135:
16126:
16121:
16117:
16107:
16089:
16085:
16078:
16068:
16064:
16058:
16055:
16052:
16048:
16037:
16025:
16021:
16014:
16004:
16000:
15994:
15991:
15988:
15984:
15970:
15965:
15962:
15957:
15953:
15944:
15939:
15927:
15923:
15917:
15914:
15911:
15907:
15902:
15888:
15875:
15870:
15866:
15862:
15857:
15853:
15849:
15844:
15831:
15821:
15817:
15812:
15803:
15791:
15787:
15781:
15778:
15775:
15771:
15766:
15757:
15750:
15746:
15742:
15737:
15733:
15724:
15719:
15706:
15696:
15692:
15687:
15678:
15666:
15662:
15656:
15653:
15650:
15646:
15641:
15630:
15626:
15620:
15617:
15612:
15608:
15599:
15596:
15593:
15588:
15584:
15554:
15551:
15548:
15545:
15535:
15532:
15529:
15525:
15495:
15492:
15476:
15460:
15456:
15452:
15449:
15445:
15439:
15424:
15414:
15409:
15403:
15388:
15383:
15376:
15373:
15367:
15363:
15352:
15349:
15323:
15320:
15317:
15314:
15308:
15302:
15299:
15274:
15265:
15251:
15242:
15223:
15219:
15215:
15212:
15201:
15198:
15195:
15191:
15187:
15184:
15178:
15175:
15172:
15167:
15152:
15148:
15138:
15121:
15118:
15115:
15111:
15107:
15104:
15098:
15095:
15092:
15087:
15073:
15069:
15065:
15055:
15043:
15040:
15037:
15034:
15003:
14997:
14994:
14978:
14975:
14972:
14968:
14962:
14953:
14947:
14943:
14926:
14923:
14920:
14916:
14912:
14909:
14904:
14895:
14891:
14875:
14860:
14847:
14839:
14835:
14826:
14819:
14815:
14811:
14808:
14800:
14795:
14780:
14776:
14766:
14754:
14751:
14748:
14743:
14739:
14718:
14715:
14707:
14703:
14687:
14683:
14650:
14647:
14644:
14641:
14631:
14627:
14603:
14600:
14590:
14586:
14566:
14556:
14551:
14545:
14539:
14532:
14527:
14522:
14516:
14511:
14495:
14491:
14486:
14481:
14467:
14463:
14427:
14424:
14421:
14418:
14408:
14404:
14380:
14377:
14367:
14363:
14343:
14333:
14328:
14322:
14314:
14309:
14302:
14299:
14293:
14288:
14272:
14268:
14264:
14261:
14257:
14252:
14243:
14239:
14201:
14197:
14183:
14179:
14173:
14164:
14160:
14146:
14142:
14138:
14135:
14130:
14123:
14069:
14017:
14004:
13996:
13992:
13983:
13976:
13972:
13968:
13965:
13957:
13952:
13948:
13922:
13918:
13914:
13911:
13903:
13900:
13897:
13892:
13889:
13886:
13878:
13875:
13872:
13856:
13847:
13845:
13835:
13833:
13828:
13813:
13809:
13805:
13800:
13796:
13792:
13789:
13786:
13783:
13780:
13775:
13759:
13756:
13752:
13734:
13731:
13727:
13712:
13709:
13706:
13700:
13694:
13674:
13665:
13640:
13637:
13633:
13600:
13597:
13593:
13577:
13553:
13550:
13546:
13535:
13531:
13522:
13519:
13515:
13501:
13496:
13491:
13488:
13483:
13474:
13471:
13461:
13443:
13423:
13420:
13410:
13391:
13376:
13372:
13368:
13365:
13360:
13357:
13354:
13350:
13344:
13341:
13336:
13333:
13311:
13308:
13303:
13300:
13278:
13275:
13272:
13267:
13217:
13212:Error sources
13188:
13184:
13172:
13167:
13163:
13159:
13156:
13152:
13148:
13139:
13135:
13106:
13102:
13075:
13071:
13038:
13034:
13019:
12998:
12969:
12965:
12936:
12932:
12923:
12910:
12895:
12891:
12879:
12870:
12864:
12860:
12845:
12841:
12829:
12820:
12814:
12810:
12795:
12787:
12778:
12775:
12771:
12737:
12734:
12730:
12697:
12686:
12666:
12663:
12659:
12649:
12646:
12637:
12633:
12600:
12571:
12568:
12564:
12531:
12527:
12512:
12491:
12487:
12448:
12447:
12445:
12428:
12424:
12390:
12374:
12370:
12343:
12339:
12306:
12302:
12286:
12241:
12234:
12228:
12225:
12220:
12217:
12213:
12185:
12178:
12173:
12170:
12167:
12162:
12134:
12127:
12121:
12118:
12113:
12110:
12106:
12085:
12076:
12063:
12054:
12044:
12029:
12024:
12020:
12017:
12014:
12007:
12001:
11997:
11994:
11991:
11986:
11980:
11975:
11972:
11967:
11964:
11960:
11956:
11947:
11944:
11941:
11931:
11913:
11910:
11895:
11892:
11887:
11883:
11857:
11854:
11846:
11820:
11813:
11786:
11783:
11775:
11749:
11742:
11732:
11712:
11705:
11700:
11695:
11689:
11680:
11676:
11671:
11667:
11662:
11635:
11628:
11622:
11619:
11613:
11607:
11598:
11594:
11589:
11584:
11581:
11575:
11566:
11550:
11546:
11541:
11518:
11513:
11510:
11504:
11489:
11475:
11468:. The number
11453:
11450:
11445:
11442:
11420:
11410:
11405:
11402:
11396:
11390:
11386:
11381:
11375:
11370:
11367:
11361:
11354:
11348:
11343:
11339:
11329:
11324:
11320:
11316:
11313:
11309:
11302:
11297:
11292:
11288:
11284:
11281:
11277:
11254:
11251:
11248:
11245:
11241:
11231:
11229:
11225:
11220:
11207:
11201:
11192:
11179:
11171:
11167:
11153:
11148:
11144:
11139:
11133:
11129:
11124:
11120:
11112:
11109:
11106:
11099:
11094:
11071:
11066:
11059:
11056:
11051:
11044:
11041:
11036:
11031:
11018:
11010:
11005:
11002:
10999:
10995:
10980:
10978:
10962:
10959:
10947:
10944:
10936:
10932:
10924:
10917:
10893:
10890:
10878:
10875:
10867:
10863:
10855:
10848:
10835:
10822:
10814:
10807:
10801:
10792:
10788:
10780:
10777:
10774:
10771:
10768:
10763:
10759:
10755:
10750:
10745:
10741:
10732:
10723:
10719:
10716:
10697:
10689:
10682:
10676:
10667:
10663:
10655:
10652:
10649:
10646:
10643:
10638:
10634:
10630:
10625:
10620:
10616:
10607:
10598:
10594:
10591:
10583:
10569:
10566:
10563:
10560:
10555:
10551:
10528:
10524:
10514:
10501:
10493:
10490:
10487:
10484:
10481:
10473:
10469:
10463:
10458:
10455:
10452:
10444:
10436:
10428:
10424:
10418:
10413:
10410:
10407:
10399:
10394:
10390:
10381:
10367:
10353:
10339:
10336:
10333:
10308:
10303:
10297:
10291:
10287:
10251:
10246:
10240:
10234:
10230:
10176:
10174:
10168:
10151:
10149:
10143:
10139:
10115:
10112:
10108:
10102:
10099:
10095:
10091:
10065:
10060:
10054:
10048:
10044:
10036:
10002:
9999:
9996:
9993:
9925:
9903:
9900:
9895:
9892:
9862:
9857:
9851:
9847:
9840:
9831:
9827:
9817:
9804:
9797:
9793:
9781:
9777:
9771:
9764:
9756:
9754:
9748:
9745:
9735:
9732:
9729:
9725:
9721:
9708:
9705:
9697:
9695:
9657:
9621:
9618:
9585:
9581:
9569:
9565:
9559:
9549:
9545:
9536:
9533:
9528:
9525:
9515:
9505:
9498:
9490:
9488:
9482:
9479:
9466:
9463:
9460:
9457:
9454:
9451:
9448:
9444:
9440:
9432:
9429:
9412:
9409:
9401:
9399:
9393:
9381:
9368:
9365:
9357:
9355:
9349:
9332:
9330:
9324:
9320:
9310:
9297:
9290:
9287:
9276:
9272:
9266:
9259:
9251:
9249:
9243:
9240:
9230:
9227:
9224:
9220:
9216:
9203:
9200:
9192:
9190:
9152:
9116:
9113:
9084:
9075:
9062:
9052:
9049:
9044:
9041:
9029:
9026:
9014:
9011:
9007:
9001:
8998:
8994:
8990:
8983:
8980:
8976:
8973:
8970:
8966:
8963:
8959:
8956:
8952:
8948:
8938:
8935:
8922:
8919:
8907:
8904:
8900:
8896:
8893:
8885:
8882:
8875:
8873:
8868:
8859:
8856:is therefore
8838:
8835:
8828:
8826:
8821:
8812:
8786:
8770:
8761:
8748:
8743:
8737:
8729:
8726:
8716:
8712:
8706:
8700:
8692:
8689:
8679:
8676:
8671:
8668:
8664:
8658:
8653:
8650:
8647:
8644:
8641:
8631:
8626:
8623:
8620:
8612:
8607:
8604:
8599:
8596:
8567:
8564:
8561:
8557:
8545:
8541:
8535:
8532:
8527:
8524:
8514:
8510:
8504:
8499:
8496:
8493:
8490:
8487:
8477:
8472:
8469:
8466:
8463:
8460:
8452:
8444:
8441:
8410:
8401:
8398:
8390:
8387:
8373:We can write
8371:
8358:
8350:
8322:
8297:
8281:
8277:
8273:
8268:
8250:
8246:
8243:
8239:
8224:
8216:
8212:
8206:
8200:
8192:
8189:
8176:
8168:
8164:
8158:
8152:
8144:
8141:
8128:
8125:
8122:
8114:
8110:
8104:
8099:
8096:
8093:
8090:
8087:
8077:
8072:
8069:
8066:
8063:
8060:
8057:
8054:
8046:
8039:
8025:
8007:
7993:
7989:
7985:
7967:
7963:
7940:
7936:
7926:
7913:
7904:
7893:
7881:
7868:
7865:
7862:
7854:
7850:
7840:
7836:
7832:
7818:
7815:
7812:
7808:
7805:
7802:
7793:
7772:
7769:
7766:
7762:
7759:
7756:
7747:
7730:
7717:
7714:
7711:
7703:
7699:
7689:
7685:
7681:
7672:
7668:
7665:
7661:
7658:
7655:
7647:
7638:
7636:
7631:
7618:
7609:
7598:
7586:
7573:
7570:
7567:
7559:
7555:
7545:
7541:
7537:
7522:
7519:
7516:
7512:
7504:
7482:
7479:
7476:
7472:
7464:
7448:
7435:
7432:
7429:
7421:
7417:
7406:
7402:
7398:
7389:
7386:
7383:
7379:
7372:
7363:
7344:
7335:
7325:
7322:
7319:
7315:
7311:
7308:
7303:
7299:
7290:
7287:
7282:
7268:
7264:
7260:
7254:
7250:
7245:
7239:
7235:
7230:
7226:
7223:
7219:
7209:
7191:
7176:
7157:
7152:
7149:
7146:
7137:
7111:
7108:
7105:
7072:
7056:
7052:
7048:
7045:
7029:
6997:
6991:
6978:
6971:
6967:
6960:
6950:
6946:
6940:
6937:
6934:
6930:
6917:
6913:
6905:
6901:
6896:
6890:
6886:
6882:
6875:
6871:
6864:
6857:
6855:
6847:
6843:
6837:
6834:
6831:
6827:
6813:
6794:
6790:
6786:
6781:
6777:
6769:
6743:
6708:'s. The only
6690:
6687:
6684:
6651:
6648:
6645:
6629:
6625:
6605:
6588:
6582:
6580:
6576:
6560:
6536:
6532:
6528:
6525:
6520:
6516:
6510:
6506:
6501:
6493:coefficients
6478:
6474:
6465:
6461:
6451:
6437:
6417:
6409:
6404:
6388:
6384:
6361:
6357:
6336:
6314:
6310:
6301:
6300:ferromagnetic
6297:
6292:
6276:
6272:
6262:
6249:
6237:
6234:
6231:
6228:
6218:
6214:
6190:
6187:
6177:
6173:
6153:
6143:
6138:
6132:
6124:
6119:
6112:
6109:
6103:
6098:
6087:
6079:
6064:
6054:
6049:
6043:
6028:
6023:
6016:
6013:
6007:
6003:
5992:
5987:
5966:
5963:
5950:
5947:
5944:
5938:
5931:
5927:
5923:
5920:
5916:
5911:
5902:
5892:
5887:
5881:
5872:
5858:
5849:
5831:
5827:
5804:
5799:
5774:
5771:
5768:
5761:
5755:
5752:
5749:
5745:
5722:
5718:
5714:
5710:
5689:
5681:
5661:
5657:
5653:
5650:
5645:
5641:
5635:
5631:
5626:
5605:
5599:
5596:
5593:
5587:
5584:
5580:
5576:
5573:
5567:
5564:
5561:
5555:
5551:
5545:
5540:
5516:
5512:
5504:, instead of
5503:
5493:
5474:
5470:
5466:
5463:
5458:
5455:
5452:
5448:
5442:
5439:
5436:
5432:
5418:
5414:
5407:
5397:
5394:
5391:
5387:
5376:
5373:
5370:
5363:
5360:
5357:
5353:
5348:
5344:
5341:
5333:
5330:
5327:
5317:
5314:
5311:
5307:
5302:
5294:
5291:
5288:
5284:
5277:
5274:
5271:
5261:
5258:
5255:
5251:
5245:
5241:
5228:
5224:
5220:
5217:
5212:
5209:
5206:
5202:
5198:
5193:
5190:
5187:
5183:
5178:
5174:
5162:
5159:
5156:
5153:
5143:
5139:
5119:
5099:
5095:
5091:
5088:
5083:
5079:
5073:
5069:
5055:
5051:
5044:
5034:
5030:
5024:
5021:
5018:
5014:
5005:
5002:
4994:
4984:
4980:
4975:
4967:
4963:
4956:
4946:
4942:
4929:
4926:
4923:
4919:
4915:
4912:
4907:
4903:
4899:
4894:
4890:
4885:
4881:
4869:
4866:
4856:
4852:
4832:
4829:
4807:
4804:
4801:
4798:
4788:
4784:
4760:
4757:
4747:
4743:
4723:
4713:
4708:
4702:
4694:
4690:
4685:
4681:
4660:
4658:
4653:
4651:
4632:
4629:
4626:
4623:
4617:
4611:
4608:
4583:
4580:
4576:
4553:
4550:
4546:
4523:
4520:
4512:
4509:
4506:
4475:
4470:
4448:
4445:
4435:
4431:
4410:
4406:
4400:
4396:
4381:
4371:
4366:
4360:
4352:
4348:
4341:
4331:
4327:
4321:
4317:
4303:
4293:
4289:
4284:
4276:
4272:
4265:
4255:
4251:
4238:
4234:
4230:
4225:
4221:
4217:
4212:
4208:
4204:
4199:
4195:
4190:
4186:
4165:
4164:
4159:
4137:
4134:
4124:
4120:
4100:
4090:
4085:
4079:
4071:
4067:
4060:
4050:
4046:
4040:
4036:
4023:
4019:
4014:
4010:
3999:
3996:
3986:
3982:
3976:
3972:
3967:
3936:
3926:
3921:
3915:
3885:
3882:
3872:
3868:
3836:
3828:
3827:
3801:
3798:
3788:
3784:
3778:
3774:
3769:
3754:
3753:
3748:
3746:
3721:
3718:
3708:
3704:
3698:
3694:
3689:
3673:
3651:
3648:
3638:
3634:
3628:
3624:
3619:
3603:
3599:
3585:
3581:
3576:
3572:
3560:
3557:
3547:
3543:
3502:
3499:
3489:
3485:
3465:
3464:
3458:
3436:
3433:
3423:
3419:
3398:
3394:
3379:
3369:
3365:
3360:
3352:
3348:
3341:
3331:
3327:
3316:
3313:
3310:
3304:
3301:
3296:
3292:
3287:
3282:
3278:
3269:
3248:
3233:
3224:
3214:
3210:
3200:
3178:
3169:
3166:
3156:
3152:
3142:
3137:
3128:
3118:
3114:
3104:
3078:
3068:
3063:
3057:
3033:
3030:
3024:
3018:
3006:
3004:
2994:
2980:
2975:
2970:
2966:
2962:
2955:
2951:
2946:
2942:
2937:
2933:
2929:
2924:
2920:
2916:
2911:
2906:
2902:
2898:
2891:
2887:
2882:
2878:
2873:
2869:
2865:
2858:
2854:
2848:
2844:
2840:
2834:
2809:
2797:
2793:
2788:
2781:
2769:
2765:
2760:
2753:
2741:
2737:
2732:
2725:
2713:
2709:
2704:
2682:
2660:
2656:
2646:
2637:
2633:
2617:
2613:
2608:
2599:
2594:
2590:
2574:
2562:
2558:
2548:
2539:
2535:
2519:
2515:
2510:
2501:
2496:
2492:
2476:
2464:
2460:
2450:
2441:
2437:
2421:
2417:
2412:
2403:
2398:
2394:
2378:
2366:
2362:
2352:
2343:
2339:
2323:
2319:
2314:
2305:
2300:
2296:
2268:
2262:
2257:
2253:
2247:
2242:
2238:
2233:
2224:
2219:
2214:
2211:
2206:
2198:
2194:
2188:
2183:
2179:
2173:
2168:
2164:
2159:
2150:
2145:
2140:
2137:
2132:
2124:
2120:
2112:
2090:
2082:
2073:
2066:
2055:
2050:
2043:
2032:
2027:
2020:
2010:
2001:
1996:
1992:
1987:
1965:
1942:
1938:
1930:
1927:
1924:
1915:
1911:
1903:
1900:
1891:
1886:
1878:
1874:
1853:
1850:
1845:
1840:
1836:
1828:
1820:
1816:
1810:
1807:
1804:
1777:
1773:
1749:
1745:
1717:
1712:
1679:
1675:
1648:
1643:
1606:
1601:
1587:
1578:
1573:
1559:
1550:
1545:
1535:
1530:
1498:
1492:
1487:
1477:
1472:
1463:
1457:
1453:
1445:
1437:
1433:
1427:
1424:
1421:
1413:
1409:
1403:
1349:
1345:
1341:
1336:
1332:
1327:
1304:
1302:
1298:
1293:
1291:
1286:
1269:
1265:
1258:
1248:
1245:
1242:
1238:
1224:
1221:
1218:
1208:
1205:
1202:
1198:
1193:
1185:
1182:
1179:
1175:
1167:
1164:
1161:
1150:
1147:
1144:
1140:
1134:
1131:
1128:
1124:
1115:
1112:
1109:
1106:
1098:
1088:
1084:
1079:
1071:
1067:
1060:
1050:
1046:
1040:
1036:
1022:
1012:
1008:
1003:
995:
991:
984:
974:
970:
964:
960:
946:
936:
932:
927:
919:
915:
908:
898:
894:
883:
878:
875:
870:
867:
864:
860:
856:
853:
850:
847:
842:
838:
829:
822:
818:
814:
811:
806:
802:
796:
792:
787:
762:
758:
754:
751:
746:
742:
736:
732:
727:
717:
715:
693:
689:
685:
680:
677:
674:
670:
666:
663:
660:
657:
652:
648:
644:
639:
635:
621:
617:
613:
610:
605:
601:
595:
591:
586:
580:
575:
572:
569:
561:
525:
521:
517:
512:
509:
506:
502:
498:
495:
492:
489:
484:
480:
476:
471:
467:
439:
435:
384:
380:
375:
371:
352:
349:
334:
332:
328:
326:
323:
319:
315:
310:
308:
304:
300:
299:ground states
295:
291:
290:we selected.
289:
285:
281:
277:
273:
269:
265:
261:
259:
253:
251:
247:
243:
238:
233:
228:
226:
222:
217:
206:
203:
195:
185:
181:
175:
174:
169:This article
167:
158:
157:
149:
147:
143:
142:Hilbert space
138:
134:
130:
126:
115:
112:
104:
93:
90:
86:
83:
79:
76:
72:
69:
65:
62: –
61:
57:
56:Find sources:
50:
46:
40:
39:
34:This article
32:
28:
23:
22:
16:
17314:
17308:
17257:
17253:
17247:
17196:
17192:
17186:
17133:
17129:
17123:
17072:
17068:
17043:
17039:
17030:
16808:
16643:
16426:
16108:
15889:
15477:
15266:
15243:
14876:
14056:. The state
13848:
13843:
13841:
13831:
13829:
13666:
13578:
13444:
13392:
13223:
13215:
12757:
12391:
12287:
12077:
11914:
11911:
11733:
11567:
11495:
11232:
11221:
11193:
10986:
10977:Hale Trotter
10836:
10584:
10515:
10382:
10359:
9829:
9823:
9311:
9076:
8858:diagonalized
8810:
8784:
8762:
8372:
8279:
8275:
8271:
8269:
8026:
7991:
7987:
7983:
7927:
7639:
7634:
7632:
7364:
7288:
7283:
7210:
7177:
7054:
7050:
7046:
7001:
6995:
6814:
6627:
6591:
6586:
6459:
6457:
6407:
6405:
6293:
6263:
5967:
5964:
5873:
5847:
5679:
5501:
5499:
5120:
4833:
4830:
4661:
4656:
4654:
4649:
4476:. The last
4473:
4471:
4162:
4161:
4160:
3825:
3824:
3751:
3750:
3749:
3675:The vectors
3674:
3462:
3461:
3459:
3249:
3007:
3002:
3000:
2683:
1966:
1310:
1294:
1287:
718:
713:
347:
345:
329:
311:
296:
292:
276:Hamiltonians
271:
264:Guifré Vidal
262:
254:
229:
213:
198:
192:October 2010
189:
170:
152:Introduction
146:entanglement
128:
124:
122:
107:
98:
88:
81:
74:
67:
55:
43:Please help
38:verification
35:
15:
13832:independent
7022:'s and the
4598:bond, i.e.
288:eigenvalues
17372:Categories
17049:References
16492:, we get:
11604:even
10673:even
10604:even
8274:to qubits
7049:on qubits
6575:polynomial
3745:normalized
303:isentropic
101:March 2011
71:newspapers
17359:118378501
17351:0075-8450
17284:0031-9007
17099:0031-9007
17008:ϵ
17001:−
16987:−
16969:∏
16965:−
16947:ϵ
16943:−
16929:ϵ
16922:−
16908:−
16890:∏
16877:ϵ
16873:−
16859:−
16841:∏
16837:−
16817:ϵ
16781:ϵ
16777:−
16763:ϵ
16756:−
16735:ϵ
16731:−
16718:ϵ
16712:−
16691:⟩
16682:ψ
16666:ψ
16661:⟨
16623:→
16613:ϵ
16586:→
16573:ϵ
16569:−
16556:ϵ
16538:−
16529:≤
16523:−
16500:ϵ
16477:−
16451:−
16435:ϵ
16358:α
16353:λ
16325:α
16316:−
16309:α
16289:χ
16271:α
16266:∑
16230:α
16220:λ
16211:⋅
16190:α
16181:−
16174:α
16154:χ
16136:α
16131:∑
16065:α
16056:−
16049:α
16044:Γ
16038:∗
16001:α
15992:−
15985:α
15980:Γ
15949:∑
15924:α
15915:−
15908:α
15818:α
15813:λ
15788:α
15779:−
15772:α
15758:χ
15747:χ
15734:α
15729:∑
15693:α
15688:λ
15663:α
15654:−
15647:α
15627:χ
15609:α
15604:∑
15564:‖
15549:−
15533:−
15526:α
15521:Φ
15516:‖
15493:−
15457:ϵ
15453:−
15410:α
15404:λ
15384:χ
15364:α
15358:∑
15220:ϵ
15216:−
15192:ϵ
15188:−
15179:−
15158:⟩
15149:ψ
15139:ψ
15135:⟨
15112:ϵ
15108:−
15099:−
15078:⟩
15066:ψ
15056:ψ
15052:⟨
15044:−
15035:ϵ
15015:⟩
15009:⊥
14995:ψ
14969:ϵ
14960:⟩
14944:ψ
14917:ϵ
14913:−
14902:⟩
14892:ψ
14840:α
14836:λ
14827:χ
14816:χ
14809:α
14805:∑
14786:⟩
14777:ψ
14767:ψ
14763:⟨
14755:−
14740:ϵ
14713:⟩
14704:ψ
14693:⊥
14684:ψ
14680:⟨
14660:⟩
14628:α
14623:Φ
14613:⟩
14587:α
14582:Φ
14552:α
14546:λ
14540:χ
14528:χ
14512:α
14505:∑
14492:ϵ
14479:⟩
14473:⊥
14464:ψ
14437:⟩
14405:α
14400:Φ
14390:⟩
14364:α
14359:Φ
14329:α
14323:λ
14310:χ
14289:α
14282:∑
14269:ϵ
14265:−
14250:⟩
14240:ψ
14213:⟩
14207:⊥
14198:ψ
14180:ϵ
14171:⟩
14161:ψ
14143:ϵ
14139:−
14128:⟩
14124:ψ
14074:⟩
14070:ψ
13997:α
13993:λ
13984:χ
13973:χ
13966:α
13962:∑
13949:ϵ
13919:ϵ
13915:−
13901:−
13883:∏
13879:−
13857:ϵ
13810:ϵ
13797:ϵ
13787:−
13766:⟩
13753:ψ
13740:~
13728:ψ
13721:⟨
13713:−
13695:ϵ
13652:⟩
13646:⊥
13634:ψ
13607:⟩
13594:ψ
13565:⟩
13559:⊥
13547:ψ
13536:ϵ
13529:⟩
13516:ψ
13497:ϵ
13492:−
13481:⟩
13465:~
13462:ψ
13430:⟩
13414:~
13411:ψ
13373:δ
13351:δ
13345:δ
13334:ϵ
13312:δ
13268:δ
13195:⟩
13185:ψ
13157:−
13146:⟩
13136:ψ
13082:⟩
13072:ψ
13045:⟩
13035:ψ
13002:~
12976:⟩
12966:ψ
12905:‖
12902:⟩
12892:ψ
12880:τ
12874:~
12865:−
12857:‖
12852:⟩
12842:ψ
12830:τ
12824:~
12815:−
12802:∞
12799:→
12796:τ
12785:⟩
12772:ψ
12744:⟩
12731:ψ
12701:~
12673:⟩
12660:ψ
12644:⟩
12634:ψ
12604:~
12578:⟩
12565:ψ
12538:⟩
12528:ψ
12498:⟩
12488:ψ
12425:χ
12350:⟩
12340:ψ
12313:⟩
12303:ψ
12226:δ
12218:−
12174:δ
12168:−
12119:δ
12111:−
12086:δ
12061:⟩
12048:~
12045:ψ
12021:δ
12015:−
11998:δ
11992:−
11973:δ
11965:−
11954:⟩
11948:δ
11935:~
11932:ψ
11888:≠
11701:δ
11686:odd
11681:∏
11668:δ
11620:δ
11599:∏
11582:δ
11547:δ
11511:δ
11454:δ
11403:δ
11387:δ
11368:δ
11349:δ
11330:δ
11314:−
11282:−
11246:−
11205:→
11202:δ
11168:δ
11145:δ
11130:δ
11100:δ
11025:∞
11022:→
10798:odd
10793:∑
10729:odd
10724:∑
10720:≡
10668:∑
10599:∑
10595:≡
10449:∑
10404:∑
10304:χ
10298:⋅
10247:χ
10241:⋅
10200:⟩
10169:β
10165:Φ
10144:β
10140:λ
10100:γ
10096:γ
10092:ρ
10061:χ
10055:⋅
10000:α
9994:γ
9986:for each
9926:β
9896:γ
9893:α
9889:Θ
9858:χ
9852:⋅
9802:⟩
9794:α
9782:α
9778:λ
9749:β
9746:α
9742:Γ
9736:α
9726:∑
9719:⟩
9691:Φ
9650:Γ
9626:⟩
9622:α
9590:⟩
9582:α
9570:α
9566:λ
9550:γ
9546:λ
9529:γ
9526:α
9522:Θ
9516:∗
9483:γ
9480:β
9476:Γ
9467:γ
9461:α
9445:∑
9438:⟩
9430:ψ
9394:β
9390:Φ
9385:⟨
9379:⟩
9350:β
9346:Φ
9325:β
9321:λ
9295:⟩
9291:γ
9277:γ
9273:λ
9244:γ
9241:β
9237:Γ
9231:γ
9221:∑
9214:⟩
9186:Φ
9145:Γ
9121:⟩
9117:γ
9085:λ
9050:γ
9037:⟨
9034:⟩
9030:γ
8999:γ
8995:γ
8991:ρ
8981:γ
8974:γ
8953:∑
8936:ψ
8931:⟨
8928:⟩
8920:ψ
8869:ρ
8822:ρ
8796:Γ
8771:λ
8730:γ
8727:β
8723:Γ
8717:β
8713:λ
8693:β
8690:α
8686:Γ
8638:∑
8632:χ
8621:β
8617:∑
8600:γ
8597:α
8593:Θ
8572:⟩
8568:γ
8558:α
8546:γ
8542:λ
8528:γ
8525:α
8521:Θ
8515:α
8511:λ
8484:∑
8478:χ
8467:γ
8461:α
8457:∑
8450:⟩
8442:ψ
8415:⟩
8411:ψ
8396:⟩
8388:ψ
8344:Γ
8323:λ
8291:Γ
8256:⟩
8251:γ
8240:α
8217:γ
8213:λ
8193:γ
8190:β
8186:Γ
8169:β
8165:λ
8145:β
8142:α
8138:Γ
8126:−
8115:α
8111:λ
8084:∑
8078:χ
8067:γ
8061:β
8055:α
8051:∑
8044:⟩
8040:ψ
8012:⟩
8008:ψ
7905:γ
7901:⟨
7898:⟩
7894:γ
7855:γ
7851:λ
7841:γ
7837:∑
7794:γ
7790:Φ
7785:⟨
7782:⟩
7748:γ
7744:Φ
7704:γ
7700:λ
7690:γ
7686:∑
7648:ρ
7610:α
7606:⟨
7603:⟩
7599:α
7571:−
7560:α
7556:λ
7546:α
7542:∑
7520:−
7505:α
7501:Φ
7496:⟨
7493:⟩
7480:−
7465:α
7461:Φ
7433:−
7422:α
7418:λ
7407:α
7403:∑
7387:−
7373:ρ
7345:ψ
7342:⟨
7339:⟩
7336:ψ
7300:ρ
7261:⊗
7246:⊗
7231:⊗
7196:⟩
7192:ψ
7153:χ
7150:⋅
7099:Γ
7066:Γ
7030:λ
7010:Γ
6947:α
6938:−
6931:α
6926:Γ
6887:∑
6844:α
6835:−
6828:α
6823:Γ
6791:χ
6787:⋅
6737:Γ
6716:Γ
6678:Γ
6649:−
6639:Γ
6626:at qubit
6610:⟩
6606:ψ
6561:χ
6438:ψ
6418:ψ
6311:χ
6273:χ
6247:⟩
6215:α
6210:Φ
6200:⟩
6174:α
6169:Φ
6139:α
6133:λ
6120:χ
6099:α
6092:∑
6088:⋅
6050:α
6044:λ
6024:χ
6004:α
5998:∑
5985:⟩
5981:Ψ
5928:α
5921:−
5912:∼
5888:α
5882:λ
5859:α
5800:χ
5772:−
5762:⋅
5690:χ
5606:χ
5597:−
5581:χ
5565:−
5552:⋅
5541:χ
5496:Rationale
5481:⟩
5395:−
5388:α
5383:Γ
5374:−
5361:−
5354:α
5349:λ
5345:⋅
5342:⋅
5308:α
5303:λ
5252:α
5242:α
5237:Γ
5225:α
5203:α
5184:α
5179:∑
5172:⟩
5140:α
5135:Φ
5106:⟩
5031:α
5022:−
5015:α
5010:Γ
5006:⋅
5003:⋅
4981:α
4976:λ
4943:α
4938:Γ
4927:−
4920:α
4904:α
4891:α
4886:∑
4879:⟩
4853:α
4848:Φ
4817:⟩
4785:α
4780:Φ
4770:⟩
4744:α
4739:Φ
4709:α
4703:λ
4691:α
4686:∑
4679:⟩
4675:Ψ
4510:−
4484:Γ
4458:⟩
4432:α
4427:Φ
4417:⟩
4367:α
4361:λ
4328:α
4318:α
4313:Γ
4290:α
4285:λ
4252:α
4247:Γ
4235:α
4222:α
4191:∑
4184:⟩
4180:Ψ
4147:⟩
4121:α
4116:Φ
4086:α
4080:λ
4047:α
4037:α
4032:Γ
4020:α
4015:∑
4008:⟩
3973:α
3968:τ
3922:α
3916:λ
3895:⟩
3869:α
3864:Φ
3837:χ
3811:⟩
3775:α
3770:τ
3731:⟩
3695:α
3690:τ
3661:⟩
3625:α
3620:τ
3610:⟩
3577:∑
3570:⟩
3544:α
3539:Φ
3512:⟩
3486:α
3481:Φ
3446:⟩
3420:α
3415:Φ
3405:⟩
3366:α
3361:λ
3328:α
3323:Γ
3317:χ
3293:α
3274:∑
3267:⟩
3263:Ψ
3211:α
3206:Φ
3153:α
3148:Φ
3115:α
3110:Φ
3064:α
3058:λ
2921:⊗
2883:−
2838:Φ
2814:⟩
2789:−
2786:⟩
2761:−
2758:⟩
2730:⟩
2702:⟩
2698:Φ
2667:⟩
2647:−
2644:⟩
2606:⟩
2591:ϕ
2569:⟩
2549:−
2546:⟩
2508:⟩
2493:ϕ
2471:⟩
2448:⟩
2410:⟩
2395:ϕ
2373:⟩
2350:⟩
2312:⟩
2297:ϕ
2254:ϕ
2239:ϕ
2212:−
2180:ϕ
2165:ϕ
2117:Ψ
2087:⟩
2071:⟩
2048:⟩
2025:⟩
1985:⟩
1981:Ψ
1931:
1904:
1801:∑
1724:⟩
1709:Φ
1655:⟩
1640:Φ
1613:⟩
1598:Φ
1588:⊗
1585:⟩
1570:Φ
1557:⟩
1542:Φ
1527:Φ
1484:Φ
1469:Φ
1418:∑
1407:Ψ
1383:⟩
1379:Ψ
1342:⊗
1328:∈
1325:⟩
1322:Ψ
1246:−
1239:α
1234:Γ
1222:−
1206:−
1199:α
1194:λ
1183:−
1165:−
1148:−
1141:α
1132:−
1125:α
1120:Γ
1116:⋅
1107:⋅
1085:α
1080:λ
1047:α
1037:α
1032:Γ
1009:α
1004:λ
971:α
961:α
956:Γ
933:α
928:λ
895:α
890:Γ
884:χ
868:−
861:α
839:α
834:∑
700:⟩
678:−
566:∑
559:⟩
556:Ψ
531:⟩
510:−
415:⟩
412:Ψ
381:⊗
372:∈
369:⟩
366:Ψ
320:atoms in
246:factoring
133:algorithm
17300:30670203
17292:15323740
17239:26736344
17231:15600965
17176:36218923
17168:15600964
17115:15188855
17107:14611555
16223:′
15074:′
14998:′
14948:′
13057:towards
11896:′
11858:′
11787:′
10948:′
10879:′
10582:, where
10177:′
10152:′
10116:′
10103:′
9757:′
9698:′
9676:'s are:
9491:′
9433:′
9402:′
9358:′
9333:′
9252:′
9193:′
9053:′
9045:′
9015:′
9002:′
8984:′
8967:′
8939:′
8923:′
8876:′
8829:′
8445:′
8391:′
7286:subspace
6858:′
6622:after a
6296:fermions
3234:⟩
3179:⟩
3138:⟩
2976:⟩
2943:⟩
2912:⟩
2879:⟩
2841:⟩
2269:⟩
2195:⟩
2121:⟩
1499:⟩
1410:⟩
307:linearly
17211:Bibcode
17148:Bibcode
12513:relate
11224:unitary
9826:tensors
4648:, from
3826:at most
280:Hubbard
268:Caltech
178:Please
137:quantum
85:scholar
17357:
17349:
17339:
17298:
17290:
17282:
17237:
17229:
17174:
17166:
17113:
17105:
17097:
16608:
16605:
16596:
16593:
15891:where
14225:where
13940:where
13579:where
11435:where
9641:, the
8584:where
7090:, and
6403:ones.
6337:0.0001
6264:where
5942:
4163:Step 3
3752:Step 2
3463:Step 1
2581:
2578:
2483:
2480:
2385:
2382:
1512:where
712:where
351:qubits
244:, for
87:
80:
73:
66:
58:
17355:S2CID
17319:arXiv
17296:S2CID
17262:arXiv
17235:S2CID
17201:arXiv
17172:S2CID
17138:arXiv
17111:S2CID
17077:arXiv
16646:trace
11228:phase
6573:as a
6464:gates
6298:in a
2282:with
92:JSTOR
78:books
17347:ISSN
17337:ISBN
17288:PMID
17280:ISSN
17227:PMID
17164:PMID
17103:PMID
17095:ISSN
15342:is:
13687:as:
13442:is:
11269:as:
9962:and
9136:the
8427:as:
8335:and
7986:and
7955:and
7284:The
6996:k, k
5948:>
5118:and
318:cold
129:TEBD
123:The
64:news
17329:doi
17272:doi
17219:doi
17156:doi
17087:doi
15552:1..
15321:1..
15300:1..
14648:1..
14601:1..
14425:1..
14378:1..
12792:lim
11015:lim
9828:in
7513:1..
7473:1..
7380:1..
6235:1..
6188:1..
5160:1..
4867:1..
4805:1..
4758:1..
4630:1..
4609:1..
4446:3..
4135:3..
3997:3..
3883:3..
3799:3..
3719:3..
3649:3..
3558:2..
3500:2..
3434:2..
3167:2..
3031:2..
3005:).
1928:dim
1901:dim
1895:min
47:by
17374::
17353:.
17345:.
17335:.
17327:.
17294:.
17286:.
17278:.
17270:.
17258:93
17256:.
17233:.
17225:.
17217:.
17209:.
17197:93
17195:.
17170:.
17162:.
17154:.
17146:.
17134:93
17132:.
17109:.
17101:.
17093:.
17085:.
17073:91
17071:.
17056:^
16106:.
13121::
12716:,
12389:.
12098:,
11533:,
10979:.
10906:,
9938:,
8860::
8315:,
8278:,
7994:,
7362::
7053:,
6998:+1
6581:.
6389:50
6362:14
5951:0.
5871::
4659::
4652:.
3954::
3747:.
2810:11
2782:10
2754:01
2726:00
2083:11
2067:10
2044:01
2021:00
779::
543::
252:.
131:)
17361:.
17331::
17321::
17302:.
17274::
17264::
17241:.
17221::
17213::
17203::
17178:.
17158::
17150::
17140::
17117:.
17089::
17079::
17017:)
17012:n
17004:2
16998:1
16995:(
16990:1
16984:N
16979:1
16976:=
16973:n
16962:1
16959:=
16951:n
16940:1
16933:n
16925:2
16919:1
16911:1
16905:N
16900:1
16897:=
16894:n
16886:)
16881:n
16870:1
16867:(
16862:1
16856:N
16851:1
16848:=
16845:n
16834:1
16831:=
16828:)
16824:D
16820:(
16785:l
16774:1
16767:l
16759:2
16753:1
16747:=
16739:l
16728:1
16722:l
16709:1
16706:=
16701:2
16696:|
16686:D
16677:|
16670:D
16657:|
16628:0
16617:l
16602:s
16599:a
16590:0
16577:l
16566:1
16560:l
16550:=
16545:R
16541:R
16535:1
16526:1
16518:R
16513:1
16509:S
16503:=
16480:1
16472:2
16468:n
16464:=
16459:1
16455:n
16446:2
16442:n
16438:=
16411:R
16406:1
16402:S
16396:=
16391:R
16385:2
16381:)
16375:]
16372:l
16369:[
16362:l
16349:(
16341:2
16337:)
16329:l
16319:1
16313:l
16304:c
16300:(
16293:c
16283:1
16280:=
16275:l
16262:=
16257:2
16253:)
16247:]
16244:l
16241:[
16234:l
16214:(
16206:2
16202:)
16194:l
16184:1
16178:l
16169:c
16165:(
16158:c
16148:1
16145:=
16140:l
16127:=
16122:2
16118:n
16090:l
16086:i
16082:]
16079:l
16076:[
16069:l
16059:1
16053:l
16034:)
16026:l
16022:i
16018:]
16015:l
16012:[
16005:l
15995:1
15989:l
15976:(
15971:d
15966:1
15963:=
15958:l
15954:i
15945:=
15940:2
15936:)
15928:l
15918:1
15912:l
15903:c
15899:(
15876:,
15871:2
15867:S
15863:+
15858:1
15854:S
15850:=
15845:2
15841:)
15835:]
15832:l
15829:[
15822:l
15809:(
15804:2
15800:)
15792:l
15782:1
15776:l
15767:c
15763:(
15751:c
15743:=
15738:l
15725:+
15720:2
15716:)
15710:]
15707:l
15704:[
15697:l
15684:(
15679:2
15675:)
15667:l
15657:1
15651:l
15642:c
15638:(
15631:c
15621:1
15618:=
15613:l
15600:=
15597:1
15594:=
15589:1
15585:n
15558:]
15555:N
15546:l
15543:[
15536:1
15530:l
15496:1
15488:l
15461:l
15450:1
15446:=
15440:2
15434:|
15428:]
15425:l
15422:[
15415:l
15399:|
15389:c
15377:1
15374:=
15368:l
15353:=
15350:R
15330:)
15327:]
15324:N
15318:+
15315:l
15312:[
15309::
15306:]
15303:l
15297:[
15294:(
15275:l
15252:D
15229:)
15224:n
15213:1
15210:(
15207:)
15202:1
15199:+
15196:n
15185:1
15182:(
15176:1
15173:=
15168:2
15163:|
15153:D
15144:|
15131:|
15127:)
15122:1
15119:+
15116:n
15105:1
15102:(
15096:1
15093:=
15088:2
15083:|
15070:D
15061:|
15048:|
15041:1
15038:=
15004:D
14987:|
14979:1
14976:+
14973:n
14963:+
14954:D
14935:|
14927:1
14924:+
14921:n
14910:1
14905:=
14896:D
14886:|
14861:2
14857:)
14851:]
14848:n
14845:[
14832:(
14820:c
14812:=
14801:=
14796:2
14791:|
14781:D
14772:|
14759:|
14752:1
14749:=
14744:n
14719:0
14716:=
14708:D
14699:|
14688:D
14654:]
14651:N
14645:+
14642:n
14639:[
14632:n
14617:|
14607:]
14604:n
14598:[
14591:n
14576:|
14570:]
14567:n
14564:[
14557:n
14533:c
14523:=
14517:n
14496:n
14487:1
14482:=
14468:D
14458:|
14431:]
14428:N
14422:+
14419:n
14416:[
14409:n
14394:|
14384:]
14381:n
14375:[
14368:n
14353:|
14347:]
14344:n
14341:[
14334:n
14315:c
14303:1
14300:=
14294:n
14273:n
14262:1
14258:1
14253:=
14244:D
14234:|
14202:D
14192:|
14184:n
14174:+
14165:D
14155:|
14147:n
14136:1
14131:=
14119:|
14096:n
14065:|
14042:n
14018:2
14014:)
14008:]
14005:n
14002:[
13989:(
13977:c
13969:=
13958:=
13953:n
13928:)
13923:n
13912:1
13909:(
13904:1
13898:N
13893:1
13890:=
13887:n
13876:1
13873:=
13870:)
13865:D
13860:(
13844:D
13814:2
13806:=
13801:2
13793:+
13790:1
13784:1
13781:=
13776:2
13771:|
13760:r
13757:T
13747:|
13735:r
13732:T
13717:|
13710:1
13707:=
13704:)
13701:T
13698:(
13675:T
13641:r
13638:T
13628:|
13601:r
13598:T
13588:|
13554:r
13551:T
13541:|
13532:+
13523:r
13520:T
13510:|
13502:2
13489:1
13484:=
13475:r
13472:T
13453:|
13424:r
13421:T
13402:|
13377:p
13369:T
13366:=
13361:1
13358:+
13355:p
13342:T
13337:=
13309:T
13304:=
13301:n
13279:1
13276:+
13273:p
13242:h
13239:t
13235:p
13189:0
13179:|
13173:T
13168:n
13164:H
13160:i
13153:e
13149:=
13140:T
13130:|
13107:n
13103:H
13076:T
13066:|
13039:0
13029:|
12999:H
12970:P
12960:|
12937:1
12933:H
12911:,
12896:P
12886:|
12871:H
12861:e
12846:P
12836:|
12821:H
12811:e
12788:=
12779:r
12776:g
12766:|
12738:r
12735:g
12725:|
12698:H
12667:r
12664:g
12654:|
12650:Q
12647:=
12638:0
12628:|
12601:H
12572:r
12569:g
12559:|
12532:0
12522:|
12492:P
12482:|
12459:D
12429:c
12402:D
12375:n
12371:H
12344:T
12334:|
12307:0
12297:|
12271:D
12245:]
12242:l
12239:[
12235:F
12229:2
12221:i
12214:e
12189:]
12186:l
12183:[
12179:G
12171:i
12163:e
12138:]
12135:l
12132:[
12128:F
12122:2
12114:i
12107:e
12064:.
12055:t
12036:|
12030:F
12025:2
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9722:=
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9217:=
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8894:=
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8497:=
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8301:]
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8097:=
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8091:,
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8070:=
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8058:,
8047:=
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7964:H
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7914:.
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7833:=
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7822:]
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7776:]
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7738:|
7731:2
7727:)
7721:]
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7682:=
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7619:.
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7527:]
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7399:=
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5218:.
5213:2
5210:+
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5191:+
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5175:=
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5163:N
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5151:[
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5096:i
5092:.
5089:.
5084:2
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4913:.
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4882:=
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4802:+
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4789:k
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4727:]
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4721:[
4714:k
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4682:=
4670:|
4657:D
4650:D
4636:]
4633:N
4627:+
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4621:[
4618::
4615:]
4612:k
4606:[
4584:h
4581:t
4577:k
4554:h
4551:t
4547:N
4524:h
4521:t
4517:)
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4504:(
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4332:2
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4218:,
4213:2
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4205:,
4200:1
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4187:=
4175:|
4141:]
4138:N
4132:[
4125:2
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4101:2
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4091:2
4072:2
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4064:]
4061:2
4058:[
4051:2
4041:1
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4011:=
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4000:N
3994:[
3987:2
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3977:1
3963:|
3940:]
3937:2
3934:[
3927:2
3889:]
3886:N
3880:[
3873:2
3858:|
3805:]
3802:N
3796:[
3789:2
3785:i
3779:1
3764:|
3725:]
3722:N
3716:[
3709:2
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3699:1
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3655:]
3652:N
3646:[
3639:2
3635:i
3629:1
3614:|
3604:2
3600:i
3594:|
3586:2
3582:i
3573:=
3564:]
3561:N
3555:[
3548:1
3533:|
3506:]
3503:N
3497:[
3490:1
3475:|
3440:]
3437:N
3431:[
3424:1
3409:|
3399:1
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3389:|
3383:]
3380:1
3377:[
3370:1
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3342:1
3339:[
3332:1
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3305:1
3302:=
3297:1
3288:,
3283:1
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3270:=
3258:|
3228:]
3225:1
3222:[
3215:1
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3173:]
3170:N
3164:[
3157:1
3143:|
3132:]
3129:1
3126:[
3119:1
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3082:]
3079:1
3076:[
3069:1
3037:]
3034:N
3028:[
3025::
3022:]
3019:1
3016:[
3003:D
2981:)
2971:B
2967:1
2963:|
2956:2
2952:1
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2938:B
2934:0
2930:|
2925:(
2917:)
2907:A
2903:1
2899:|
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2859:3
2855:1
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2845:=
2835:|
2805:|
2798:6
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2777:|
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2742:6
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2705:=
2693:|
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2306:A
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2243:2
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2125:=
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2033:3
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1925:,
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1892:=
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1854:1
1851:=
1846:2
1841:i
1837:a
1829:B
1825:|
1821:A
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1808:=
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1634:|
1607:B
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1560:=
1551:B
1546:i
1536:A
1531:i
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1266:i
1262:]
1259:N
1256:[
1249:1
1243:N
1228:]
1225:1
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1216:[
1209:1
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1186:1
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1158:[
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
