4337:(this regime is readily accessible experimentally, for example by introducing temporal delay between photons). The opportunity then exists to tune between ideally indistinguishable (quantum) and perfectly distinguishable (classical) data and measure the change in a suitably constructed metric. This scenario can be addressed by a statistical test which performs a one-on-one likelihood comparison of the output probabilities. This test requires the calculation of a small number of permanents, but does not need the calculation of the full expected probability distribution. Experimental implementation of the test has been successfully reported in integrated laser-written circuits for both the standard boson sampling (3 photons in 7-, 9- and 13-mode interferometers) and the scattershot version (3 photons in 9- and 13-mode interferometers with different input states). Another possibility is based on the bunching property of indinguishable photons. One can analyze the probability to find a
4333:
impossible (roughly speaking a symmetric measurement scheme does not allow for labeling the output modes of the optical circuit). However, within current technologies the assumption of a symmetric setting is not justified (the tracking of the measurement statistics is fully accessible), and therefore the above argument does not apply. It is then possible to define a rigorous and efficient test to discriminate the boson sampling statistics from an unbiased probability distribution. The corresponding discriminator is correlated to the permanent of the submatrix associated with a given measurement pattern, but can be efficiently calculated. This test has been applied experimentally to distinguish between a boson sampling and a uniform distribution in the 3-photon regime with integrated circuits of 5, 7, 9 and 13 modes.
4341:-fold coincidence measurement outcomes (without any multiply populated input mode), which is significantly higher for distinguishable particles than for bosons due to the bunching tendency of the latters. Finally, leaving the space of random matrices one may focus on specific multimode setups with certain features. In particular, the analysis of the effect of bosonic clouding (the tendency for bosons to favor events with all particles in the same half of the output array of a continuous-time many-particle quantum walk) has been proven to discriminate the behavior of distinguishable and indistinguishable particles in this specific platform.
7617:
4094:). Such a model deals with continuous-variable measurement outcome, which, under certain conditions, is a computationally hard task. Finally, a linear optics platform for implementing a boson sampling experiment where input single-photons undergo an active (non-linear) Gaussian transformation is also available. This setting makes use of a set of two-mode squeezed vacuum states as a prior resource, with no need of single-photon sources or in-line nonlinear amplification medium. This variant uses the
7607:
3217:(Haar random matrices can be directly implemented in optical circuits by mapping independent probability density functions for their parameters, to optical circuit components, i.e., beam splitters and phase shifters). Therefore, if the linear optical circuit implements a Haar random unitary matrix, the adversarial sampler will not be able to detect which of the exponentially many probabilities
4021:
important leap towards a convincing experimental demonstration of the quantum computational supremacy. The scattershot boson sampling model can be further generalized to the case where both legs of PDC sources are subject to linear optical transformations (in the original scattershot case, one of the arms is used for heralding, i.e., it goes through the identity channel). Such a
3893:. In other words, in order to generate the input state for the boson sampling machine, one would have to wait for exponentially long time, which would kill the advantage of the quantum setup over a classical machine. Subsequently, this characteristic restricted the use of PDC sources to proof-of-principle demonstrations of a boson sampling device.
1904:
3877:) mechanism. The main advantages of PDC sources are the high photon indistinguishability, collection efficiency and relatively simple experimental setups. However, one of the drawbacks of this approach is its non-deterministic (heralded) nature. Specifically, suppose the probability of generating a single photon by means of a PDC crystal is
4086:
the same complexity assumption as can approximate ordinary or scattershot boson sampling. Gaussian resources can be employed at the measurement stage, as well. Namely, one can define a boson sampling model, where a linear optical evolution of input single-photon states is concluded by
Gaussian measurements (more specifically, by eight-port
3847:
order to collect enough statistics to approximate its value, one has to run the quantum experiment for exponentially long time. Therefore, the estimate obtained from a boson sampler is not more efficient that running the classical polynomial-time algorithm by
Gurvits for approximating the permanent of any matrix to within additive error.
3769:
feature facilitates the implementation of a restricted boson sampling device. Namely, if the probability of having more than one photon at the output of a linear optical circuit is negligible, one does not require photon-number-resolving detectors anymore: on-off detectors will be sufficient for the realization of the setup.
4345:
reasonable assumption is that the system maintains correct operation as the circuit is continuously reconfigured to implement a random unitary operation. To this end, one can exploit quantum suppression laws (the probability of specific input-output combinations is suppressed when the linear interferometer is described by a
69:. Moreover, while not universal, the boson sampling scheme is strongly believed to implement computing tasks which are hard to implement with classical computers by using far fewer physical resources than a full linear-optical quantum computing setup. This advantage makes it an ideal candidate for demonstrating the power of
2382:
4299:
There are several other proposals for the implementation of photonic boson sampling. This includes, e.g., the scheme for arbitrarily scalable boson sampling using two nested fiber loops. In this case, the architecture employs time-bin encoding, whereby the incident photons form a pulse train entering
3846:
of a specific measurement outcome at the output of the interferometer is related to the permanent of submatrices of a unitary matrix, a boson sampling machine does not allow its estimation. The main reason behind is that the corresponding detection probability is usually exponentially small. Thus, in
3768:
modes of a linear interferometer with no two bosons in the same mode, then with high probability two bosons will not be found in the same output mode either. This property has been experimentally observed with two and three photons in integrated interferometers of up to 16 modes. On the one hand this
4303:
Another approach relies on the realization of unitary transformations on temporal modes based on dispersion and pulse shaping. Namely, passing consecutively heralded photons through time-independent dispersion and measuring the output time of the photons is equivalent to a boson sampling experiment.
3751:
By making use of the above two conjectures (which have several evidences of being true), the final proof eventually states that the existence of a classical polynomial-time algorithm for the approximate boson sampling task implies the collapse of the polynomial hierarchy. It is also worth mentioning
4434:
It has also been suggested to use a superconducting resonator network Boson
Sampling device as an interferometer. This application is assumed to be practical, as small changes in the couplings between the resonators will change the sampling results. Sensing of variation in the parameters capable of
4336:
The test above does not distinguish between more complex distributions, such as quantum and classical, or between fermionic and bosonic statistics. A physically motivated scenario to be addressed is the unwanted introduction of distinguishability between photons, which destroys quantum interference
4287:
A first scattershot boson sampling experiment has been recently implemented using six photon-pair sources coupled to integrated photonic circuits with 13 modes. The 6 photon-pair sources were obtained via type-II PDC processes in 3 different nonlinear crystals (exploiting the polarization degree of
4278:
Later on, more complex boson sampling experiments have been performed, increasing the number of spatial modes of random interferometers up to 13 and 9 modes, and realizing a 6-mode fully reconfigurable integrated circuit. These experiments altogether constitute the proof-of-principle demonstrations
4085:
This is precisely equivalent to scattershot boson sampling, except for the fact that our measurement of the herald photons has been deferred till the end of the experiment, instead of happening at the beginning. Therefore, approximate
Gaussian boson sampling can be argued to be hard under precisely
4457:
Coarse-grained boson sampling has been proposed as a resource of decision and function problems that are computationally hard, and may thus have cryptographic applications. The first related proof-of-principle experiment was performed with a photonic boson-sampling machine (fabricated by a direct
2839:
The above hardness proofs are not applicable to the realistic implementation of a boson sampling device, due to the imperfection of any experimental setup (including the presence of noise, decoherence, photon losses, etc.). Therefore, for practical needs one necessitates the hardness proof for the
4332:
A first relevant question is whether it is possible or not to distinguish between uniform and boson-sampling distributions by performing a polynomial number of measurements. The initial argument introduced in Ref. stated that as long as one makes use of symmetric measurement settings the above is
2522:
All current proofs of the hardness of simulating boson sampling on a classical computer rely on the strong computational consequences that its efficient simulation by a classical algorithm would have. Namely, these proofs show that an efficient classical simulation would imply the collapse of the
2498:
The main reason of the growing interest towards the model of boson sampling is that despite being non-universal it is strongly believed to perform a computational task that is intractable for a classical computer. One of the main reasons behind this is that the probability distribution, which the
4344:
A different approach to confirm that the boson sampling machine behaves as the theory predicts is to make use of fully reconfigurable optical circuits. With large-scale single-photon and multiphoton interference verified with predictable multimode correlations in a fully characterized circuit, a
4041:
is a
Gaussian one. The hardness of the corresponding sampling task can be linked to that of scattershot boson sampling. Namely, the latter can be embedded into the conventional boson sampling setup with Gaussian inputs. For this, one has to generate two-mode entangled Gaussian states and apply a
97:
crystals), as well as a linear interferometer. The latter can be fabricated, e.g., with fused-fiber beam splitters, through silica-on-silicon or laser-written integrated interferometers, or electrically and optically interfaced optical chips. Finally, the scheme also necessitates high efficiency
4106:
The above results state that the existence of a polynomial-time classical algorithm for the original boson sampling scheme with indistinguishable single photons (in the exact and approximate cases), for scattershot, as well as for the general
Gaussian boson sampling problems is highly unlikely.
3860:
As already mentioned above, for the implementation of a boson sampling machine one necessitates a reliable source of many indistinguishable photons, and this requirement currently remains one of the main difficulties in scaling up the complexity of the device. Namely, despite recent advances in
2713:
On the other hand, the alternative proof is inspired by a similar result for another restricted model of quantum computation – the model of instantaneous quantum computing. Namely, the proof uses the KLM scheme, which says that linear optics with adaptive measurements is universal for the class
4020:
PDC crystals generated single photons. Therefore, the proof can be constructed here similar to the original one. Furthermore, scattershot boson sampling has been also recently implemented with six photon-pair sources coupled to integrated photonic circuits of nine and thirteen modes, being an
93:). Then, the photonic implementation of the boson sampling task consists of generating a sample from the probability distribution of single-photon measurements at the output of the circuit. Specifically, this requires reliable sources of single photons (currently the most widely used ones are
4328:
complexity class), it is not understood how to verify correct operation for large versions of the setup. Specifically, the naive verification of the output of a boson sampler by computing the corresponding measurement probabilities represents a problem intractable for a classical computer.
4349:
or other matrices with relevant symmetries). These suppression laws can be classically predicted in efficient ways. This approach allows also to exclude other physical models, such as mean-field states, which mimic some collective multiparticle properties (including bosonic clouding). The
5786:
Spagnolo, Nicolo; Vitelli, Chiara; Bentivegna, Marco; Brod, Daniel; Crespi, Andrea; Flamini, Fulvio; Giacomini, Sandro; Milani, Giorgio; Ramponi, Roberta; Mataloni, Paolo; Osellame, Roberto; GalvĂŁo, Ernesto; Sciarrino, Fabio (2014). "Experimental validation of photonic boson sampling".
4403:
spins. One necessitates several additional assumptions here, including small boson bunching probability and efficient error postselection. This scalable scheme, however, is rather promising, in the light of considerable development in the construction and manipulation of coupled
4618:
Spring, Justin; Metcalf, Benjamin; Humphreys, Peter; Kolthammer, Steven; Jin, Xian-Min; Barbieri, Marco; Datta, Animesh; Thomas-Peter, Nicholas; Langford, Nathan; Kundys, Dmytro; Gates, James; Smith, Brian; Smith, Peter; Walmsley, Ian (2013). "Boson sampling on a photonic chip".
1623:
4792:
Crespi, Andrea; Osellame, Roberto; Ramponi, Roberta; Brod, Daniel; Galvao, Ernesto; Spagnolo, Nicolò; Vitelli, Chiara; Maiorino, Enrico; Mataloni, Paolo; Sciarrino, Fabio (2013). "Integrated multimode interferometers with arbitrary designs for photonic boson sampling".
5847:
Carolan, Jacques; Meinecke, Jasmin; Shadbolt, Pete; Russell, Nicholas; Ismail, Nur; Wörhoff, Kerstin; Rudolph, Terry; Thompson, Mark; O'Brien, Jeremy; Matthews, Jonathan; Laing, Anthony (2014). "On the experimental verification of quantum complexity in linear optics".
4255:
The above requirements for the photonic boson sampling machine allow for its small-scale construction by means of existing technologies. Consequently, shortly after the theoretical model was introduced, four different groups simultaneously reported its realization.
4107:
Nevertheless, there are some non-trivial realizations of the boson sampling problem that allow for its efficient classical simulation. One such example is when the optical circuit is injected with distinguishable single photons. In this case, instead of summing the
4430:
states into a linear interferometer that is determined by the properties of the molecule of interest. Therefore, this prominent observation makes the interest towards the implementation of the boson sampling task to get spread well beyond the fundamental basis.
4300:
the loops. Meanwhile, dynamically controlled loop coupling ratios allow the construction of arbitrary linear interferometers. Moreover, the architecture employs only a single point of interference and may thus be easier to stabilize than other implementations.
4273:
three photons in a femtosecond laser-written five-mode interferometer implementing a Haar-random unitary transformation, by a collaboration between Milan's
Institute of Photonics and Nanotechnology, Universidade Federal Fluminense and Sapienza University of
376:
4215:
problem, its approximation can be performed efficiently on a classical computer, due to the seminal algorithm by Jerrum, Sinclaire and Vigoda. In other words, approximate boson sampling with distinguishable photons is efficiently classically simulable.
2118:
2749:
Again, the combination of these three results, as in the previous case, results in the collapse of the polynomial hierarchy. This makes the existence of a classical polynomial-time algorithm for the exact boson sampling problem highly unlikely.
5370:
Bentivegna, Marco; Spagnolo, Nicolo; Vitelli, Chiara; Flamini, Fulvio; Viggianiello, Niko; Latmiral, Ludovico; Mataloni, Paolo; Brod, Daniel; GalvĂŁo, Ernesto; Crespi, Andrea; Ramponi, Roberta; Osellame, Roberto; Sciarrino, Fabio (2015).
4263:
two and three photons scattered by a six-mode linear unitary transformation (represented by two orthogonal polarizations in 3Ă—3 spatial modes of a fused-fiber beam splitter) by a collaboration between the
University of Queensland and
1291:
1036:
4304:
With time-dependent dispersion, it is also possible to implement arbitrary single-particle unitaries. This scheme requires a much smaller number of sources and detectors and do not necessitate a large system of beam splitters.
2709:
result in the collapse of the polynomial hierarchy, which as mentioned above is highly unlikely to occur. This leads to the conclusion that there is no classical polynomial-time algorithm for the exact boson sampling problem.
4377:. This scheme is scalable and relies on the recent advances in ion trapping techniques (several dozens of ions can be successfully trapped, for example, in linear Paul traps by making use of anharmonic axial potentials).
4350:
implementation of a
Fourier matrix circuit in a fully reconfigurable 6-mode device has been reported, and experimental observations of the suppression law have been shown for 2 photons in 4- and 8-mode Fourier matrices.
3746:
2489:
describing the linear-optical circuit as input. As detailed below, the appearance of the permanent in the corresponding statistics of single-photon measurements contributes to the hardness of the boson sampling problem.
6320:
Goldstein, Samuel; Korenblit, Simcha; Bendor, Ydan; You, Hao; Geller, Michael R.; Katz, Nadav (17 January 2017). "Decoherence and interferometric sensitivity of boson sampling in superconducting resonator networks".
3659:
3524:
3432:
2740:
The existence of a classical boson sampling algorithm implies the simulability of postselected linear optics in the PostBPP class (that is, classical polynomial-time with postselection, known also as the class
1388:
1899:{\displaystyle p(t_{1},t_{2},...,t_{N})=|\langle t_{1},t_{2},...,t_{N}|\psi _{\text{out}}\rangle |^{2}={\frac {|{\text{Perm}}\,U_{S,T}|^{2}}{t_{1}!\cdot \cdot \cdot t_{N}!s_{1}!\cdot \cdot \cdot s_{N}!}}.}
4324:(NP) complexity class. It is however not clear that a similar structure exists for the boson sampling scheme. Namely, as the latter is related to the problem of estimating matrix permanents (falling into
6599:
Wang, Xiao-Wei; Zhou, Wen-Hao; Fu, Yu-Xuan; Gao, Jun; Lu, Yong-Heng; Chang, Yi-Jun; Qiao, Lu-Feng; Ren, Ruo-Jing; Jiang, Ze-Kun; Jiao, Zhi-Qiang; Nikolopoulos, Georgios M.; Jin, Xian-Min (2023-02-09).
3576:
which the given probability of a specific measurement outcome is proportional to. However to establish this link one has to rely on another conjecture – the permanent anticoncentration conjecture:
1484:
1182:
3976:
4358:
Apart from the photonic realization of the boson sampling task, several other setups have been proposed. This includes, e.g., the encoding of bosons into the local transverse phonon modes of
4288:
freedom). This allowed to sample simultaneously between 8 different input states. The 13-mode interferometer was realized by femtosecond laser-writing technique on alumino-borosilicate glas.
4537:
Troyansky, Lidror; Tishby, Naftali (1996). “Permanent uncertainty: On the quantum evaluation of the determinant and the permanent of a matrix”. Proceedings of PhysComp, 1996: 314-318.
795:
4114:
corresponding to photonic many-particle paths, one has to sum the corresponding probabilities (i.e. the squared absolute values of the amplitudes). Consequently, the detection probability
2527:
to its third level, a possibility that is considered very unlikely by the computer science community, due to its strong computational implications (in line with the strong implications of
917:
723:
2941:
Specifically, the proofs of the exact boson sampling problem cannot be directly applied here, since they are based on the #P-hardness of estimating the exponentially-small probability
4186:
3844:
3363:
3289:
3169:
3091:
3013:
2932:
2696:
2619:
1584:
224:
3574:
1946:
849:
4552:
Broome, Matthew; Fedrizzi, Alessandro; Rahimi-Keshari, Saleh; Dove, Justin; Aaronson, Scott; Ralph, Timothy; White, Andrew (2013). "Photonic boson sampling in a tunable circuit".
658:
4227:
among the modes. More precisely, only their amplitudes are transformed, and the transformation can be efficiently calculated on a classical computer (the computation comprises
2377:{\displaystyle p(t_{1},t_{2},...,t_{N})=|\langle t_{1},t_{2},...,t_{N}|\varphi _{M}(U)|1_{M}\rangle |^{2}={\frac {|{\text{Perm}}\,U_{T}|^{2}}{t_{1}!\cdot \cdot \cdot t_{N}!}},}
2810:
419:
182:
2106:
1092:
517:
4062:
to their "right halves", while doing nothing to the others. Then we can measure the "left halves" to find out which of the input states contained a photon before we applied
2621:
of a specific measurement outcome at the output of a linear interferometer to within a multiplicative constant is a #P-hard problem (due to the complexity of the permanent)
2858:
564:
453:
216:
4267:
three photons in different modes of a six-mode silica-on-silicon waveguide circuit, by a collaboration between
Universities of Oxford, Shanghai, London and Southampton
1986:
878:
2463:
2412:
2064:
2033:
1611:
65:
version is currently considered as the most promising platform for a scalable implementation of a boson sampling device, which makes it a non-universal approach to
6389:
Chakhmakhchyan, Levon; Cerf, Nicolas; Garcia-Patron, Raul (2017). "A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices".
4209:
4083:
3455:
1191:
142:
4060:
3197:
2487:
2432:
2006:
1508:
1056:
925:
682:
588:
537:
481:
4979:
Bremner, Michael; Jozsa, Richard; Shepherd, Dan (2011). "Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy".
5901:
Motes, Keith; Gilchrist, Alexei; Dowling, Jonathan; Rohde, Peter (2014). "Scalable boson sampling with time-bin encoding using a loop-based architecture".
5622:
Jerrum, Mark; Sinclair, Alistair; Vigoda, Eric (2001). "A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries".
102:, which perform the measurements at the output of the circuit. Therefore, based on these three ingredients, the boson sampling setup does not require any
4223:
injected into the linear interferometer. The reason is that at the output of a linear optical circuit coherent states remain such, and do not create any
4461:
Gaussian boson sampling has been analyzed as a search component for computing binding propensity between molecules of pharmacological interest as well.
114:
scheme). This makes it a non-universal model of quantum computation, and reduces the amount of physical resources needed for its practical realization.
6267:
Huh, Joonsuk; Giacomo Guerreschi, Gian; Peropadre, Borja; McClean, Jarrod; Aspuru-Guzik, Alan (2015). "Boson sampling for molecular vibronic spectra".
2539:
The hardness proof of the exact boson sampling problem can be achieved following two distinct paths. Specifically, the first one uses the tools of the
3457:
These arguments bring us to the first conjecture of the hardness proof of approximate boson sampling problem – the permanent-of-Gaussians conjecture:
3896:
Recently, however, a new scheme has been proposed to make the best use of PDC sources for the needs of boson sampling, greatly enhancing the rate of
2469:
th row. Subsequently, the task of boson sampling is to sample either exactly or approximately from the above output distribution, given the unitary
3668:
99:
6806:
2938:. The understanding of the complexity of this problem relies then on several additional assumptions, as well as on two yet unproven conjectures.
2823:
output modes. This algorithm leads to an estimate of 50 photons required to demonstrate quantum supremacy with boson sampling. There is also an
4380:
Another platform for implementing the boson sampling setup is a system of interacting spins: recent observation show that boson sampling with
6768:
3986:, this results in an exponential improvement in the single photon generation rate with respect to the usual, fixed-input boson sampling with
5962:
Pant, Mihir; Englund, Dirk (2016). "High dimensional unitary transformations and boson sampling on temporal modes using dispersive optics".
3093:
we wanted to estimate, then it could adversarially choose to corrupt it (as long as the task is approximate). That is why, the idea is to "
4734:
Tillmann, Max; Dakic, Borivoje; Heilmann, Rene; Nolte, Stefan; Szameit, Alexander; Walther, Philip (2013). "Experimental boson sampling".
4004:
Scattershot boson sampling is still intractable for a classical computer: in the conventional setup we fixed the columns that defined our
3599:
3464:
3372:
7498:
5204:
Spagnolo, Nicolò; Vitelli, Chiara; Sanson, Linda; et al. (2013). "General Rules for Bosonic Bunching in Multimode Interferometers".
6057:
Tichy, Malte; Mayer, Klaus; Buchleitner, Andreas; Mølmer, Klaus (2014). "Stringent and Efficient Assessment of Boson-Sampling Devices".
7399:
7060:
1296:
5106:
Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony (2017). "Direct dialling of Haar random unitary matrices".
3752:
another fact important to the proof of this statement, namely the so-called bosonic birthday paradox (in analogy with the well-known
6961:
5667:
Rahimi-Keshari, Saleh; Lund, Austin; Ralph, Timothy (2015). "What can quantum optics say about computational complexity theory?".
4419:: a feasible modification of the boson sampling scheme results in a setup that can be used for the reconstruction of a molecule's
7286:
4291:
This experimental implementation represents a leap towards an experimental demonstration of the quantum computational supremacy.
4270:
three photons in a femtosecond laser-written five-mode interferometer, by a collaboration between universities of Vienna and Jena
2698:
could have been approximated to within a multiplicative constant in the BPPcomplexity class, i.e. within the third level of the
4231:). This fact can be used to perform corresponding sampling tasks from another set of states: so-called classical states, whose
1397:
7641:
7610:
6796:
2512:
4219:
Another instance of classically simulable boson sampling setups consists of sampling from the probability distribution of
7568:
5497:
Hamilton, Craig S.; Kruse, Regina; Sansoni, Linda; Barkhofen, Sonja; Silberhorn, Christine; Jex, Igor (23 October 2017).
1104:
7144:
4232:
4211:
The latter is now a non-negative matrix. Therefore, although the exact computation of the corresponding permanent is a
3927:
2070:
th row. Usually, in the context of the boson sampling problem the input state is taken of a standard form, denoted as
7620:
7508:
6761:
4037:
Another photonic implementation of boson sampling concerns Gaussian input states, i.e. states whose quasiprobability
727:
7436:
7094:
4480:
4442:
computational algorithms, aimed, e.g., at the estimation of certain matrix permanents (for instance, permanents of
4239:
is a well-defined probability distribution. These states can be represented as a mixture of coherent states due to
66:
5728:
Rahimi-Keshari, Saleh; Ralph, Timothy; Carlton, Caves (2016). "Efficient classical simulation of quantum optics".
7431:
7159:
7139:
5280:
Lund, Austin; Laing, Anthony; Rahimi-Keshari, Saleh; et al. (2014). "Boson sampling from a Gaussian state".
4423:(for which no efficient classical algorithm is currently known). Specifically, the task now is to input specific
2540:
887:
690:
371:{\displaystyle b_{j}^{\dagger }=\sum _{i=1}^{N}U_{ji}a_{i}^{\dagger }\;\;(b_{j}=\sum _{i=1}^{N}U_{ji}^{*}a_{i}).}
7426:
4188:
will be proportional to the permanent of submatrices of (component-wise) squared absolute value of the unitary
6938:
6442:
Nikolopoulos, Georgios M.; Brougham, Thomas (2016). "Decision and function problems based on boson sampling".
4247:
function, one can perform efficient classical simulation of boson sampling from this set of classical states.
4117:
3775:
3294:
3220:
3100:
3022:
2944:
2863:
2627:
2550:
1515:
594:
of the system: simple counting arguments show that the size of the Hilbert space corresponding to a system of
7459:
7281:
7184:
6845:
6015:
Gogolin, C.; Kliesch, M.; Aolita, L.; Eisert, J. (2013). "Boson-Sampling in the light of sample complexity".
3916:) heralded single-photon sources to different input ports of the linear interferometer. Then, by pumping all
3533:
2726:, i.e. quantum polynomial-time class with postselection (a straightforward corollary of the KLM construction)
1912:
800:
7464:
7332:
6923:
6754:
4957:
4038:
2504:
5569:
Chakhmakhchyan, Levon; Cerf, Nicolas (2018). "Simulating arbitrary Gaussian circuits with linear optics".
4025:
scattershot boson sampling model is also computationally hard, as proven by making use of the symmetry of
608:
7651:
7488:
7244:
7104:
6878:
6833:
6777:
4420:
4240:
6185:
Shen, C.; Zhang, Z.; Duan, L.-M. (2014). "Scalable implementation of boson sampling with trapped ions".
4435:
altering the couplings is thus achieved, when comparing the sampling results to an unaltered reference.
2840:
corresponding approximate task. The latter consists of sampling from a probability distribution that is
7360:
7232:
7129:
7005:
6933:
6840:
5498:
4851:
Carolan, Jacques; Harrold, Christopher; Sparrow, Chris; et al. (2015). "Universal linear optics".
4475:
4393:
4313:
3870:
2112:
modes of the interferometer is injected with a single photon. In this case the above expression reads:
94:
2760:
2624:
If a polynomial-time classical algorithm for exact boson sampling existed, then the above probability
392:
155:
7169:
7134:
7030:
6973:
2073:
1061:
486:
6246:
Peropadre, Borja; Aspuru-Guzik, Alan; Garcia-Ripoll, Juan (2015). "Spin models and boson sampling".
7342:
7315:
7291:
7254:
7045:
6978:
6913:
6898:
6868:
6791:
5636:
4443:
2935:
2828:
50:
6654:
Banchi, Leonardo; Fingerhuth, Mark; Babej, Tomas; Ing, Christopher; Arrazola, Juan Miguel (2020).
2843:
542:
7646:
7493:
7227:
7119:
7089:
6888:
4470:
4091:
3994:
3866:
1949:
42:
4680:
Szameit, Alexander; Dreisow, Felix; Pertsch, Thomas; Nolte, Stefan; TĂĽnnermann, Andreas (2007).
423:
186:
7563:
7327:
7320:
7067:
5631:
5085:
Clifford, Peter; Clifford, Raphaël (2017-06-05). "The Classical Complexity of Boson Sampling".
5032:
Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time".
4405:
4362:. The scheme allows deterministic preparation and high-efficiency readout of the corresponding
1955:
7483:
7109:
7035:
7000:
4416:
4279:
of an operational boson sampling device, and route towards its larger-scale implementations.
4228:
4108:
2734:
149:
6495:
Nikolopoulos, Georgios M. (2019). "Cryptographic one-way function based on boson sampling".
4446:
related to the corresponding open problem in computer science) by combining tools proper to
4012:
submatrix and only varied the rows, whereas now we vary the columns too, depending on which
853:
7273:
7022:
6873:
6677:
6612:
6514:
6461:
6408:
6375:
6340:
6286:
6204:
6141:
6076:
5981:
5920:
5867:
5806:
5747:
5686:
5588:
5520:
5460:
5394:
5299:
5223:
5125:
5051:
4998:
4923:
4812:
4753:
4693:
4638:
4571:
4224:
2699:
2524:
2441:
2390:
2042:
2011:
1589:
1286:{\displaystyle )\equiv x_{1}^{s_{1}}{\cdot }x_{2}^{s_{2}}{\cdot \cdot \cdot }x_{N}^{s_{N}}}
603:
6600:
5441:
Chakhmakhchyan, Levon; Cerf, Nicolas (2017). "Boson sampling with Gaussian measurements".
1031:{\displaystyle |\psi _{\text{out}}\rangle =\varphi _{M}(U)|s_{1},s_{2},...,s_{N}\rangle .}
8:
7592:
7545:
7375:
7149:
6903:
6883:
6818:
6813:
5265:
Gurvits, Leonid (2005). "On the complexity of mixed discriminants and related problems".
4458:
femtosecond laser-writing technique), and confirmed many of the theoretical predictions.
4370:
4317:
3210:
2528:
70:
46:
23:
6681:
6616:
6518:
6465:
6412:
6344:
6290:
6208:
6145:
6120:"Quantum suppression law in a 3-D photonic chip implementing the fast Fourier transform"
6080:
5985:
5924:
5871:
5810:
5751:
5690:
5592:
5524:
5464:
5398:
5303:
5227:
5129:
5055:
5002:
4927:
4816:
4757:
4697:
4642:
4575:
4415:
The task of boson sampling shares peculiar similarities with the problem of determining
4191:
4065:
3437:
124:
7308:
7154:
6956:
6893:
6700:
6667:
6655:
6636:
6561:
6530:
6504:
6477:
6451:
6424:
6398:
6356:
6330:
6302:
6276:
6247:
6228:
6194:
6162:
6131:
6119:
6100:
6066:
6037:
6016:
5997:
5971:
5944:
5910:
5883:
5857:
5822:
5796:
5763:
5737:
5710:
5676:
5649:
5604:
5578:
5544:
5510:
5476:
5450:
5415:
5384:
5372:
5323:
5289:
5247:
5213:
5186:
5168:
5141:
5115:
5086:
5067:
5041:
5014:
4988:
4939:
4913:
4886:
4860:
4828:
4802:
4769:
4743:
4662:
4628:
4595:
4561:
4243:. Therefore, picking random coherent states distributed according to the corresponding
4087:
4045:
3182:
2472:
2417:
1991:
1493:
1041:
667:
573:
522:
466:
7572:
7217:
7124:
7081:
7012:
6928:
6908:
6863:
6823:
6801:
6705:
6640:
6628:
6581:
6534:
6360:
6220:
6167:
6092:
5936:
5826:
5702:
5608:
5536:
5480:
5420:
5315:
5239:
4943:
4878:
4832:
4773:
4711:
4654:
4587:
2706:
145:
6428:
6232:
6104:
5948:
5887:
5767:
5653:
5327:
5251:
5190:
5145:
5018:
4890:
4666:
4599:
7239:
7189:
6966:
6695:
6685:
6624:
6620:
6571:
6522:
6481:
6469:
6416:
6348:
6294:
6216:
6212:
6157:
6149:
6088:
6084:
6001:
5989:
5932:
5928:
5875:
5814:
5755:
5698:
5694:
5641:
5596:
5548:
5532:
5528:
5468:
5410:
5402:
5311:
5307:
5235:
5231:
5178:
5133:
5071:
5059:
5006:
4931:
4870:
4820:
4761:
4701:
4646:
4579:
4518:
3753:
2824:
6306:
5714:
4320:, can be efficiently verified classically, as is the case for all problems in the
7365:
7303:
6993:
6988:
5346:"Scattershot BosonSampling: a new approach to scalable BosonSampling experiments"
4451:
4427:
4369:
and universal manipulation of the phonon modes through a combination of inherent
4220:
3207:
7474:
7451:
7418:
7222:
6576:
6549:
6473:
6420:
6352:
5993:
5600:
5472:
5137:
4447:
4424:
4409:
3291:
we care about, and thus will not be able to avoid its estimation. In this case
2500:
590:-dimensional unitary matrices, and unitaries acting on the exponentially large
58:
31:
27:
6746:
6741:
6736:
6731:
6526:
5759:
4935:
4523:
4506:
7635:
7296:
7114:
7040:
6585:
6298:
6036:
Aaronson, Scott; Arkhipov, Alex (2013). "BosonSampling is far from uniform".
5879:
5818:
4824:
4765:
3741:{\displaystyle |\,{\text{Perm}}\,X|<{\frac {\sqrt {M!}}{Q(M,1/\delta )}}.}
687:
Suppose the interferometer is injected with an input state of single photons
591:
5645:
5345:
4874:
4650:
4583:
7516:
7441:
6709:
6690:
6632:
6224:
6171:
6096:
5940:
5706:
5540:
5424:
5406:
5319:
5243:
5063:
5010:
4882:
4715:
4682:"Control of directional evanescent coupling in fs laser written waveguides"
4658:
4591:
4485:
4374:
3920:
PDC crystals with simultaneous laser pulses, the probability of generating
3862:
3200:
2516:
2508:
661:
567:
107:
5182:
7526:
7380:
6918:
6118:
Crespi, Andrea; Osellame, Roberto; Ramponi, Roberta; et al. (2016).
5046:
4918:
4706:
4681:
4366:
2499:
boson sampling device has to sample from, is related to the permanent of
1098:
103:
6153:
5159:
Arkhipov, Alex; Kuperberg, Greg (2012). "The bosonic birthday paradox".
7587:
7521:
7385:
6737:
Quantum Information Lab – Sapienza: video on scattershot boson sampling
4346:
4259:
Specifically, this included the implementation of boson sampling with:
4026:
38:
3365:
is proportional to the squared absolute value of the permanent of the
61:. Although the problem is well defined for any bosonic particles, its
7370:
6601:"Experimental Boson Sampling Enabling Cryptographic One-Way Function"
6266:
5105:
4965:
3526:
of i.i.d. Gaussians to within multiplicative error is a #P-hard task.
2754:
3654:{\displaystyle X\sim {\mathcal {N}}(0,1)_{\mathcal {C}}^{M\times M}}
3519:{\displaystyle X\sim {\mathcal {N}}(0,1)_{\mathcal {C}}^{M\times M}}
3427:{\displaystyle X\sim {\mathcal {N}}(0,1)_{\mathcal {C}}^{M\times M}}
7555:
7531:
7390:
7355:
6672:
6566:
6509:
6456:
6403:
6335:
6252:
6136:
5976:
5742:
5583:
5515:
5455:
5389:
5120:
5091:
4865:
4359:
4321:
455:
denotes the creation (annihilation) operators of the output modes (
117:
Specifically, suppose the linear interferometer is described by an
106:, adaptive measurements or entangling operations, as does e.g. the
6281:
6245:
6199:
6071:
6042:
6021:
5915:
5862:
5801:
5681:
5294:
5218:
5173:
4993:
4807:
4748:
4633:
4566:
3530:
Moreover, the above conjecture can be linked to the estimation of
30:
and Alex Arkhipov after the original work of Lidror Troyansky and
7582:
7199:
4438:
Variants of the boson sampling model have been used to construct
4325:
4212:
4095:
2730:
2723:
5369:
4904:
Scheel, Stefan (2008). "Permanents in linear optical networks".
4617:
4551:
3990:
sources. This setting can also be seen as a problem of sampling
1383:{\displaystyle P_{{\varphi (U)}|s_{1},s_{2},...,s_{N}\rangle }(}
7559:
7055:
6388:
6376:"Shtetl Optimized: Introducing some British people to P vs. NP"
4363:
4353:
4294:
4282:
3204:
2816:
2507:
is in the general case an extremely hard task: it falls in the
62:
5846:
2722:
Linear optics with postselected measurements is universal for
6828:
5785:
5496:
4101:
54:
35:
2737:(i.e. the probabilistic polynomial-time class): PostBQP = PP
7577:
7050:
6983:
6732:
Quantum Information Lab – Sapienza: video on boson sampling
6319:
6056:
6014:
5900:
5727:
4733:
4679:
98:
single photon-counting detectors, such as those based on
7194:
7179:
4791:
3175:
random unitary matrix. This can be done knowing that any
2715:
6653:
6550:"Computational indistinguishability and boson sampling*"
6726:
5666:
5279:
3015:
of a specific measurement outcome. Thus, if a sampler "
108:
universal optical scheme by Knill, Laflamme and Milburn
3932:
1479:{\displaystyle )=P_{|s_{1},s_{2},...,s_{N}\rangle }(U}
613:
6117:
5436:
5434:
5203:
4850:
4388:
modes is equivalent to the short-time evolution with
4194:
4120:
4068:
4048:
3930:
3881:. Then, the probability of generating simultaneously
3861:
photon generation techniques using atoms, molecules,
3778:
3671:
3602:
3536:
3467:
3440:
3375:
3297:
3223:
3185:
3103:
3025:
2947:
2866:
2846:
2763:
2630:
2553:
2475:
2444:
2420:
2393:
2121:
2076:
2045:
2014:
1994:
1958:
1915:
1626:
1592:
1518:
1496:
1400:
1299:
1194:
1107:
1064:
1044:
928:
890:
856:
803:
730:
693:
670:
611:
576:
545:
525:
489:
469:
426:
395:
227:
189:
158:
127:
16:
Restricted model of non-universal quantum computation
6441:
5621:
4978:
3661:
of the following inequality to hold is smaller than
1038:
A simple way to understand the homomorphism between
7021:
1177:{\displaystyle P_{|s_{1},s_{2},...,s_{N}\rangle }(}
463:). An interferometer characterized by some unitary
5568:
5440:
5431:
4203:
4180:
4077:
4054:
3970:
3838:
3740:
3653:
3568:
3518:
3449:
3426:
3357:
3283:
3191:
3163:
3085:
3007:
2926:
2852:
2804:
2690:
2613:
2481:
2457:
2426:
2406:
2376:
2100:
2058:
2027:
2000:
1980:
1940:
1898:
1605:
1578:
1502:
1478:
1382:
1285:
1176:
1086:
1050:
1030:
911:
872:
843:
789:
717:
676:
652:
582:
558:
531:
511:
475:
447:
413:
370:
210:
176:
136:
5365:
5363:
5361:
5359:
5158:
3971:{\displaystyle {\tbinom {N}{M}}\varepsilon ^{M}.}
922:the output of the circuit can be written down as
7633:
6656:"Molecular docking with Gaussian Boson Sampling"
6035:
5084:
4504:
3203:, is close in variation distance to a matrix of
6776:
5781:
5779:
5777:
4507:"The computational complexity of linear optics"
4250:
81:Consider a multimode linear-optical circuit of
5842:
5840:
5838:
5836:
5356:
4729:
4727:
4725:
790:{\displaystyle |s_{1},s_{2},...,s_{N}\rangle }
144:which performs a linear transformation of the
6762:
6598:
6184:
4846:
4844:
4842:
3948:
3935:
3900:-photon events. This approach has been named
3855:
643:
616:
6547:
6494:
5774:
5267:Mathematical Foundations of Computer Science
4787:
4785:
4783:
4613:
4611:
4609:
4547:
4545:
4543:
4354:Alternative implementations and applications
4295:Proposals with alternative photonic platform
4283:Implementation of scattershot boson sampling
2513:approximation to within multiplicative error
2493:
2277:
2187:
2092:
1758:
1692:
1465:
1372:
1166:
1022:
944:
906:
784:
709:
598:indistinguishable photons distributed among
5961:
5833:
5492:
5490:
5339:
5337:
4722:
912:{\displaystyle |\psi _{\text{out}}\rangle }
880:is the number of photons injected into the
718:{\displaystyle |\psi _{\text{in}}\rangle =}
6769:
6755:
4839:
4102:Classically simulable boson sampling tasks
4032:
3869:, the most widely used method remains the
296:
295:
6699:
6689:
6671:
6575:
6565:
6508:
6455:
6402:
6334:
6280:
6251:
6198:
6161:
6135:
6070:
6041:
6020:
5975:
5914:
5861:
5800:
5741:
5680:
5635:
5582:
5514:
5454:
5414:
5388:
5373:"Experimental scattershot boson sampling"
5293:
5217:
5172:
5119:
5090:
5045:
4992:
4917:
4864:
4806:
4780:
4747:
4705:
4632:
4606:
4565:
4540:
4522:
3683:
3677:
3547:
2718:. It also relies on the following facts:
2705:When combined these two facts along with
2308:
1921:
1789:
5487:
5343:
5334:
5031:
4505:Aaronson, Scott; Arkhipov, Alex (2013).
4181:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3839:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3461:Approximating the permanent of a matrix
3358:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3284:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3164:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3086:{\displaystyle p(t_{1},t_{2},...,t_{N})}
3008:{\displaystyle p(t_{1},t_{2},...,t_{N})}
2927:{\displaystyle p(t_{1},t_{2},...,t_{N})}
2691:{\displaystyle p(t_{1},t_{2},...,t_{N})}
2614:{\displaystyle p(t_{1},t_{2},...,t_{N})}
1579:{\displaystyle p(t_{1},t_{2},...,t_{N})}
100:current-biased superconducting nanowires
7287:Continuous-variable quantum information
6548:Nikolopoulos, Georgios M (2022-11-29).
5264:
3569:{\displaystyle |{\text{Perm}}\,X|^{2},}
2834:
389:) labels the input (output) modes, and
22:is a restricted model of non-universal
7634:
4903:
4090:that projects each output mode onto a
2757:for exact boson sampling runs in time
2543:and combines the following two facts:
1941:{\displaystyle {\text{Perm}}\,U_{S,T}}
1293:, and get the following result :
844:{\displaystyle \sum _{k=1}^{N}s_{k}=M}
483:naturally induces a unitary evolution
6750:
4098:, a generalization of the permanent.
4027:quantum mechanics under time reversal
3889:, which decreases exponentially with
3760:identical bosons are scattered among
3434:of i.i.d. Gaussians, smuggled inside
5163:. Proceedings of the Freedman Fest.
653:{\displaystyle {\tbinom {M+N-1}{M}}}
5344:Aaronson, Scott (8 November 2013).
5078:
3199:, randomly chosen according to the
1988:which is obtained from the unitary
13:
5161:Geometry & Topology Monographs
3939:
3634:
3611:
3499:
3476:
3407:
3384:
620:
539:-photon states. Moreover, the map
89:indistinguishable single photons (
41:to evaluate expectation values of
34:, that explored possible usage of
14:
7663:
6720:
4322:non-deterministic polynomial-time
2534:
7616:
7615:
7606:
7605:
4481:Linear optical quantum computing
4307:
2805:{\displaystyle O(n2^{n}+mn^{2})}
2511:complexity class. Moreover, its
414:{\displaystyle b_{j}^{\dagger }}
177:{\displaystyle a_{i}^{\dagger }}
67:linear optical quantum computing
6647:
6592:
6541:
6488:
6435:
6382:
6367:
6313:
6260:
6239:
6178:
6111:
6050:
6029:
6008:
5955:
5894:
5721:
5660:
5615:
5562:
5273:
5258:
5197:
5152:
5099:
5025:
4972:
4950:
4241:the optical equivalence theorem
3995:two-mode squeezed vacuum states
3904:, which consists of connecting
2541:computational complexity theory
2101:{\displaystyle |1_{M}\rangle ,}
1512:Consequently, the probability
1087:{\displaystyle \varphi _{M}(U)}
512:{\displaystyle \varphi _{M}(U)}
6742:The Qubit Lab – Boson Sampling
6625:10.1103/PhysRevLett.130.060802
6497:Quantum Information Processing
6217:10.1103/PhysRevLett.112.050504
6089:10.1103/PhysRevLett.113.020502
5933:10.1103/PhysRevLett.113.120501
5699:10.1103/PhysRevLett.114.060501
5533:10.1103/PhysRevLett.119.170501
5312:10.1103/PhysRevLett.113.100502
5236:10.1103/PhysRevLett.111.130503
4897:
4673:
4531:
4498:
4444:positive-semidefinite matrices
4175:
4124:
3833:
3782:
3729:
3709:
3688:
3673:
3629:
3616:
3553:
3538:
3494:
3481:
3402:
3389:
3352:
3301:
3278:
3227:
3158:
3107:
3080:
3029:
3002:
2951:
2921:
2870:
2799:
2767:
2685:
2634:
2608:
2557:
2547:Approximating the probability
2321:
2299:
2282:
2263:
2259:
2253:
2239:
2183:
2176:
2125:
2078:
1808:
1780:
1763:
1744:
1688:
1681:
1630:
1573:
1522:
1497:
1470:
1413:
1401:
1377:
1320:
1315:
1309:
1195:
1171:
1114:
1081:
1075:
970:
966:
960:
930:
892:
857:
732:
695:
506:
500:
442:
427:
362:
297:
218:of the circuit's input modes:
205:
190:
76:
1:
7282:Adiabatic quantum computation
4491:
3756:). The latter states that if
7333:Topological quantum computer
4251:Experimental implementations
4039:Wigner distribution function
2853:{\displaystyle \varepsilon }
2753:The best proposed classical
2505:computation of the permanent
2108:for which each of the first
559:{\displaystyle \varphi _{M}}
85:modes that is injected with
7:
7642:Quantum information science
7611:Quantum information science
6778:Quantum information science
4958:"Polynomial-time hierarchy"
4464:
3850:
3592:>0 the probability over
1617:th output mode is given as
10:
7668:
7006:quantum gate teleportation
6474:10.1103/PhysRevA.94.012315
6421:10.1103/PhysRevA.96.022329
6353:10.1103/PhysRevB.95.020502
5994:10.1103/PhysRevA.93.043803
5601:10.1103/PhysRevA.98.062314
5473:10.1103/PhysRevA.96.032326
4476:Cross-entropy benchmarking
4417:molecular vibronic spectra
4318:Shor's factoring algorithm
4314:universal quantum computer
3902:scattershot boson sampling
3871:parametric down-conversion
3856:Scattershot boson sampling
3580:There exists a polynomial
2860:close to the one given by
2825:open-source implementation
884:th mode). Then, the state
664:exists, not all values of
448:{\displaystyle (b_{j}^{})}
211:{\displaystyle (a_{i}^{})}
95:parametric down-conversion
7601:
7544:
7507:
7473:
7450:
7417:
7408:
7341:
7270:
7208:
7168:
7135:Quantum Fourier transform
7080:
7031:Post-quantum cryptography
6974:Entanglement distillation
6947:
6856:
6784:
6527:10.1007/s11128-019-2372-9
5760:10.1103/PhysRevX.6.021039
5499:"Gaussian Boson Sampling"
4936:10.2478/v10155-010-0092-x
4524:10.4086/toc.2013.v009a004
3924:photons will be given as
3867:color centers in diamonds
3772:Although the probability
3211:random Gaussian variables
2494:Complexity of the problem
7621:Quantum mechanics topics
7316:Quantum machine learning
7292:One-way quantum computer
7145:Quantum phase estimation
7046:Quantum key distribution
6979:Monogamy of entanglement
6577:10.1088/1402-4896/aca1ed
6373:See open problem (4) at
6299:10.1038/NPHOTON.2015.153
5880:10.1038/nphoton.2014.152
5819:10.1038/nphoton.2014.135
5138:10.1088/1367-2630/aa60ed
4825:10.1038/nphoton.2013.112
4766:10.1038/nphoton.2013.102
4452:computational complexity
3097:" the above probability
2936:total variation distance
1981:{\displaystyle U_{S,T},}
1909:In the above expression
1094:is the following :
660:(notice that since this
51:probability distribution
45:. The model consists of
7228:Randomized benchmarking
7090:Amplitude amplification
6605:Physical Review Letters
5646:10.1145/1008731.1008738
5503:Physical Review Letters
4875:10.1126/science.aab3642
4651:10.1126/science.1231692
4584:10.1126/science.1231440
4471:Quantum random circuits
4092:squeezed coherent state
4033:Gaussian boson sampling
3179:submatrix of a unitary
7328:Quantum Turing machine
7321:quantum neural network
7068:Quantum secret sharing
6691:10.1126/sciadv.aax1950
5407:10.1126/sciadv.1400255
5064:10.1098/rspa.2005.1546
5011:10.1098/rspa.2010.0301
4421:Franck–Condon profiles
4406:superconducting qubits
4316:running, for example,
4205:
4182:
4079:
4056:
3972:
3840:
3742:
3655:
3570:
3520:
3451:
3428:
3359:
3285:
3193:
3165:
3087:
3009:
2928:
2854:
2806:
2692:
2615:
2483:
2459:
2438:columns and repeating
2428:
2408:
2378:
2102:
2060:
2029:
2002:
1982:
1942:
1900:
1607:
1580:
1504:
1480:
1384:
1287:
1178:
1101:for the basis states:
1088:
1052:
1032:
913:
874:
873:{\displaystyle (s_{k}}
845:
824:
791:
719:
678:
654:
602:modes is given by the
584:
560:
533:
513:
477:
449:
415:
372:
333:
266:
212:
178:
138:
57:scattered by a linear
43:permanents of matrices
7400:Entanglement-assisted
7361:quantum convolutional
7036:Quantum coin flipping
7001:Quantum teleportation
6962:entanglement-assisted
6792:DiVincenzo's criteria
6124:Nature Communications
5183:10.2140/gtm.2012.18.1
4968:on February 14, 2014.
4408:and specifically the
4229:matrix multiplication
4206:
4183:
4080:
4057:
3973:
3841:
3743:
3656:
3571:
3521:
3452:
3429:
3360:
3286:
3194:
3166:
3088:
3010:
2929:
2855:
2807:
2693:
2616:
2484:
2460:
2458:{\displaystyle t_{j}}
2434:by keeping its first
2429:
2409:
2407:{\displaystyle U_{T}}
2379:
2103:
2061:
2059:{\displaystyle t_{j}}
2030:
2028:{\displaystyle s_{i}}
2003:
1983:
1943:
1901:
1608:
1606:{\displaystyle t_{k}}
1581:
1505:
1481:
1385:
1288:
1179:
1089:
1053:
1033:
914:
875:
846:
804:
792:
720:
679:
655:
585:
561:
534:
514:
478:
450:
416:
373:
313:
246:
213:
179:
139:
7211:processor benchmarks
7140:Quantum optimization
7023:Quantum cryptography
6834:physical vs. logical
4906:Acta Physica Slovaca
4707:10.1364/OE.15.001579
4225:quantum entanglement
4192:
4118:
4066:
4046:
4042:Haar-random unitary
3928:
3776:
3669:
3600:
3534:
3465:
3438:
3373:
3295:
3221:
3183:
3101:
3023:
2945:
2864:
2844:
2835:Approximate sampling
2761:
2700:polynomial hierarchy
2628:
2551:
2525:polynomial hierarchy
2473:
2442:
2418:
2391:
2119:
2074:
2043:
2012:
1992:
1956:
1913:
1624:
1590:
1516:
1494:
1398:
1297:
1192:
1105:
1062:
1042:
926:
888:
854:
801:
728:
691:
668:
609:
604:binomial coefficient
574:
543:
523:
487:
467:
424:
393:
225:
187:
156:
125:
6924:Quantum speed limit
6819:Quantum programming
6814:Quantum information
6682:2020SciA....6.1950B
6617:2023PhRvL.130f0802W
6519:2019QuIP...18..259N
6466:2016PhRvA..94a2315N
6413:2017PhRvA..96b2329C
6345:2017PhRvB..95b0502G
6291:2015NaPho...9..615H
6209:2014PhRvL.112e0504S
6154:10.1038/ncomms10469
6146:2015arXiv150800782C
6081:2014PhRvL.113b0502T
5986:2016PhRvA..93d3803P
5925:2014PhRvL.113l0501M
5872:2014NaPho...8..621C
5811:2014NaPho...8..615S
5752:2016PhRvX...6b1039R
5691:2015PhRvL.114f0501R
5593:2018PhRvA..98f2314C
5525:2017PhRvL.119q0501H
5465:2017PhRvA..96c2326C
5399:2015SciA....1E0255B
5304:2014PhRvL.113j0502L
5228:2013PhRvL.111m0503S
5130:2017NJPh...19c3007R
5056:2005RSPSA.461.3473A
5040:(2063): 3473–3482.
5003:2011RSPSA.467..459B
4928:2004quant.ph..6127S
4817:2013NaPho...7..545C
4758:2013NaPho...7..540T
4698:2007OExpr..15.1579S
4643:2013Sci...339..798S
4576:2013Sci...339..794B
4511:Theory of Computing
4392:excitations in the
4371:Coulomb interaction
3650:
3515:
3423:
1282:
1249:
1222:
441:
410:
351:
294:
242:
204:
173:
71:quantum computation
24:quantum computation
7652:Quantum algorithms
7573:Forest/Rigetti QCS
7309:quantum logic gate
7095:Bernstein–Vazirani
7082:Quantum algorithms
6957:Classical capacity
6841:Quantum processors
6824:Quantum simulation
5624:Journal of the ACM
4233:Glauber-Sudarshan
4204:{\displaystyle U.}
4201:
4178:
4088:homodyne detection
4078:{\displaystyle U.}
4075:
4052:
3968:
3953:
3885:single photons is
3836:
3738:
3651:
3628:
3584:such that for any
3566:
3516:
3493:
3450:{\displaystyle U.}
3447:
3424:
3401:
3355:
3281:
3189:
3161:
3083:
3005:
2934:, in terms of the
2924:
2850:
2812:for a system with
2802:
2688:
2611:
2479:
2455:
2424:
2404:
2374:
2098:
2056:
2025:
1998:
1978:
1938:
1896:
1603:
1576:
1500:
1476:
1380:
1283:
1261:
1228:
1201:
1174:
1084:
1048:
1028:
909:
870:
841:
787:
715:
674:
650:
648:
580:
556:
529:
509:
473:
445:
430:
411:
396:
368:
334:
280:
228:
208:
193:
174:
159:
137:{\displaystyle U,}
134:
73:in the near term.
7629:
7628:
7540:
7539:
7437:Linear optical QC
7218:Quantum supremacy
7172:complexity theory
7125:Quantum annealing
7076:
7075:
7013:Superdense coding
6802:Quantum computing
6444:Physical Review A
5964:Physical Review A
5730:Physical Review X
4987:(2126): 459–472.
4859:(6249): 711–716.
4627:(6121): 798–801.
4560:(6121): 794–798.
4055:{\displaystyle U}
3946:
3733:
3704:
3681:
3545:
3192:{\displaystyle U}
2733:is equivalent to
2519:problem as well.
2482:{\displaystyle U}
2427:{\displaystyle U}
2414:is obtained from
2387:where the matrix
2369:
2306:
2001:{\displaystyle U}
1919:
1891:
1787:
1755:
1503:{\displaystyle )}
1051:{\displaystyle U}
941:
903:
706:
677:{\displaystyle W}
641:
583:{\displaystyle N}
532:{\displaystyle M}
476:{\displaystyle U}
7659:
7619:
7618:
7609:
7608:
7415:
7414:
7345:error correction
7274:computing models
7240:Relaxation times
7130:Quantum counting
7019:
7018:
6967:quantum capacity
6914:No-teleportation
6899:No-communication
6771:
6764:
6757:
6748:
6747:
6714:
6713:
6703:
6693:
6675:
6666:(23): eaax1950.
6660:Science Advances
6651:
6645:
6644:
6596:
6590:
6589:
6579:
6569:
6545:
6539:
6538:
6512:
6492:
6486:
6485:
6459:
6439:
6433:
6432:
6406:
6386:
6380:
6379:
6371:
6365:
6364:
6338:
6317:
6311:
6310:
6284:
6269:Nature Photonics
6264:
6258:
6257:
6255:
6243:
6237:
6236:
6202:
6182:
6176:
6175:
6165:
6139:
6115:
6109:
6108:
6074:
6054:
6048:
6047:
6045:
6033:
6027:
6026:
6024:
6012:
6006:
6005:
5979:
5959:
5953:
5952:
5918:
5898:
5892:
5891:
5865:
5850:Nature Photonics
5844:
5831:
5830:
5804:
5789:Nature Photonics
5783:
5772:
5771:
5745:
5725:
5719:
5718:
5684:
5664:
5658:
5657:
5639:
5619:
5613:
5612:
5586:
5566:
5560:
5559:
5557:
5555:
5518:
5494:
5485:
5484:
5458:
5438:
5429:
5428:
5418:
5392:
5377:Science Advances
5367:
5354:
5353:
5350:Shtetl-Optimized
5341:
5332:
5331:
5297:
5277:
5271:
5270:
5262:
5256:
5255:
5221:
5201:
5195:
5194:
5176:
5156:
5150:
5149:
5123:
5103:
5097:
5096:
5094:
5082:
5076:
5075:
5049:
5047:quant-ph/0412187
5029:
5023:
5022:
4996:
4976:
4970:
4969:
4964:. Archived from
4954:
4948:
4947:
4921:
4919:quant-ph/0406127
4901:
4895:
4894:
4868:
4848:
4837:
4836:
4810:
4795:Nature Photonics
4789:
4778:
4777:
4751:
4736:Nature Photonics
4731:
4720:
4719:
4709:
4692:(4): 1579–1587.
4677:
4671:
4670:
4636:
4615:
4604:
4603:
4569:
4549:
4538:
4535:
4529:
4528:
4526:
4502:
4312:The output of a
4210:
4208:
4207:
4202:
4187:
4185:
4184:
4179:
4174:
4173:
4149:
4148:
4136:
4135:
4084:
4082:
4081:
4076:
4061:
4059:
4058:
4053:
3978:Therefore, for
3977:
3975:
3974:
3969:
3964:
3963:
3954:
3952:
3951:
3938:
3845:
3843:
3842:
3837:
3832:
3831:
3807:
3806:
3794:
3793:
3754:birthday paradox
3747:
3745:
3744:
3739:
3734:
3732:
3725:
3697:
3696:
3691:
3682:
3679:
3676:
3660:
3658:
3657:
3652:
3649:
3638:
3637:
3615:
3614:
3575:
3573:
3572:
3567:
3562:
3561:
3556:
3546:
3543:
3541:
3525:
3523:
3522:
3517:
3514:
3503:
3502:
3480:
3479:
3456:
3454:
3453:
3448:
3433:
3431:
3430:
3425:
3422:
3411:
3410:
3388:
3387:
3364:
3362:
3361:
3356:
3351:
3350:
3326:
3325:
3313:
3312:
3290:
3288:
3287:
3282:
3277:
3276:
3252:
3251:
3239:
3238:
3213:, provided that
3198:
3196:
3195:
3190:
3170:
3168:
3167:
3162:
3157:
3156:
3132:
3131:
3119:
3118:
3092:
3090:
3089:
3084:
3079:
3078:
3054:
3053:
3041:
3040:
3014:
3012:
3011:
3006:
3001:
3000:
2976:
2975:
2963:
2962:
2933:
2931:
2930:
2925:
2920:
2919:
2895:
2894:
2882:
2881:
2859:
2857:
2856:
2851:
2811:
2809:
2808:
2803:
2798:
2797:
2782:
2781:
2697:
2695:
2694:
2689:
2684:
2683:
2659:
2658:
2646:
2645:
2620:
2618:
2617:
2612:
2607:
2606:
2582:
2581:
2569:
2568:
2488:
2486:
2485:
2480:
2464:
2462:
2461:
2456:
2454:
2453:
2433:
2431:
2430:
2425:
2413:
2411:
2410:
2405:
2403:
2402:
2383:
2381:
2380:
2375:
2370:
2368:
2364:
2363:
2342:
2341:
2331:
2330:
2329:
2324:
2318:
2317:
2307:
2304:
2302:
2296:
2291:
2290:
2285:
2276:
2275:
2266:
2252:
2251:
2242:
2237:
2236:
2212:
2211:
2199:
2198:
2186:
2175:
2174:
2150:
2149:
2137:
2136:
2107:
2105:
2104:
2099:
2091:
2090:
2081:
2065:
2063:
2062:
2057:
2055:
2054:
2034:
2032:
2031:
2026:
2024:
2023:
2007:
2005:
2004:
1999:
1987:
1985:
1984:
1979:
1974:
1973:
1947:
1945:
1944:
1939:
1937:
1936:
1920:
1917:
1905:
1903:
1902:
1897:
1892:
1890:
1886:
1885:
1864:
1863:
1851:
1850:
1829:
1828:
1818:
1817:
1816:
1811:
1805:
1804:
1788:
1785:
1783:
1777:
1772:
1771:
1766:
1757:
1756:
1753:
1747:
1742:
1741:
1717:
1716:
1704:
1703:
1691:
1680:
1679:
1655:
1654:
1642:
1641:
1612:
1610:
1609:
1604:
1602:
1601:
1585:
1583:
1582:
1577:
1572:
1571:
1547:
1546:
1534:
1533:
1509:
1507:
1506:
1501:
1490:
1485:
1483:
1482:
1477:
1469:
1468:
1464:
1463:
1439:
1438:
1426:
1425:
1416:
1394:
1389:
1387:
1386:
1381:
1376:
1375:
1371:
1370:
1346:
1345:
1333:
1332:
1323:
1318:
1292:
1290:
1289:
1284:
1281:
1280:
1279:
1269:
1260:
1248:
1247:
1246:
1236:
1227:
1221:
1220:
1219:
1209:
1188:
1183:
1181:
1180:
1175:
1170:
1169:
1165:
1164:
1140:
1139:
1127:
1126:
1117:
1093:
1091:
1090:
1085:
1074:
1073:
1057:
1055:
1054:
1049:
1037:
1035:
1034:
1029:
1021:
1020:
996:
995:
983:
982:
973:
959:
958:
943:
942:
939:
933:
918:
916:
915:
910:
905:
904:
901:
895:
879:
877:
876:
871:
869:
868:
850:
848:
847:
842:
834:
833:
823:
818:
796:
794:
793:
788:
783:
782:
758:
757:
745:
744:
735:
724:
722:
721:
716:
708:
707:
704:
698:
683:
681:
680:
675:
659:
657:
656:
651:
649:
647:
646:
637:
619:
589:
587:
586:
581:
565:
563:
562:
557:
555:
554:
538:
536:
535:
530:
518:
516:
515:
510:
499:
498:
482:
480:
479:
474:
454:
452:
451:
446:
440:
438:
420:
418:
417:
412:
409:
404:
377:
375:
374:
369:
361:
360:
350:
345:
332:
327:
309:
308:
293:
288:
279:
278:
265:
260:
241:
236:
217:
215:
214:
209:
203:
201:
183:
181:
180:
175:
172:
167:
143:
141:
140:
135:
7667:
7666:
7662:
7661:
7660:
7658:
7657:
7656:
7632:
7631:
7630:
7625:
7597:
7547:
7536:
7509:Superconducting
7503:
7469:
7460:Neutral atom QC
7452:Ultracold atoms
7446:
7411:implementations
7410:
7404:
7344:
7337:
7304:Quantum circuit
7272:
7266:
7260:
7250:
7210:
7204:
7171:
7164:
7120:Hidden subgroup
7072:
7061:other protocols
7017:
6994:quantum network
6989:Quantum channel
6949:
6943:
6889:No-broadcasting
6879:Gottesman–Knill
6852:
6780:
6775:
6723:
6718:
6717:
6652:
6648:
6597:
6593:
6554:Physica Scripta
6546:
6542:
6493:
6489:
6440:
6436:
6387:
6383:
6378:. 22 July 2015.
6374:
6372:
6368:
6318:
6314:
6265:
6261:
6244:
6240:
6187:Phys. Rev. Lett
6183:
6179:
6116:
6112:
6059:Phys. Rev. Lett
6055:
6051:
6034:
6030:
6013:
6009:
5960:
5956:
5903:Phys. Rev. Lett
5899:
5895:
5845:
5834:
5784:
5775:
5726:
5722:
5669:Phys. Rev. Lett
5665:
5661:
5620:
5616:
5567:
5563:
5553:
5551:
5495:
5488:
5439:
5432:
5383:(3): e1400255.
5368:
5357:
5342:
5335:
5282:Phys. Rev. Lett
5278:
5274:
5263:
5259:
5206:Phys. Rev. Lett
5202:
5198:
5157:
5153:
5104:
5100:
5083:
5079:
5034:Proc. R. Soc. A
5030:
5026:
4981:Proc. R. Soc. A
4977:
4973:
4956:
4955:
4951:
4902:
4898:
4849:
4840:
4790:
4781:
4732:
4723:
4678:
4674:
4616:
4607:
4550:
4541:
4536:
4532:
4503:
4499:
4494:
4467:
4373:and individual
4356:
4310:
4297:
4285:
4253:
4221:coherent states
4193:
4190:
4189:
4169:
4165:
4144:
4140:
4131:
4127:
4119:
4116:
4115:
4104:
4067:
4064:
4063:
4047:
4044:
4043:
4035:
3997:generated from
3959:
3955:
3947:
3934:
3933:
3931:
3929:
3926:
3925:
3858:
3853:
3827:
3823:
3802:
3798:
3789:
3785:
3777:
3774:
3773:
3721:
3705:
3695:
3687:
3678:
3672:
3670:
3667:
3666:
3639:
3633:
3632:
3610:
3609:
3601:
3598:
3597:
3557:
3552:
3551:
3542:
3537:
3535:
3532:
3531:
3504:
3498:
3497:
3475:
3474:
3466:
3463:
3462:
3439:
3436:
3435:
3412:
3406:
3405:
3383:
3382:
3374:
3371:
3370:
3346:
3342:
3321:
3317:
3308:
3304:
3296:
3293:
3292:
3272:
3268:
3247:
3243:
3234:
3230:
3222:
3219:
3218:
3184:
3181:
3180:
3152:
3148:
3127:
3123:
3114:
3110:
3102:
3099:
3098:
3074:
3070:
3049:
3045:
3036:
3032:
3024:
3021:
3020:
2996:
2992:
2971:
2967:
2958:
2954:
2946:
2943:
2942:
2915:
2911:
2890:
2886:
2877:
2873:
2865:
2862:
2861:
2845:
2842:
2841:
2837:
2793:
2789:
2777:
2773:
2762:
2759:
2758:
2744:
2679:
2675:
2654:
2650:
2641:
2637:
2629:
2626:
2625:
2602:
2598:
2577:
2573:
2564:
2560:
2552:
2549:
2548:
2537:
2496:
2474:
2471:
2470:
2449:
2445:
2443:
2440:
2439:
2419:
2416:
2415:
2398:
2394:
2392:
2389:
2388:
2359:
2355:
2337:
2333:
2332:
2325:
2320:
2319:
2313:
2309:
2303:
2298:
2297:
2295:
2286:
2281:
2280:
2271:
2267:
2262:
2247:
2243:
2238:
2232:
2228:
2207:
2203:
2194:
2190:
2182:
2170:
2166:
2145:
2141:
2132:
2128:
2120:
2117:
2116:
2086:
2082:
2077:
2075:
2072:
2071:
2050:
2046:
2044:
2041:
2040:
2019:
2015:
2013:
2010:
2009:
1993:
1990:
1989:
1963:
1959:
1957:
1954:
1953:
1948:stands for the
1926:
1922:
1916:
1914:
1911:
1910:
1881:
1877:
1859:
1855:
1846:
1842:
1824:
1820:
1819:
1812:
1807:
1806:
1794:
1790:
1784:
1779:
1778:
1776:
1767:
1762:
1761:
1752:
1748:
1743:
1737:
1733:
1712:
1708:
1699:
1695:
1687:
1675:
1671:
1650:
1646:
1637:
1633:
1625:
1622:
1621:
1613:photons at the
1597:
1593:
1591:
1588:
1587:
1567:
1563:
1542:
1538:
1529:
1525:
1517:
1514:
1513:
1495:
1492:
1491:
1486:
1459:
1455:
1434:
1430:
1421:
1417:
1412:
1411:
1407:
1399:
1396:
1395:
1390:
1366:
1362:
1341:
1337:
1328:
1324:
1319:
1305:
1304:
1300:
1298:
1295:
1294:
1275:
1271:
1270:
1265:
1250:
1242:
1238:
1237:
1232:
1223:
1215:
1211:
1210:
1205:
1193:
1190:
1189:
1184:
1160:
1156:
1135:
1131:
1122:
1118:
1113:
1112:
1108:
1106:
1103:
1102:
1069:
1065:
1063:
1060:
1059:
1043:
1040:
1039:
1016:
1012:
991:
987:
978:
974:
969:
954:
950:
938:
934:
929:
927:
924:
923:
900:
896:
891:
889:
886:
885:
864:
860:
855:
852:
851:
829:
825:
819:
808:
802:
799:
798:
778:
774:
753:
749:
740:
736:
731:
729:
726:
725:
703:
699:
694:
692:
689:
688:
684:are possible).
669:
666:
665:
642:
621:
615:
614:
612:
610:
607:
606:
575:
572:
571:
550:
546:
544:
541:
540:
524:
521:
520:
494:
490:
488:
485:
484:
468:
465:
464:
439:
434:
425:
422:
421:
405:
400:
394:
391:
390:
356:
352:
346:
338:
328:
317:
304:
300:
289:
284:
271:
267:
261:
250:
237:
232:
226:
223:
222:
202:
197:
188:
185:
184:
168:
163:
157:
154:
153:
126:
123:
122:
121:unitary matrix
79:
17:
12:
11:
5:
7665:
7655:
7654:
7649:
7647:Quantum optics
7644:
7627:
7626:
7624:
7623:
7613:
7602:
7599:
7598:
7596:
7595:
7593:many others...
7590:
7585:
7580:
7575:
7566:
7552:
7550:
7542:
7541:
7538:
7537:
7535:
7534:
7529:
7524:
7519:
7513:
7511:
7505:
7504:
7502:
7501:
7496:
7491:
7486:
7480:
7478:
7471:
7470:
7468:
7467:
7465:Trapped-ion QC
7462:
7456:
7454:
7448:
7447:
7445:
7444:
7439:
7434:
7429:
7423:
7421:
7419:Quantum optics
7412:
7406:
7405:
7403:
7402:
7397:
7396:
7395:
7388:
7383:
7378:
7373:
7368:
7363:
7358:
7349:
7347:
7339:
7338:
7336:
7335:
7330:
7325:
7324:
7323:
7313:
7312:
7311:
7301:
7300:
7299:
7289:
7284:
7278:
7276:
7268:
7267:
7265:
7264:
7263:
7262:
7258:
7252:
7248:
7237:
7236:
7235:
7225:
7223:Quantum volume
7220:
7214:
7212:
7206:
7205:
7203:
7202:
7197:
7192:
7187:
7182:
7176:
7174:
7166:
7165:
7163:
7162:
7157:
7152:
7147:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7107:
7102:
7100:Boson sampling
7097:
7092:
7086:
7084:
7078:
7077:
7074:
7073:
7071:
7070:
7065:
7064:
7063:
7058:
7053:
7043:
7038:
7033:
7027:
7025:
7016:
7015:
7010:
7009:
7008:
6998:
6997:
6996:
6986:
6981:
6976:
6971:
6970:
6969:
6964:
6953:
6951:
6945:
6944:
6942:
6941:
6936:
6934:Solovay–Kitaev
6931:
6926:
6921:
6916:
6911:
6906:
6901:
6896:
6891:
6886:
6881:
6876:
6871:
6866:
6860:
6858:
6854:
6853:
6851:
6850:
6849:
6848:
6838:
6837:
6836:
6826:
6821:
6816:
6811:
6810:
6809:
6799:
6794:
6788:
6786:
6782:
6781:
6774:
6773:
6766:
6759:
6751:
6745:
6744:
6739:
6734:
6729:
6727:QUCHIP project
6722:
6721:External links
6719:
6716:
6715:
6646:
6591:
6540:
6487:
6434:
6381:
6366:
6312:
6275:(9): 615–620.
6259:
6238:
6177:
6110:
6049:
6028:
6007:
5954:
5909:(12): 120501.
5893:
5856:(8): 621–626.
5832:
5795:(8): 615–620.
5773:
5720:
5659:
5637:10.1.1.18.9466
5630:(4): 671–697.
5614:
5561:
5509:(17): 170501.
5486:
5430:
5355:
5333:
5288:(10): 100502.
5272:
5257:
5212:(13): 130503.
5196:
5151:
5098:
5077:
5024:
4971:
4962:Complexity Zoo
4949:
4896:
4838:
4801:(7): 545–549.
4779:
4742:(7): 540–544.
4721:
4686:Optics Express
4672:
4605:
4539:
4530:
4496:
4495:
4493:
4490:
4489:
4488:
4483:
4478:
4473:
4466:
4463:
4448:quantum optics
4410:D-Wave machine
4355:
4352:
4347:Fourier matrix
4309:
4306:
4296:
4293:
4284:
4281:
4276:
4275:
4271:
4268:
4265:
4252:
4249:
4200:
4197:
4177:
4172:
4168:
4164:
4161:
4158:
4155:
4152:
4147:
4143:
4139:
4134:
4130:
4126:
4123:
4103:
4100:
4074:
4071:
4051:
4034:
4031:
3967:
3962:
3958:
3950:
3945:
3942:
3937:
3857:
3854:
3852:
3849:
3835:
3830:
3826:
3822:
3819:
3816:
3813:
3810:
3805:
3801:
3797:
3792:
3788:
3784:
3781:
3749:
3748:
3737:
3731:
3728:
3724:
3720:
3717:
3714:
3711:
3708:
3703:
3700:
3694:
3690:
3686:
3675:
3648:
3645:
3642:
3636:
3631:
3627:
3624:
3621:
3618:
3613:
3608:
3605:
3565:
3560:
3555:
3550:
3540:
3528:
3527:
3513:
3510:
3507:
3501:
3496:
3492:
3489:
3486:
3483:
3478:
3473:
3470:
3446:
3443:
3421:
3418:
3415:
3409:
3404:
3400:
3397:
3394:
3391:
3386:
3381:
3378:
3354:
3349:
3345:
3341:
3338:
3335:
3332:
3329:
3324:
3320:
3316:
3311:
3307:
3303:
3300:
3280:
3275:
3271:
3267:
3264:
3261:
3258:
3255:
3250:
3246:
3242:
3237:
3233:
3229:
3226:
3188:
3160:
3155:
3151:
3147:
3144:
3141:
3138:
3135:
3130:
3126:
3122:
3117:
3113:
3109:
3106:
3082:
3077:
3073:
3069:
3066:
3063:
3060:
3057:
3052:
3048:
3044:
3039:
3035:
3031:
3028:
3004:
2999:
2995:
2991:
2988:
2985:
2982:
2979:
2974:
2970:
2966:
2961:
2957:
2953:
2950:
2923:
2918:
2914:
2910:
2907:
2904:
2901:
2898:
2893:
2889:
2885:
2880:
2876:
2872:
2869:
2849:
2836:
2833:
2801:
2796:
2792:
2788:
2785:
2780:
2776:
2772:
2769:
2766:
2747:
2746:
2742:
2738:
2727:
2707:Toda's theorem
2703:
2702:
2687:
2682:
2678:
2674:
2671:
2668:
2665:
2662:
2657:
2653:
2649:
2644:
2640:
2636:
2633:
2622:
2610:
2605:
2601:
2597:
2594:
2591:
2588:
2585:
2580:
2576:
2572:
2567:
2563:
2559:
2556:
2536:
2535:Exact sampling
2533:
2503:matrices. The
2495:
2492:
2478:
2452:
2448:
2423:
2401:
2397:
2385:
2384:
2373:
2367:
2362:
2358:
2354:
2351:
2348:
2345:
2340:
2336:
2328:
2323:
2316:
2312:
2301:
2294:
2289:
2284:
2279:
2274:
2270:
2265:
2261:
2258:
2255:
2250:
2246:
2241:
2235:
2231:
2227:
2224:
2221:
2218:
2215:
2210:
2206:
2202:
2197:
2193:
2189:
2185:
2181:
2178:
2173:
2169:
2165:
2162:
2159:
2156:
2153:
2148:
2144:
2140:
2135:
2131:
2127:
2124:
2097:
2094:
2089:
2085:
2080:
2053:
2049:
2039:th column and
2022:
2018:
1997:
1977:
1972:
1969:
1966:
1962:
1952:of the matrix
1935:
1932:
1929:
1925:
1907:
1906:
1895:
1889:
1884:
1880:
1876:
1873:
1870:
1867:
1862:
1858:
1854:
1849:
1845:
1841:
1838:
1835:
1832:
1827:
1823:
1815:
1810:
1803:
1800:
1797:
1793:
1782:
1775:
1770:
1765:
1760:
1751:
1746:
1740:
1736:
1732:
1729:
1726:
1723:
1720:
1715:
1711:
1707:
1702:
1698:
1694:
1690:
1686:
1683:
1678:
1674:
1670:
1667:
1664:
1661:
1658:
1653:
1649:
1645:
1640:
1636:
1632:
1629:
1600:
1596:
1575:
1570:
1566:
1562:
1559:
1556:
1553:
1550:
1545:
1541:
1537:
1532:
1528:
1524:
1521:
1499:
1475:
1472:
1467:
1462:
1458:
1454:
1451:
1448:
1445:
1442:
1437:
1433:
1429:
1424:
1420:
1415:
1410:
1406:
1403:
1379:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1349:
1344:
1340:
1336:
1331:
1327:
1322:
1317:
1314:
1311:
1308:
1303:
1278:
1274:
1268:
1264:
1259:
1256:
1253:
1245:
1241:
1235:
1231:
1226:
1218:
1214:
1208:
1204:
1200:
1197:
1173:
1168:
1163:
1159:
1155:
1152:
1149:
1146:
1143:
1138:
1134:
1130:
1125:
1121:
1116:
1111:
1097:We define the
1083:
1080:
1077:
1072:
1068:
1047:
1027:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
999:
994:
990:
986:
981:
977:
972:
968:
965:
962:
957:
953:
949:
946:
937:
932:
908:
899:
894:
867:
863:
859:
840:
837:
832:
828:
822:
817:
814:
811:
807:
786:
781:
777:
773:
770:
767:
764:
761:
756:
752:
748:
743:
739:
734:
714:
711:
702:
697:
673:
645:
640:
636:
633:
630:
627:
624:
618:
579:
553:
549:
528:
508:
505:
502:
497:
493:
472:
444:
437:
433:
429:
408:
403:
399:
379:
378:
367:
364:
359:
355:
349:
344:
341:
337:
331:
326:
323:
320:
316:
312:
307:
303:
299:
292:
287:
283:
277:
274:
270:
264:
259:
256:
253:
249:
245:
240:
235:
231:
207:
200:
196:
192:
171:
166:
162:
133:
130:
78:
75:
59:interferometer
32:Naftali Tishby
28:Scott Aaronson
26:introduced by
20:Boson sampling
15:
9:
6:
4:
3:
2:
7664:
7653:
7650:
7648:
7645:
7643:
7640:
7639:
7637:
7622:
7614:
7612:
7604:
7603:
7600:
7594:
7591:
7589:
7586:
7584:
7581:
7579:
7576:
7574:
7570:
7567:
7565:
7561:
7557:
7554:
7553:
7551:
7549:
7543:
7533:
7530:
7528:
7525:
7523:
7520:
7518:
7515:
7514:
7512:
7510:
7506:
7500:
7497:
7495:
7492:
7490:
7489:Spin qubit QC
7487:
7485:
7482:
7481:
7479:
7476:
7472:
7466:
7463:
7461:
7458:
7457:
7455:
7453:
7449:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7424:
7422:
7420:
7416:
7413:
7407:
7401:
7398:
7394:
7393:
7389:
7387:
7384:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7353:
7351:
7350:
7348:
7346:
7340:
7334:
7331:
7329:
7326:
7322:
7319:
7318:
7317:
7314:
7310:
7307:
7306:
7305:
7302:
7298:
7297:cluster state
7295:
7294:
7293:
7290:
7288:
7285:
7283:
7280:
7279:
7277:
7275:
7269:
7261:
7257:
7253:
7251:
7247:
7243:
7242:
7241:
7238:
7234:
7231:
7230:
7229:
7226:
7224:
7221:
7219:
7216:
7215:
7213:
7207:
7201:
7198:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7178:
7177:
7175:
7173:
7167:
7161:
7158:
7156:
7153:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7105:Deutsch–Jozsa
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7087:
7085:
7083:
7079:
7069:
7066:
7062:
7059:
7057:
7054:
7052:
7049:
7048:
7047:
7044:
7042:
7041:Quantum money
7039:
7037:
7034:
7032:
7029:
7028:
7026:
7024:
7020:
7014:
7011:
7007:
7004:
7003:
7002:
6999:
6995:
6992:
6991:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6968:
6965:
6963:
6960:
6959:
6958:
6955:
6954:
6952:
6950:communication
6946:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6870:
6867:
6865:
6862:
6861:
6859:
6855:
6847:
6844:
6843:
6842:
6839:
6835:
6832:
6831:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6808:
6805:
6804:
6803:
6800:
6798:
6795:
6793:
6790:
6789:
6787:
6783:
6779:
6772:
6767:
6765:
6760:
6758:
6753:
6752:
6749:
6743:
6740:
6738:
6735:
6733:
6730:
6728:
6725:
6724:
6711:
6707:
6702:
6697:
6692:
6687:
6683:
6679:
6674:
6669:
6665:
6661:
6657:
6650:
6642:
6638:
6634:
6630:
6626:
6622:
6618:
6614:
6611:(6): 060802.
6610:
6606:
6602:
6595:
6587:
6583:
6578:
6573:
6568:
6563:
6560:(1): 014001.
6559:
6555:
6551:
6544:
6536:
6532:
6528:
6524:
6520:
6516:
6511:
6506:
6502:
6498:
6491:
6483:
6479:
6475:
6471:
6467:
6463:
6458:
6453:
6450:(1): 012315.
6449:
6445:
6438:
6430:
6426:
6422:
6418:
6414:
6410:
6405:
6400:
6397:(2): 022329.
6396:
6392:
6385:
6377:
6370:
6362:
6358:
6354:
6350:
6346:
6342:
6337:
6332:
6329:(2): 020502.
6328:
6324:
6316:
6308:
6304:
6300:
6296:
6292:
6288:
6283:
6278:
6274:
6270:
6263:
6254:
6249:
6242:
6234:
6230:
6226:
6222:
6218:
6214:
6210:
6206:
6201:
6196:
6193:(5): 050504.
6192:
6188:
6181:
6173:
6169:
6164:
6159:
6155:
6151:
6147:
6143:
6138:
6133:
6129:
6125:
6121:
6114:
6106:
6102:
6098:
6094:
6090:
6086:
6082:
6078:
6073:
6068:
6065:(2): 020502.
6064:
6060:
6053:
6044:
6039:
6032:
6023:
6018:
6011:
6003:
5999:
5995:
5991:
5987:
5983:
5978:
5973:
5970:(4): 043803.
5969:
5965:
5958:
5950:
5946:
5942:
5938:
5934:
5930:
5926:
5922:
5917:
5912:
5908:
5904:
5897:
5889:
5885:
5881:
5877:
5873:
5869:
5864:
5859:
5855:
5851:
5843:
5841:
5839:
5837:
5828:
5824:
5820:
5816:
5812:
5808:
5803:
5798:
5794:
5790:
5782:
5780:
5778:
5769:
5765:
5761:
5757:
5753:
5749:
5744:
5739:
5736:(2): 021039.
5735:
5731:
5724:
5716:
5712:
5708:
5704:
5700:
5696:
5692:
5688:
5683:
5678:
5675:(6): 060501.
5674:
5670:
5663:
5655:
5651:
5647:
5643:
5638:
5633:
5629:
5625:
5618:
5610:
5606:
5602:
5598:
5594:
5590:
5585:
5580:
5577:(6): 062314.
5576:
5572:
5565:
5550:
5546:
5542:
5538:
5534:
5530:
5526:
5522:
5517:
5512:
5508:
5504:
5500:
5493:
5491:
5482:
5478:
5474:
5470:
5466:
5462:
5457:
5452:
5449:(3): 032326.
5448:
5444:
5437:
5435:
5426:
5422:
5417:
5412:
5408:
5404:
5400:
5396:
5391:
5386:
5382:
5378:
5374:
5366:
5364:
5362:
5360:
5351:
5347:
5340:
5338:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5296:
5291:
5287:
5283:
5276:
5268:
5261:
5253:
5249:
5245:
5241:
5237:
5233:
5229:
5225:
5220:
5215:
5211:
5207:
5200:
5192:
5188:
5184:
5180:
5175:
5170:
5166:
5162:
5155:
5147:
5143:
5139:
5135:
5131:
5127:
5122:
5117:
5114:(3): 033007.
5113:
5109:
5102:
5093:
5088:
5081:
5073:
5069:
5065:
5061:
5057:
5053:
5048:
5043:
5039:
5035:
5028:
5020:
5016:
5012:
5008:
5004:
5000:
4995:
4990:
4986:
4982:
4975:
4967:
4963:
4959:
4953:
4945:
4941:
4937:
4933:
4929:
4925:
4920:
4915:
4911:
4907:
4900:
4892:
4888:
4884:
4880:
4876:
4872:
4867:
4862:
4858:
4854:
4847:
4845:
4843:
4834:
4830:
4826:
4822:
4818:
4814:
4809:
4804:
4800:
4796:
4788:
4786:
4784:
4775:
4771:
4767:
4763:
4759:
4755:
4750:
4745:
4741:
4737:
4730:
4728:
4726:
4717:
4713:
4708:
4703:
4699:
4695:
4691:
4687:
4683:
4676:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4640:
4635:
4630:
4626:
4622:
4614:
4612:
4610:
4601:
4597:
4593:
4589:
4585:
4581:
4577:
4573:
4568:
4563:
4559:
4555:
4548:
4546:
4544:
4534:
4525:
4520:
4516:
4512:
4508:
4501:
4497:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4468:
4462:
4459:
4455:
4453:
4449:
4445:
4441:
4436:
4432:
4429:
4426:
4422:
4418:
4413:
4411:
4407:
4402:
4398:
4396:
4391:
4387:
4384:particles in
4383:
4378:
4376:
4372:
4368:
4365:
4361:
4351:
4348:
4342:
4340:
4334:
4330:
4327:
4323:
4319:
4315:
4308:Certification
4305:
4301:
4292:
4289:
4280:
4272:
4269:
4266:
4262:
4261:
4260:
4257:
4248:
4246:
4242:
4238:
4236:
4230:
4226:
4222:
4217:
4214:
4198:
4195:
4170:
4166:
4162:
4159:
4156:
4153:
4150:
4145:
4141:
4137:
4132:
4128:
4121:
4113:
4112:
4099:
4097:
4093:
4089:
4072:
4069:
4049:
4040:
4030:
4028:
4024:
4019:
4015:
4011:
4007:
4002:
4001:PDC sources.
4000:
3996:
3993:
3989:
3985:
3981:
3965:
3960:
3956:
3943:
3940:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3894:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3864:
3848:
3828:
3824:
3820:
3817:
3814:
3811:
3808:
3803:
3799:
3795:
3790:
3786:
3779:
3770:
3767:
3763:
3759:
3755:
3735:
3726:
3722:
3718:
3715:
3712:
3706:
3701:
3698:
3692:
3684:
3664:
3646:
3643:
3640:
3625:
3622:
3619:
3606:
3603:
3595:
3591:
3587:
3583:
3579:
3578:
3577:
3563:
3558:
3548:
3511:
3508:
3505:
3490:
3487:
3484:
3471:
3468:
3460:
3459:
3458:
3444:
3441:
3419:
3416:
3413:
3398:
3395:
3392:
3379:
3376:
3368:
3347:
3343:
3339:
3336:
3333:
3330:
3327:
3322:
3318:
3314:
3309:
3305:
3298:
3273:
3269:
3265:
3262:
3259:
3256:
3253:
3248:
3244:
3240:
3235:
3231:
3224:
3216:
3212:
3209:
3206:
3202:
3186:
3178:
3174:
3153:
3149:
3145:
3142:
3139:
3136:
3133:
3128:
3124:
3120:
3115:
3111:
3104:
3096:
3075:
3071:
3067:
3064:
3061:
3058:
3055:
3050:
3046:
3042:
3037:
3033:
3026:
3018:
2997:
2993:
2989:
2986:
2983:
2980:
2977:
2972:
2968:
2964:
2959:
2955:
2948:
2939:
2937:
2916:
2912:
2908:
2905:
2902:
2899:
2896:
2891:
2887:
2883:
2878:
2874:
2867:
2847:
2832:
2830:
2826:
2822:
2818:
2815:
2794:
2790:
2786:
2783:
2778:
2774:
2770:
2764:
2756:
2751:
2739:
2736:
2732:
2728:
2725:
2721:
2720:
2719:
2717:
2711:
2708:
2701:
2680:
2676:
2672:
2669:
2666:
2663:
2660:
2655:
2651:
2647:
2642:
2638:
2631:
2623:
2603:
2599:
2595:
2592:
2589:
2586:
2583:
2578:
2574:
2570:
2565:
2561:
2554:
2546:
2545:
2544:
2542:
2532:
2530:
2526:
2520:
2518:
2514:
2510:
2506:
2502:
2491:
2476:
2468:
2450:
2446:
2437:
2421:
2399:
2395:
2371:
2365:
2360:
2356:
2352:
2349:
2346:
2343:
2338:
2334:
2326:
2314:
2310:
2292:
2287:
2272:
2268:
2256:
2248:
2244:
2233:
2229:
2225:
2222:
2219:
2216:
2213:
2208:
2204:
2200:
2195:
2191:
2179:
2171:
2167:
2163:
2160:
2157:
2154:
2151:
2146:
2142:
2138:
2133:
2129:
2122:
2115:
2114:
2113:
2111:
2095:
2087:
2083:
2069:
2051:
2047:
2038:
2020:
2016:
2008:by repeating
1995:
1975:
1970:
1967:
1964:
1960:
1951:
1933:
1930:
1927:
1923:
1893:
1887:
1882:
1878:
1874:
1871:
1868:
1865:
1860:
1856:
1852:
1847:
1843:
1839:
1836:
1833:
1830:
1825:
1821:
1813:
1801:
1798:
1795:
1791:
1773:
1768:
1749:
1738:
1734:
1730:
1727:
1724:
1721:
1718:
1713:
1709:
1705:
1700:
1696:
1684:
1676:
1672:
1668:
1665:
1662:
1659:
1656:
1651:
1647:
1643:
1638:
1634:
1627:
1620:
1619:
1618:
1616:
1598:
1594:
1586:of detecting
1568:
1564:
1560:
1557:
1554:
1551:
1548:
1543:
1539:
1535:
1530:
1526:
1519:
1510:
1489:
1473:
1460:
1456:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1427:
1422:
1418:
1408:
1404:
1393:
1367:
1363:
1359:
1356:
1353:
1350:
1347:
1342:
1338:
1334:
1329:
1325:
1312:
1306:
1301:
1276:
1272:
1266:
1262:
1257:
1254:
1251:
1243:
1239:
1233:
1229:
1224:
1216:
1212:
1206:
1202:
1198:
1187:
1161:
1157:
1153:
1150:
1147:
1144:
1141:
1136:
1132:
1128:
1123:
1119:
1109:
1100:
1095:
1078:
1070:
1066:
1045:
1025:
1017:
1013:
1009:
1006:
1003:
1000:
997:
992:
988:
984:
979:
975:
963:
955:
951:
947:
935:
920:
897:
883:
865:
861:
838:
835:
830:
826:
820:
815:
812:
809:
805:
779:
775:
771:
768:
765:
762:
759:
754:
750:
746:
741:
737:
712:
700:
685:
671:
663:
638:
634:
631:
628:
625:
622:
605:
601:
597:
593:
592:Hilbert space
577:
569:
551:
547:
526:
503:
495:
491:
470:
462:
458:
435:
431:
406:
401:
397:
388:
384:
365:
357:
353:
347:
342:
339:
335:
329:
324:
321:
318:
314:
310:
305:
301:
290:
285:
281:
275:
272:
268:
262:
257:
254:
251:
247:
243:
238:
233:
229:
221:
220:
219:
198:
194:
169:
164:
160:
151:
147:
131:
128:
120:
115:
113:
109:
105:
101:
96:
92:
88:
84:
74:
72:
68:
64:
60:
56:
53:of identical
52:
48:
44:
40:
37:
33:
29:
25:
21:
7517:Charge qubit
7442:KLM protocol
7391:
7255:
7245:
7099:
6939:Purification
6869:Eastin–Knill
6663:
6659:
6649:
6608:
6604:
6594:
6557:
6553:
6543:
6500:
6496:
6490:
6447:
6443:
6437:
6394:
6391:Phys. Rev. A
6390:
6384:
6369:
6326:
6323:Phys. Rev. B
6322:
6315:
6272:
6268:
6262:
6241:
6190:
6186:
6180:
6127:
6123:
6113:
6062:
6058:
6052:
6031:
6010:
5967:
5963:
5957:
5906:
5902:
5896:
5853:
5849:
5792:
5788:
5733:
5729:
5723:
5672:
5668:
5662:
5627:
5623:
5617:
5574:
5571:Phys. Rev. A
5570:
5564:
5552:. Retrieved
5506:
5502:
5446:
5443:Phys. Rev. A
5442:
5380:
5376:
5349:
5285:
5281:
5275:
5266:
5260:
5209:
5205:
5199:
5164:
5160:
5154:
5111:
5107:
5101:
5080:
5037:
5033:
5027:
4984:
4980:
4974:
4966:the original
4961:
4952:
4909:
4905:
4899:
4856:
4852:
4798:
4794:
4739:
4735:
4689:
4685:
4675:
4624:
4620:
4557:
4553:
4533:
4514:
4510:
4500:
4486:KLM protocol
4460:
4456:
4439:
4437:
4433:
4414:
4400:
4394:
4389:
4385:
4381:
4379:
4375:phase shifts
4360:trapped ions
4357:
4343:
4338:
4335:
4331:
4311:
4302:
4298:
4290:
4286:
4277:
4258:
4254:
4244:
4234:
4218:
4110:
4109:probability
4105:
4036:
4022:
4017:
4013:
4009:
4005:
4003:
3998:
3991:
3987:
3983:
3979:
3921:
3917:
3913:
3909:
3905:
3901:
3897:
3895:
3890:
3886:
3882:
3878:
3874:
3863:quantum dots
3859:
3771:
3765:
3761:
3757:
3750:
3662:
3593:
3589:
3585:
3581:
3529:
3366:
3214:
3201:Haar measure
3176:
3172:
3094:
3016:
2940:
2838:
2820:
2813:
2752:
2748:
2712:
2704:
2538:
2521:
2497:
2466:
2435:
2386:
2109:
2067:
2036:
1908:
1614:
1511:
1487:
1391:
1185:
1096:
921:
881:
686:
662:homomorphism
599:
595:
568:homomorphism
460:
456:
386:
382:
380:
152:) operators
150:annihilation
118:
116:
111:
90:
86:
82:
80:
19:
18:
7548:programming
7527:Phase qubit
7432:Circuit QED
6904:No-deleting
6846:cloud-based
5108:New J. Phys
4517:: 143–252.
4367:Fock states
4213:#P-complete
1099:isomorphism
77:Description
7636:Categories
7588:libquantum
7522:Flux qubit
7427:Cavity QED
7376:Bacon–Shor
7366:stabilizer
6894:No-cloning
6673:1902.00462
6567:2211.04420
6510:1607.02987
6503:(8): 259.
6457:1607.02987
6404:1609.02416
6336:1701.00714
6253:1509.02703
6137:1508.00782
5977:1505.03103
5743:1511.06526
5584:1803.11534
5554:22 January
5516:1612.01199
5456:1705.05299
5390:1505.03708
5269:: 447–458.
5121:1506.06220
5092:1706.01260
4912:(5): 675.
4866:1505.01182
4492:References
4111:amplitudes
2729:The class
2531:problem).
2465:times its
2066:times its
2035:times its
39:scattering
7494:NV center
6929:Threshold
6909:No-hiding
6874:Gleason's
6641:256757275
6586:0031-8949
6535:195791867
6361:119077553
6282:1412.8427
6200:1310.4860
6130:: 10469.
6072:1312.3080
6043:1309.7460
6022:1306.3995
5916:1403.4007
5863:1311.2913
5827:120825561
5802:1311.1622
5682:1408.3712
5632:CiteSeerX
5609:119227039
5481:119431211
5295:1305.4346
5219:1305.3188
5174:1106.0849
4994:1005.1407
4944:121606171
4833:121093296
4808:1212.2783
4774:119241050
4749:1212.2240
4634:1212.2622
4567:1212.2234
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