Quantum error correction

2279: 179: 77:. A repetition code stores the desired (logical) information as multiple copies, and—if these copies are later found to disagree due to errors introduced to the system-determines the most likely value for the original data by majority vote. E.g. suppose we copy a bit in the one (on) state three times. Suppose further that noise in the system introduces an error which corrupts the three-bit state so that one of the copied bits becomes zero (off) but the other two remain equal to one. Assuming that errors are independent and occur with some sufficiently low probability 986: 8419: 81:, it is most likely that the error is a single-bit error and the intended message is three bits in the one state. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. In this example, the logical information is a single bit in the one state and the physical information are the three duplicate bits. Creating a physical state that represents the logical state is called 9226: 8409: 4546: 1355: 9528: 2780: 4258: 5424:

rapidly add up under constructive interference of coherent noise. This is a circuit-level calibration scheme that traces deviations (e.g. sharp dips or notches) in the decoherence curve to detect and localize the coherent error, but does not require encoding or parity measurements. However, further investigation is needed to establish the effectiveness of this method for the incoherent noise.

1055: 3738: 5384:. 2021 also saw the first experimental demonstration of fault-tolerant Bacon-Shor code in a single logical qubit of a trapped-ion system, i.e. a demonstration for which the addition of error correction is able to suppress more errors than is introduced by the overhead required to implement the error correction as well as fault tolerant 4541:{\displaystyle {\begin{aligned}|0_{L}^{+}\rangle &\equiv |\alpha \rangle +|-\alpha \rangle ,\\|1_{L}^{+}\rangle &\equiv |i\alpha \rangle +|-i\alpha \rangle ,\\|0_{L}^{-}\rangle &\equiv |\alpha \rangle -|-\alpha \rangle ,\\|1_{L}^{-}\rangle &\equiv |i\alpha \rangle -|-i\alpha \rangle .\end{aligned}}} 5423:

In 2022, research at University of Engineering and Technology Lahore demonstrated error-cancellation by inserting single-qubit Z-axis rotation gates into strategically chosen locations of the superconductor quantum circuits. The scheme has been shown to effectively correct errors that would otherwise

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is used to diagnose which error was the likely source of corruption on an encoded state. An error can then be reversed by applying a corrective operation based on the syndrome. Quantum error correction also employs syndrome measurements. It performs a multi-qubit measurement that does not disturb the

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is small enough that the probability of more than a single qubit being flipped is negligible. One can then detect whether a qubit was flipped, without also querying for the values being transmitted, by asking whether one of the qubits differs from the others. This amounts to performing a measurement

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has infinitely many energy levels in a single physical system. Codes for these systems include cat, Gottesman-Kitaev-Preskill (GKP), and binomial codes. One insight offered by these codes is to take advantage of the redundancy within a single system, rather than to duplicate many two-level qubits.

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capable of performing real-time feedback operations including read-out of the quantum information, its analysis, and the correction of its detected errors. The work demonstrated how the quantum-error-corrected system reaches the break-even point at which the lifetime of a logical qubit exceeds the

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The error channel may induce either a bit flip, a sign flip (i.e., a phase flip), or both. It is possible to correct for both types of errors on a logical qubit using a well-designed QEC code. One example of a code that does this is the Shor code, published in 1995. Since these two types of errors

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Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected. For example, if the first and second qubits are flipped, then the syndrome measurement gives the outcome

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This qubit virtualization system was used to create 4 logical qubits with 30 of the 32 qubits on Quantinuum’s trapped-ion hardware. The system uses an active syndrome extraction technique to diagnose errors and correct them while calculations are underway without destroying the logical qubits.

1350:{\displaystyle {\begin{aligned}P_{0}&=|000\rangle \langle 000|+|111\rangle \langle 111|,\\P_{1}&=|100\rangle \langle 100|+|011\rangle \langle 011|,\\P_{2}&=|010\rangle \langle 010|+|101\rangle \langle 101|,\\P_{3}&=|001\rangle \langle 001|+|110\rangle \langle 110|.\end{aligned}}} 2178: 6919:

Erhard, Alexander; Poulsen Nautrup, Hendrik; Meth, Michael; Postler, Lukas; Stricker, Roman; Stadler, Martin; Negnevitsky, Vlad; Ringbauer, Martin; Schindler, Philipp; Briegel, Hans J.; Blatt, Rainer; Friis, Nicolai; Monz, Thomas (2021-01-13). "Entangling logical qubits with lattice surgery".

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Postler, Lukas; Heußen, Sascha; Pogorelov, Ivan; Rispler, Manuel; Feldker, Thomas; Meth, Michael; Marciniak, Christian D.; Stricker, Roman; Ringbauer, Martin; Blatt, Rainer; Schindler, Philipp; Müller, Markus; Monz, Thomas (2022-05-25). "Demonstration of fault-tolerant universal quantum gate

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that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum state preparation, and faulty measurements. Effective quantum error correction would allow quantum computers with low qubit fidelity to execute algorithms of higher complexity or greater

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Sun, L.; Petrenko, A.; Leghtas, Z.; Vlastakis, B.; Kirchmair, G.; Sliwa, K. M.; Narla, A.; Hatridge, M.; Shankar, S.; Blumoff, J.; Frunzio, L.; Mirrahimi, M.; Devoret, M. H.; Schoelkopf, R. J. (July 2014). "Tracking photon jumps with repeated quantum non-demolition parity measurements".

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Ofek, Nissim; Petrenko, Andrei; Heeres, Reinier; Reinhold, Philip; Leghtas, Zaki; Vlastakis, Brian; Liu, Yehan; Frunzio, Luigi; Girvin, S. M.; Jiang, L.; Mirrahimi, Mazyar (August 2016). "Extending the lifetime of a quantum bit with error correction in superconducting circuits".

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quantum information in the encoded state but retrieves information about the error. Depending on the QEC code used, syndrome measurement can determine the occurrence, location and type of errors. In most QEC codes, the type of error is either a bit flip, or a sign (of the

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have demonstrated a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. They have performed a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code, and fault-tolerantly prepared a logical

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Albert, Victor V.; Noh, Kyungjoo; Duivenvoorden, Kasper; Young, Dylan J.; Brierley, R. T.; Reinhold, Philip; Vuillot, Christophe; Li, Linshu; Shen, Chao; Girvin, S. M.; Terhal, Barbara M.; Jiang, Liang (2018). "Performance and structure of single-mode bosonic codes".

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parity of the system. Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit. However, the particular binomial code above is not robust to two-photon loss.

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Egan, Laird; Debroy, Dripto M.; Noel, Crystal; Risinger, Andrew; Zhu, Daiwei; Biswas, Debopriyo; Newman, Michael; Li, Muyuan; Brown, Kenneth R.; Cetina, Marko; Monroe, Christopher (2021-10-04). "Fault-tolerant control of an error-corrected qubit".

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is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code. If distinct of the set of correctable errors produce orthogonal results, the code is considered

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In April 2024, researchers at Microsoft claimed to have successfully tested a quantum error correction code that allowed them to achieve an error rate with logical qubits that is 800 times better than the underlying physical error rate.

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If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. This means that the Shor code can also repair a sign flip error for a single qubit.

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that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three groups of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state

2984: 5307:, of which various types exist. As of June 2024, the 2D planar surface code is generally considered the most well-studied type of quantum error correction, and one of the leading contenders for practical use in quantum computing. 484:

the state into a greater number of qubits, in such a way that errors in the corresponding logical qubits can be detected and corrected. In the case of the simple three-qubit repetition code, the encoding consists in the mappings

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Mirrahimi, Mazyar; Leghtas, Zaki; Albert, Victor V; Touzard, Steven; Schoelkopf, Robert J; Jiang, Liang; Devoret, Michel H (2014-04-22). "Dynamically protected cat-qubits: a new paradigm for universal quantum computation".

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Chiaverini, J.; Leibfried, D.; Schaetz, T.; Barrett, M. D.; Blakestad, R. B.; Britton, J.; Itano, W. M.; Jost, J. D.; Knill, E.; Langer, C.; Ozeri, R.; Wineland, D. J. (2004). "Realization of quantum error correction".

2889: 2573: 1979: 1673: 452: 3733:{\displaystyle {\begin{aligned}X&={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\\Y&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}};\\Z&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\end{aligned}}} 2289:

Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with quantum computers, the sign flip. Through the transmission in a channel the relative sign between

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Lassen, M.; Sabuncu, M.; Huck, A.; Niset, J.; Leuchs, G.; Cerf, N. J.; Andersen, U. L. (2010). "Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code".

4947: 5323:, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels— 1438: 2495: 2418: 3870: 373: 4698: 4247: 4175: 5949:

Michael, Marios H.; Silveri, Matti; Brierley, R. T.; Albert, Victor V.; Salmilehto, Juha; Jiang, Liang; Girvin, S. M. (2016-07-14). "New Class of Quantum Error-Correcting Codes for a Bosonic Mode".

4634:, the error takes the logical states from the even parity subspace to the odd one, and vice versa. Single-photon-loss errors can therefore be detected by measuring the photon number parity operator 3303:

If a bit flip error happens to a qubit, the syndrome analysis will be performed on each block of qubits (1,2,3), (4,5,6), and (7,8,9) to detect and correct at most one bit flip error in each block.

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acts equally and independently on each of the three qubits in which the state is now encoded. The problem is now how to detect and correct such errors, while not corrupting the transmitted state

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The syndrome measurement provides information about the error that has happened, but not about the information that is stored in the logical qubit—as otherwise the measurement would destroy any

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Reed, M. D.; DiCarlo, L.; Nigg, S. E.; Sun, L.; Frunzio, L.; Girvin, S. M.; Schoelkopf, R. J. (2012). "Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits".

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Leghtas, Zaki; Kirchmair, Gerhard; Vlastakis, Brian; Schoelkopf, Robert J.; Devoret, Michel H.; Mirrahimi, Mazyar (2013-09-20). "Hardware-Efficient Autonomous Quantum Memory Protection".

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The Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit

1856: 4105:, superpositions of coherent states, can also be used as logical states for error correction codes. Cat code, realized by Ofek et al. in 2016, defined two sets of logical states: 163:

The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. This approach does not work for a quantum channel in which, due to the

2578: 627: 213: 5206:. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits. 5066: 2765: 2710: 1357:

This reveals which qubits are different from the others, without at the same time giving information about the state of the qubits themselves. If the outcome corresponding to

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A practical quantum computer that is capable of large circuit depth, therefore, ultimately calls for operations on logical qubits protected by quantum error correction

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Schindler, P.; Barreiro, J. T.; Monz, T.; Nebendahl, V.; Nigg, D.; Chwalla, M.; Hennrich, M.; Blatt, R. (2011). "Experimental Repetitive Quantum Error Correction".

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respectively. Since the codewords involve only even photon number, and the error states involve only odd photon number, errors can be detected by measuring the

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Fowler, Austin G.; Mariantoni, Matteo; Martinis, John M.; Cleland, Andrew N. (2012-09-18). "Surface codes: Towards practical large-scale quantum computation".

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Other error correcting codes have also been implemented, such as one aimed at correcting for photon loss, the dominant error source in photonic qubit schemes.

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is below a certain threshold; as otherwise, the attempts to measure the syndrome and correct the errors would introduce more new errors than they correct for.

3294:{\displaystyle |1_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )} 3137:{\displaystyle |0_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )} 89:. Similar to classical error correction, QEC codes do not always correctly decode logical qubits, but instead reduce the effect of noise on the logical state. 5701:

Cochrane, P. T.; Milburn, G. J.; Munro, W. J. (1999-04-01). "Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping".

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where the subscript L indicates a "logically encoded" state. Then if the dominant error mechanism of the system is the stochastic application of the bosonic

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then both a bit flip error and a sign flip error occur. In other words, the Shor code can correct any combination of bit or phase errors on a single qubit.

5362: 1602:, and the third qubit is flipped, instead of the first two. To assess the performance of this error-correcting scheme for a general input we can study the 6505:

Cory, D. G.; Price, M. D.; Maas, W.; Knill, E.; Laflamme, R.; Zurek, W. H.; Havel, T. F.; Somaroo, S. S. (1998). "Experimental Quantum Error Correction".

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In February 2023 researchers at Google claimed to have decreased quantum errors by increasing the qubit number in experiments, they used a fault tolerant

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be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability

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indicate they have odd parity. Similar to the binomial code, if the dominant error mechanism of the system is the stochastic application of the bosonic

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In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code. The error-correction demonstration was performed on

5316: 5285: 1754:{\displaystyle \rho _{\operatorname {out} }\equiv {\mathcal {E}}_{\operatorname {corr} }({\mathcal {E}}(\vert \psi '\rangle \langle \psi '\vert ))} 9262: 9023: 7608: 7253: 5234: 144:). To correct the error, the Pauli operator corresponding to the type of error is used on the corrupted qubit to revert the effect of the error. 2818: 381: 5281: 8656: 7570: 5151:'s 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit. 8300: 5347: 2173:{\displaystyle F(\psi ')=\langle \psi '\vert \rho _{\operatorname {out} }\vert \psi '\rangle \geq (1-p)^{3}+3p(1-p)^{2}=1-3p^{2}+2p^{3}.} 7355:

Ahsan, Muhammad; Naqvi, Syed Abbas Zilqurnain; Anwer, Haider (2022-02-18). "Quantum circuit engineering for correcting coherent noise".

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Horsman, Dominic; Fowler, Austin G; Devitt, Simon; Meter, Rodney Van (2012-12-01). "Surface code quantum computing by lattice surgery".

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is to be compared with the corresponding fidelity obtained when no error-correcting protocol is used, which was shown before to equal

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As of late 2004, estimates for this threshold indicate that it could be as high as 1–3%, provided that there are sufficiently many

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Several proposals have been made for storing error-correctable quantum information in bosonic modes. Unlike a two-level system, a

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are the only types of errors that can result after a projective measurement, a Shor code corrects arbitrary single-qubit errors.

1435:-th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel: 306: 4637: 8890: 8448: 4180: 4108: 8824: 4865: 167:, it is not possible to repeat a single qubit three times. To overcome this, a different method has to be used, such as the 8747: 8412: 7598: 5032:{\displaystyle ({\mathcal {R}}\circ {\mathcal {E}})(\rho )=\rho \quad \forall \rho =P_{\mathcal {C}}\rho P_{\mathcal {C}},} 2786:

to encode a single logical qubit with the Shor code and then perform bit flip error correction on each of the three blocks.

9255: 8370: 1565:{\displaystyle {\mathcal {E}}_{\operatorname {corr} }(\rho )=P_{0}\rho P_{0}+\sum _{i=1}^{3}X_{i}P_{i}\rho \,P_{i}X_{i}.} 6192:

Calderbank, A. R.; Rains, E. M.; Shor, P. W.; Sloane, N. J. A. (1998). "Quantum Error Correction via Codes over GF(4)".

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Pittman, T. B.; Jacobs, B. C.; Franson, J. D. (2005). "Demonstration of quantum error correction using linear optics".

96:. This theorem seems to present an obstacle to formulating a theory of quantum error correction. But it is possible to 3977:{\displaystyle |0_{\rm {L}}\rangle ={\frac {|0\rangle +|4\rangle }{\sqrt {2}}},\quad |1_{\rm {L}}\rangle =|2\rangle ,} 8478: 8441: 8422: 8310: 7563: 5407:

measuring an error rate of 3.028% and 2.914% for a distance-3 qubit array and a distance-5 qubit array respectively.

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Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories".

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fidelities, with (red) and without (blue) error correcting via the three qubit bit flip code. Notice how, for

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be a quantum channel that can cause at most one phase flip. Then the bit flip code from above can recover

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and collaborators found a class of 5-qubit codes that do the same, which also have the property of being

3840: 2250:. Note that this is consistent with the working assumption that was made while deriving the protocol (of 606: 192: 129: 9554: 9509: 9309: 9229: 8991: 8799: 8770: 8162: 8034: 7931: 7807: 7735: 7642: 5346:

There have been several experimental realizations of CSS-based codes. The first demonstration was with

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Guo, Qihao; Zhao, Yuan-Yuan; Grassl, Markus; Nie, Xinfang; Xiang, Guo-Yong; Xin, Tao; Yin, Zhang-Qi;

5350:. Subsequently, demonstrations have been made with linear optics, trapped ions, and superconducting ( 5312: 711: 54: 17: 5099: 5075: 4923: 4899: 4841: 4706: 3313: 2715: 958: 262: 218: 9482: 9477: 9406: 9381: 9366: 9361: 9080: 9060: 9050: 8996: 8556: 8117: 8093: 8056: 7847: 7780: 7715: 7700: 7670: 7593: 5397: 5303:

The ideas of stabilizer codes, CSS codes, and topological codes can be expanded into the 2D planar

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That these codes allow indeed for quantum computations of arbitrary length is the content of the

4050: 3990: 886: 841: 239:. Let us moreover assume that this channel either flips the state of the qubit, with probability 9189: 8737: 4608: 4022: 2321: 2293: 704:. This mapping can be realized for example using two CNOT gates, entangling the system with two 9499: 9421: 9100: 8875: 8855: 8780: 8633: 8365: 8129: 8122: 7869: 5175:

is the smallest possible code that protects a single logical qubit against single-qubit errors.

1608: 996: 802: 141: 137: 136:, so even if the error due to the noise was arbitrary, it can be expressed as a combination of 8676: 6990: 2219: 697:{\displaystyle \vert \psi '\rangle =\alpha _{0}\vert 000\rangle +\alpha _{1}\vert 111\rangle } 9472: 9446: 9271: 9125: 8623: 8603: 8285: 7911: 7837: 7802: 5069: 799:

by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability

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Daniel Gottesman; Alexei Kitaev; John Preskill (2001). "Encoding a qubit in an oscillator".

2676:{\displaystyle |\psi '\rangle =\alpha _{0}|{+}{+}{+}\rangle +\alpha _{1}|{-}{-}{-}\rangle .} 596:{\displaystyle \vert 1\rangle \rightarrow \vert 1_{\rm {L}}\rangle \equiv \vert 111\rangle } 537:{\displaystyle \vert 0\rangle \rightarrow \vert 0_{\rm {L}}\rangle \equiv \vert 000\rangle } 9411: 9334: 9156: 9085: 9030: 8760: 8568: 8526: 8075: 7824: 7675: 7510: 7491:

Freedman, Michael H.; Meyer, David A. (1998). "Projective plane and planar quantum codes".

7374: 7198: 7143: 7087: 7024: 6939: 6852: 6795: 6733: 6672: 6621: 6577: 6524: 6463: 6410: 6336: 6291: 6256: 6142: 6079: 6022: 5968: 5910: 5849: 5776: 5720: 5648: 5569: 5526: 5480: 5297: 5293: 3526: 3499: 3472: 3445: 2187: 1578: 1387: 1360: 931: 286: 172: 7418:"What's the difference between error suppression, error mitigation, and error correction?" 6370:

Gidney, Craig; Newman, Michael; Brooks, Peter; Jones, Cody (2023). "Yoked surface codes".

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with four different outcomes, corresponding to the following four projective measurements:

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Quantum Error Correction: Symmetric, Asymmetric, Synchronizable, and Convolutional Codes

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In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. Let

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Those two sets of states differ from the photon number parity, as states denoted with

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by storing the information of one qubit onto a highly entangled state of nine qubits.

9441: 9329: 9299: 9294: 9211: 9120: 9090: 9018: 8981: 8976: 8958: 8923: 8913: 8717: 8613: 8578: 8561: 8464: 8374: 8019: 7926: 7883: 7814: 7730: 7710: 7665: 7625: 7603: 7402: 7390: 7234: 7216: 7171: 7159: 7109: 7060: 7048: 7040: 6991:"Foundational step shows quantum computers can be better than the sum of their parts" 6975: 6963: 6955: 6880: 6868: 6749: 6688: 6637: 6479: 6426: 6352: 6276: 6166: 6158: 6103: 6095: 5865: 5800: 5792: 5740: 5684: 5672: 5664: 5585: 5542: 5218: 5195: 4603: 3985: 175:

and syndrome measurements and is comparable in performance with the repetition code.

6815: 6700: 6597: 6544: 6042: 5988: 5930: 5812: 9346: 8938: 8933: 8790: 8671: 8041: 7991: 7768: 7479: 7382: 7224: 7206: 7151: 7095: 7032: 6947: 6860: 6803: 6761: 6741: 6680: 6629: 6585: 6532: 6491: 6471: 6418: 6344: 6299: 6264: 6223: 6211: 6150: 6087: 6030: 5976: 5918: 5877: 5857: 5788: 5784: 5728: 5656: 5626: 5577: 5534: 5488: 5214: 5164: 2979:{\displaystyle |\psi '\rangle =\alpha _{0}|0_{S}\rangle +\alpha _{1}|1_{S}\rangle } 152: 66: 38: 6649: 6178: 6115: 9416: 9401: 9371: 9319: 9314: 9168: 9095: 9075: 9045: 9008: 9003: 8908: 8732: 8167: 8105: 7795: 7790: 5607: 5230: 5210: 4836: 2811: 2783: 2282: 182: 74: 7386: 6536: 955:. Note that a further assumption about the channel is made here: we assume that 189:

Consider the situation in which we want to transmit the state of a single qubit

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operations called the error basis (which is given by the Pauli matrices and the

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correct when no more than one qubit is flipped, which happens with probability

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and determining which logical state is encoded in the physical state is called

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by transforming into the Hadamard basis before and after transmission through

2008:

resulting from errors not properly corrected by the protocol. It follows that

9548: 9487: 9386: 9065: 8918: 8809: 8712: 8628: 8598: 8551: 8098: 7916: 7842: 7394: 7220: 7044: 6959: 6807: 6430: 6356: 6162: 6099: 5869: 5796: 5732: 5668: 5581: 5514: 5366:

lifetime of the underlying constituents of the system (the physical qubits).

5259: 5203: 5179: 5154: 4079: 1384:

is obtained, no correction is applied, while if the outcome corresponding to

121: 50: 6684: 5459:"Bosonic quantum error correction codes in superconducting quantum circuits" 9467: 8948: 8546: 8541: 8318: 8243: 7444: 7238: 7163: 7100: 7075: 7052: 6967: 6872: 6753: 6692: 6641: 6483: 6170: 6107: 5804: 5676: 5404: 5328: 5304: 5187: 4920:

is the state Hilbert space, such that there exists another quantum channel

6444:

Knill, Emanuel (2004-11-02). "Quantum Computing with Very Noisy Devices".

5890: 5589: 5546: 2884:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 2568:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 447:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 9304: 8986: 8328: 8182: 7720: 7505: 6572: 6519: 6458: 6251: 6206: 5905: 5715: 5385: 5183: 5158: 46: 9240: 6833:(2021). "Testing a quantum error-correcting code on various platforms". 6745: 6633: 6475: 6154: 6091: 155:, which would prevent it from being used to convey quantum information. 8389: 8323: 8187: 5438: 5377: 5267: 5263: 5226: 5222: 5199: 5148: 3864: 105: 6215: 5517:(1995). "Scheme for reducing decoherence in quantum computer memory". 8840: 8521: 8172: 5358: 5277: 4102: 8433: 7187:"Suppressing quantum errors by scaling a surface code logical qubit" 5190:, led to the construction of an important class of codes called the 8357: 8333: 8192: 8157: 7369: 7138: 6934: 6847: 6830: 6376: 6017: 5963: 5756: 5475: 5351: 5191: 6790: 6728: 6405: 6331: 6074: 5844: 5771: 5643: 8384: 8001: 7541: 7279:"Google's quantum computer reached an error-correcting milestone" 6610: 6918: 5209:

A more general class of codes (encompassing the former) are the

5157:

found a code that does the same with 7 instead of 9 qubits, see

1974:{\displaystyle \,\vert \psi '\rangle \langle \psi '\vert +(...)} 53:. Quantum error correction is theorised as essential to achieve 8361: 7857: 6316: 5560:

Peres, Asher (1985). "Reversible Logic and Quantum Computers".

2490:{\displaystyle |+\rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}.} 7304:"Microsoft And Quantinuum Improve Quantum Error Rates By 800x" 6662: 2413:{\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}} 7630: 7467:

Quantum Error Correction and Fault Tolerant Quantum Computing

7122: 5828: 5625:

Devitt, Simon J; Munro, William J; Nemoto, Kae (2013-06-20).

5335: 2779: 101: 7455: 5948: 4703:

Still, cat qubits are not protected against two-photon loss

104:

onto a highly entangled state of several (physical) qubits.

8379: 7852: 7785: 7474:

Freedman, Michael H.; Meyer, David A.; Luo, Feng (2002). "Z

6895:"Error-protected quantum bits entangled for the first time" 6058: 6001: 5419:

Quantum error-correction without encoding and parity-checks

368:{\displaystyle {\mathcal {E}}(\rho )=(1-p)\rho +p\ X\rho X} 171:

first proposed by Asher Peres in 1985. This technique uses

7443: 6131: 4693:{\displaystyle \exp(i\pi {\hat {a}}^{\dagger }{\hat {a}})} 7996: 7981: 6191: 4242:{\displaystyle \{|0_{L}^{-}\rangle ,|1_{L}^{-}\rangle \}} 4170:{\displaystyle \{|0_{L}^{+}\rangle ,|1_{L}^{+}\rangle \}} 6390: 5144:

Over time, researchers have come up with several codes:

2346:

can become inverted. For instance, a qubit in the state

6369: 5361:

encoded in a superconducting resonator, and employed a

92:

Copying quantum information is not possible due to the

7013: 6774: 4889:{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}} 4576:

only occupy even photon number states and states with

3689: 3633: 3580: 128:

X, Z, and Y). The measurement of the syndrome has the

5102: 5078: 5045: 4950: 4926: 4902: 4868: 4844: 4793: 4745: 4709: 4640: 4611: 4582: 4556: 4261: 4183: 4111: 4053: 4025: 3993: 3873: 3808: 3782: 3756: 3560: 3529: 3502: 3475: 3448: 3364: 3344: 3316: 3149: 2992: 2897: 2821: 2796: 2746: 2718: 2691: 2581: 2505: 2426: 2352: 2324: 2296: 2256: 2222: 2216:

error correction is greater than the one without for

2190: 2014: 1987: 1864: 1794: 1767: 1676: 1645: 1611: 1581: 1441: 1421: 1390: 1363: 1058: 1037: 999: 961: 934: 889: 844: 805: 774: 740: 714: 635: 609: 550: 491: 460: 384: 309: 289: 265: 245: 221: 195: 6557: 5700: 1027:, the error correction scheme improves the fidelity. 928:, and all three qubits are flipped with probability 7823: 7076:"Real-Time Error Correction for Quantum Computing" 6713: 6504: 5112: 5088: 5060: 5031: 4936: 4912: 4888: 4854: 4815: 4779: 4731: 4692: 4626: 4594: 4568: 4540: 4241: 4169: 4070: 4039: 4011: 3976: 3823: 3794: 3768: 3732: 3550:are complex constants, I is the identity, and the 3542: 3515: 3488: 3461: 3434: 3350: 3330: 3293: 3136: 2978: 2883: 2802: 2759: 2732: 2704: 2675: 2567: 2489: 2412: 2338: 2310: 2262: 2242: 2204: 2172: 2000: 1973: 1850: 1780: 1753: 1662: 1631: 1594: 1564: 1427: 1403: 1376: 1349: 1043: 1019: 971: 947: 920: 875: 830: 791: 757: 726: 696: 621: 595: 536: 472: 446: 367: 295: 275: 251: 231: 207: 73:. The simplest albeit inefficient approach is the 7184: 6296:Quantum Communication, Computing, and Measurement 5624: 4249:, where each of the states is a superposition of 2891:will be transformed into the product of 9 qubits 765:is what is now passed through the noisy channel. 9546: 7473: 6828: 5244:are a family of codes parameterized by integers 2212:. A little algebra then shows that the fidelity 7578: 7354: 6292:"Quantum Error Correction with Imperfect Gates" 4780:{\displaystyle {\hat {a}}^{\dagger }{\hat {a}}} 151:of this logical qubit with other qubits in the 6054: 6052: 4700:using a dispersively coupled ancillary qubit. 108:first discovered this method of formulating a 9256: 8449: 7564: 7490: 7301: 5602: 3435:{\displaystyle U=c_{0}I+c_{1}X+c_{2}Y+c_{3}Z} 838:, a single qubit is flipped with probability 7540:from the original on 2021-12-22 – via 7464: 4528: 4508: 4487: 4457: 4440: 4422: 4392: 4372: 4351: 4321: 4304: 4286: 4236: 4233: 4207: 4184: 4164: 4161: 4135: 4112: 4062: 4034: 3968: 3954: 3922: 3908: 3891: 3325: 3285: 3271: 3251: 3237: 3217: 3203: 3167: 3128: 3114: 3094: 3080: 3060: 3046: 3010: 2973: 2942: 2911: 2878: 2854: 2830: 2727: 2667: 2631: 2595: 2562: 2538: 2514: 2466: 2452: 2435: 2392: 2378: 2361: 2333: 2305: 2070: 2059: 2046: 2035: 2001:{\displaystyle \rho _{\operatorname {out} }} 1950: 1939: 1936: 1925: 1781:{\displaystyle \rho _{\operatorname {out} }} 1742: 1731: 1728: 1717: 1657: 1646: 1329: 1326: 1304: 1301: 1258: 1255: 1233: 1230: 1187: 1184: 1162: 1159: 1116: 1113: 1091: 1088: 786: 775: 752: 741: 721: 715: 691: 685: 669: 663: 647: 636: 616: 610: 590: 584: 578: 563: 557: 551: 531: 525: 519: 504: 498: 492: 441: 417: 393: 202: 196: 6049: 5612:Quantum Computation and Quantum Information 5380:has first been realized using 10 ions in a 5341: 480:. We can however improve on this number by 9527: 9263: 9249: 8456: 8442: 7571: 7557: 7456:La Guardia, Giuliano Gadioli, ed. (2020). 5509: 5507: 5378:topological quantum error-correction codes 5290:entanglement-assisted stabilizer formalism 5178:A generalisation of the technique used by 883:, two qubits are flipped with probability 9270: 7504: 7368: 7228: 7210: 7137: 7099: 6933: 6846: 6789: 6727: 6571: 6518: 6457: 6404: 6375: 6330: 6250: 6205: 6073: 6016: 5962: 5904: 5843: 5770: 5714: 5642: 5492: 5474: 3867:basis, the simplest binomial encoding is 1924: 1538: 100:the (logical) information of one logical 7251: 5627:"Quantum error correction for beginners" 2778: 2277: 984: 259:, or leaves it unchanged. The action of 177: 8089:Continuous-variable quantum information 7536:. University of Sheffield. 2018-09-28. 7415: 7276: 6988: 6194:IEEE Transactions on Information Theory 5504: 5300:shared between a sender and a receiver. 14: 9547: 7530:"Topological Quantum Error Correction" 6289: 5376:between two logical qubits encoded in 1981:, where the dots denote components of 124:) flip, or both (corresponding to the 9244: 8463: 8437: 7552: 7350: 7348: 7326: 6443: 6236: 6127: 6125: 5944: 5942: 5940: 5559: 5456: 5274:are the basis for various code types. 4816:{\displaystyle {\hat {a}}^{\dagger }} 1851:{\displaystyle (1-p)^{3}+3p(1-p)^{2}} 7073: 5824: 5822: 5752: 5750: 5696: 5694: 5513: 5256:qubits arranged in a square lattice. 1663:{\displaystyle \vert \psi '\rangle } 792:{\displaystyle \vert \psi '\rangle } 758:{\displaystyle \vert \psi '\rangle } 4019:the corresponding error states are 622:{\displaystyle \vert \psi \rangle } 208:{\displaystyle \vert \psi \rangle } 24: 7484:Mathematics of quantum computation 7437: 7345: 7302:Smith-Goodson, Paul (2024-04-18). 6122: 5937: 5327:that the error rate of individual 5105: 5081: 5052: 5020: 5005: 4990: 4966: 4956: 4929: 4905: 4881: 4871: 4847: 3948: 3885: 3161: 3004: 1709: 1693: 1445: 1031:Let us assume for simplicity that 964: 572: 513: 312: 268: 224: 25: 9571: 7522: 5819: 5747: 5691: 5348:nuclear magnetic resonance qubits 5270:, and the more general idea of a 2273: 37:) is a set of techniques used in 9526: 9225: 9224: 8418: 8417: 8408: 8407: 7416:Steffen, Matthias (2022-10-20). 7252:Boerkamp, Martijn (2023-03-20). 7185:Google Quantum AI (2023-02-22). 5457:Cai, Weizhou; Ma, Yuwei (2021). 5292:as an extension of the standard 5061:{\displaystyle P_{\mathcal {C}}} 4826: 3847: 3834: 2760:{\displaystyle E_{\text{phase}}} 2705:{\displaystyle E_{\text{phase}}} 2499:The original state of the qubit 158: 55:fault tolerant quantum computing 9560:Fault-tolerant computer systems 7409: 7320: 7295: 7270: 7245: 7178: 7116: 7067: 7007: 6982: 6912: 6887: 6822: 6768: 6707: 6656: 6604: 6551: 6498: 6437: 6384: 6363: 6310: 6283: 6230: 6185: 5995: 4989: 3936: 3802:, a sign flip error occurs. If 2575:will be changed into the state 727:{\displaystyle \vert 0\rangle } 115:In classical error correction, 9174:Relativistic quantum mechanics 6989:Bedford, Bailey (2021-10-04). 6423:10.1088/1367-2630/14/12/123011 6298:. Springer. pp. 181–188. 5884: 5789:10.1103/physrevlett.111.120501 5631:Reports on Progress in Physics 5618: 5596: 5553: 5450: 5434:Error detection and correction 5113:{\displaystyle {\mathcal {R}}} 5089:{\displaystyle {\mathcal {C}}} 4980: 4974: 4971: 4951: 4937:{\displaystyle {\mathcal {R}}} 4913:{\displaystyle {\mathcal {H}}} 4855:{\displaystyle {\mathcal {E}}} 4801: 4771: 4753: 4732:{\displaystyle {\hat {a}}^{2}} 4717: 4687: 4681: 4663: 4647: 4618: 4515: 4498: 4468: 4447: 4433: 4403: 4379: 4362: 4332: 4311: 4297: 4267: 4214: 4188: 4142: 4116: 4055: 4027: 4000: 3961: 3938: 3915: 3901: 3875: 3776:, a bit flip error occurs. If 3331:{\displaystyle |\psi \rangle } 3318: 3288: 3278: 3264: 3260: 3254: 3244: 3230: 3226: 3220: 3210: 3196: 3192: 3151: 3131: 3121: 3107: 3103: 3097: 3087: 3073: 3069: 3063: 3053: 3039: 3035: 2994: 2959: 2928: 2899: 2871: 2847: 2823: 2733:{\displaystyle |\psi \rangle } 2720: 2648: 2612: 2583: 2555: 2531: 2507: 2469: 2459: 2445: 2441: 2428: 2395: 2385: 2371: 2367: 2354: 2326: 2298: 2120: 2107: 2089: 2076: 2029: 2018: 1968: 1956: 1921: 1912: 1899: 1881: 1868: 1865: 1839: 1826: 1808: 1795: 1748: 1745: 1714: 1704: 1626: 1615: 1462: 1456: 1336: 1319: 1311: 1294: 1265: 1248: 1240: 1223: 1194: 1177: 1169: 1152: 1123: 1106: 1098: 1081: 972:{\displaystyle {\mathcal {E}}} 915: 903: 864: 851: 819: 806: 560: 501: 434: 410: 386: 341: 329: 323: 317: 276:{\displaystyle {\mathcal {E}}} 232:{\displaystyle {\mathcal {E}}} 13: 1: 9152:Quantum statistical mechanics 8929:Quantum differential calculus 8851:Delayed-choice quantum eraser 8619:Symmetry in quantum mechanics 8084:Adiabatic quantum computation 7451:. Cambridge University Press. 7277:Conover, Emily (2023-02-22). 6290:Kitaev, Alexei (1997-07-31). 5862:10.1088/1367-2630/16/4/045014 5661:10.1088/0034-4885/76/7/076001 5614:. Cambridge University Press. 5444: 5194:, named for their inventors: 3358:can be described in the form 110:quantum error correcting code 8135:Topological quantum computer 5391:In 2022, researchers at the 5382:trapped-ion quantum computer 5272:topological quantum computer 5266:, introduced in 1997 as the 2770: 1411:is observed, then the Pauli 303:can therefore be written as 27:Process in quantum computing 7: 8954:Quantum stochastic calculus 8944:Quantum measurement problem 8866:Mach–Zehnder interferometer 8413:Quantum information science 7580:Quantum information science 7447:and Todd Brun, ed. (2013). 7387:10.1103/physreva.105.022428 7074:Ball, Philip (2021-12-23). 6537:10.1103/PhysRevLett.81.2152 5427: 4086: 4071:{\displaystyle |1\rangle ,} 4012:{\displaystyle {\hat {a}},} 3841:quantum harmonic oscillator 3750:, then no error occurs. If 921:{\displaystyle 3p^{2}(1-p)} 876:{\displaystyle 3p(1-p)^{2}} 10: 9576: 9510:Thermoacoustic heat engine 7808:quantum gate teleportation 7212:10.1038/s41586-022-05434-1 7156:10.1038/s41586-022-04721-1 7037:10.1038/s41586-021-03928-y 6952:10.1038/s41586-020-03079-6 6865:10.1016/j.scib.2020.07.033 6590:10.1103/PhysRevA.71.052332 6349:10.1103/PhysRevA.86.032324 6269:10.1103/PhysRevA.73.012340 6035:10.1103/PhysRevA.97.032346 5923:10.1103/PhysRevA.64.012310 5494:10.1016/j.fmre.2020.12.006 4627:{\displaystyle {\hat {a}}} 4040:{\displaystyle |3\rangle } 2420:may have its sign flip to 2339:{\displaystyle |1\rangle } 2311:{\displaystyle |0\rangle } 629:is encoded into the state 9522: 9495:Immersive virtual reality 9455: 9285: 9278: 9220: 9182: 9134: 9014:Quantum complexity theory 8992:Quantum cellular automata 8967: 8899: 8833: 8746: 8695: 8682:Path integral formulation 8649: 8514: 8471: 8403: 8346: 8309: 8275: 8252: 8219: 8210: 8143: 8072: 8010: 7970: 7937:Quantum Fourier transform 7882: 7833:Post-quantum cryptography 7776:Entanglement distillation 7749: 7658: 7586: 7327:Yirka, Bob (2024-04-05). 6304:10.1007/978-1-4615-5923-8 5981:10.1103/PhysRevX.6.031006 5539:10.1103/PhysRevA.52.R2493 5313:quantum threshold theorem 5264:topological quantum codes 5139: 1761:. Being the output state 1632:{\displaystyle F(\psi ')} 1020:{\displaystyle p\leq 1/2} 831:{\displaystyle (1-p)^{3}} 708:initialized in the state 169:three-qubit bit flip code 9478:Digital scent technology 9081:Quantum machine learning 9061:Quantum key distribution 9051:Quantum image processing 9041:Quantum error correction 8891:Wheeler's delayed choice 8423:Quantum mechanics topics 8118:Quantum machine learning 8094:One-way quantum computer 7947:Quantum phase estimation 7848:Quantum key distribution 7781:Monogamy of entanglement 7449:Quantum Error Correction 6808:10.1038/nphoton.2010.168 5733:10.1103/PhysRevA.59.2631 5582:10.1103/PhysRevA.32.3266 5342:Experimental realization 5233:; these are also called 2243:{\displaystyle p<1/2} 215:through a noisy channel 31:Quantum error correction 8997:Quantum finite automata 8030:Randomized benchmarking 7892:Amplitude amplification 7469:. Taylor & Francis. 6685:10.1126/science.1203329 5759:Physical Review Letters 5393:University of Innsbruck 5188:classical Hamming code 1415:gate is applied to the 9500:Magnetic refrigeration 9101:Quantum neural network 8130:Quantum Turing machine 8123:quantum neural network 7870:Quantum secret sharing 7101:10.1103/Physics.14.184 6393:New Journal of Physics 5832:New Journal of Physics 5359:Schrodinger-cat states 5114: 5090: 5062: 5033: 4938: 4914: 4890: 4856: 4817: 4781: 4733: 4694: 4628: 4596: 4570: 4542: 4243: 4171: 4103:Schrödinger cat states 4072: 4041: 4013: 3978: 3825: 3796: 3770: 3734: 3544: 3517: 3490: 3463: 3436: 3352: 3332: 3295: 3138: 2980: 2885: 2804: 2787: 2761: 2734: 2706: 2677: 2569: 2491: 2414: 2340: 2312: 2286: 2285:of the phase flip code 2264: 2244: 2206: 2174: 2002: 1975: 1852: 1782: 1755: 1664: 1633: 1596: 1566: 1514: 1429: 1405: 1378: 1351: 1045: 1028: 1021: 973: 949: 922: 877: 832: 793: 759: 728: 698: 623: 597: 538: 474: 448: 369: 297: 277: 253: 233: 209: 186: 9473:Cloak of invisibility 9272:Emerging technologies 9126:Quantum teleportation 8639:Wave–particle duality 8202:Entanglement-assisted 8163:quantum convolutional 7838:Quantum coin flipping 7803:Quantum teleportation 7764:entanglement-assisted 7594:DiVincenzo's criteria 7465:Frank Gaitan (2008). 5288:also constructed the 5115: 5091: 5070:orthogonal projection 5063: 5034: 4939: 4915: 4891: 4857: 4818: 4782: 4734: 4695: 4629: 4597: 4571: 4543: 4244: 4172: 4073: 4042: 4014: 3979: 3826: 3797: 3771: 3735: 3545: 3543:{\displaystyle c_{3}} 3518: 3516:{\displaystyle c_{2}} 3491: 3489:{\displaystyle c_{1}} 3464: 3462:{\displaystyle c_{0}} 3437: 3353: 3333: 3296: 3139: 2981: 2886: 2805: 2782: 2762: 2735: 2707: 2678: 2570: 2492: 2415: 2341: 2313: 2281: 2270:being small enough). 2265: 2245: 2207: 2205:{\displaystyle {1-p}} 2175: 2003: 1976: 1858:, we can write it as 1853: 1783: 1756: 1665: 1634: 1597: 1595:{\displaystyle P_{3}} 1567: 1494: 1430: 1406: 1404:{\displaystyle P_{i}} 1379: 1377:{\displaystyle P_{0}} 1352: 1046: 1022: 989:Comparison of output 988: 974: 950: 948:{\displaystyle p^{3}} 923: 878: 833: 794: 760: 729: 699: 624: 598: 539: 475: 449: 370: 298: 296:{\displaystyle \rho } 278: 254: 234: 210: 181: 149:quantum superposition 9157:Quantum field theory 9086:Quantum metamaterial 9031:Quantum cryptography 8761:Consistent histories 8013:processor benchmarks 7942:Quantum optimization 7825:Quantum cryptography 7636:physical vs. logical 7482:and quantum codes". 5463:Fundamental Research 5298:quantum entanglement 5294:stabilizer formalism 5122:correction operation 5100: 5076: 5043: 4948: 4924: 4900: 4866: 4842: 4791: 4787:, photon-gain error 4743: 4707: 4638: 4609: 4595:{\displaystyle ^{-}} 4580: 4569:{\displaystyle ^{+}} 4554: 4259: 4181: 4109: 4051: 4023: 3991: 3871: 3824:{\displaystyle U=iY} 3806: 3780: 3754: 3558: 3527: 3500: 3473: 3446: 3362: 3342: 3314: 3147: 2990: 2895: 2819: 2794: 2744: 2716: 2689: 2579: 2503: 2424: 2350: 2322: 2294: 2254: 2220: 2188: 2012: 1985: 1862: 1792: 1765: 1674: 1643: 1609: 1579: 1439: 1419: 1388: 1361: 1056: 1035: 997: 959: 932: 887: 842: 803: 772: 768:The channel acts on 738: 734:. The encoded state 712: 633: 607: 548: 489: 458: 382: 307: 287: 263: 243: 219: 193: 185:of the bit flip code 9505:Phased-array optics 9463:Acoustic levitation 9142:Quantum fluctuation 9111:Quantum programming 9071:Quantum logic gates 9056:Quantum information 9036:Quantum electronics 8496:Classical mechanics 7726:Quantum speed limit 7621:Quantum programming 7616:Quantum information 7515:1998quant.ph.10055F 7493:Found. Comput. Math 7379:2022PhRvA.105b2428A 7203:2023Natur.614..676G 7148:2022Natur.605..675P 7092:2021PhyOJ..14..184B 7029:2021Natur.598..281E 6944:2021Natur.589..220E 6857:2021SciBu..66...29G 6800:2010NaPho...4..700L 6746:10.1038/nature10786 6738:2012Natur.482..382R 6677:2011Sci...332.1059S 6671:(6033): 1059–1061. 6634:10.1038/nature03074 6626:2004Natur.432..602C 6582:2005PhRvA..71e2332P 6529:1998PhRvL..81.2152C 6476:10.1038/nature03350 6468:2005Natur.434...39K 6415:2012NJPh...14l3011H 6341:2012PhRvA..86c2324F 6261:2006PhRvA..73a2340B 6155:10.1038/nature18949 6147:2016Natur.536..441O 6092:10.1038/nature13436 6084:2014Natur.511..444S 6027:2018PhRvA..97c2346A 5973:2016PhRvX...6c1006M 5915:2001PhRvA..64a2310G 5854:2014NJPh...16d5014M 5781:2013PhRvL.111l0501L 5725:1999PhRvA..59.2631C 5653:2013RPPh...76g6001D 5604:Nielsen, Michael A. 5574:1985PhRvA..32.3266P 5531:1995PhRvA..52.2493S 5485:2021FunRe...1...50C 5129:non-degenerate code 4739:, dephasing noise 4486: 4421: 4350: 4285: 4232: 4206: 4160: 4134: 3795:{\displaystyle U=Z} 3769:{\displaystyle U=X} 473:{\displaystyle 1-p} 283:on a general input 134:quantum measurement 45:from errors due to 43:quantum information 9195:in popular culture 8977:Quantum algorithms 8825:Von Neumann–Wigner 8805:Objective collapse 8501:Old quantum theory 8375:Forest/Rigetti QCS 8111:quantum logic gate 7897:Bernstein–Vazirani 7884:Quantum algorithms 7759:Classical capacity 7643:Quantum processors 7626:Quantum simulation 7460:. Springer Nature. 5525:(4): R2493–R2496. 5363:quantum controller 5296:that incorporates 5110: 5086: 5058: 5029: 4934: 4910: 4886: 4852: 4813: 4777: 4729: 4690: 4624: 4592: 4566: 4538: 4536: 4472: 4407: 4336: 4271: 4239: 4218: 4192: 4167: 4146: 4120: 4068: 4037: 4009: 3974: 3821: 3792: 3766: 3730: 3728: 3717: 3661: 3605: 3540: 3513: 3486: 3459: 3432: 3348: 3328: 3291: 3134: 2976: 2881: 2800: 2788: 2757: 2730: 2702: 2673: 2565: 2487: 2410: 2336: 2308: 2287: 2260: 2240: 2202: 2170: 1998: 1971: 1848: 1778: 1751: 1660: 1639:between the input 1629: 1592: 1562: 1425: 1401: 1374: 1347: 1345: 1041: 1029: 1017: 969: 945: 918: 873: 828: 789: 755: 724: 694: 619: 603:. The input state 593: 534: 470: 444: 365: 293: 273: 249: 229: 205: 187: 165:no-cloning theorem 94:no-cloning theorem 9555:Quantum computing 9542: 9541: 9518: 9517: 9325:complexity theory 9310:cellular automata 9238: 9237: 9212:Quantum mysticism 9190:Schrödinger's cat 9121:Quantum simulator 9091:Quantum metrology 9019:Quantum computing 8982:Quantum amplifier 8959:Quantum spacetime 8924:Quantum cosmology 8914:Quantum chemistry 8614:Scattering theory 8562:Zero-point energy 8557:Degenerate levels 8465:Quantum mechanics 8431: 8430: 8342: 8341: 8239:Linear optical QC 8020:Quantum supremacy 7974:complexity theory 7927:Quantum annealing 7878: 7877: 7815:Superdense coding 7604:Quantum computing 7422:IBM Research Blog 7357:Physical Review A 7197:(7949): 676–681. 7132:(7911): 675–680. 7023:(7880): 281–286. 6928:(7841): 220–224. 6722:(7385): 382–385. 6620:(7017): 602–605. 6513:(10): 2152–2155. 6319:Physical Review A 6239:Physical Review A 6216:10.1109/18.681315 6141:(7617): 441–445. 6068:(7510): 444–448. 6005:Physical Review A 5951:Physical Review X 5893:Physical Review A 5703:Physical Review A 5562:Physical Review A 5519:Physical Review A 5219:Robert Calderbank 5196:Robert Calderbank 5182:, to develop the 4804: 4774: 4756: 4720: 4684: 4666: 4621: 4604:lowering operator 4003: 3986:lowering operator 3931: 3930: 3351:{\displaystyle U} 3190: 3187: 3033: 3030: 2803:{\displaystyle E} 2754: 2699: 2482: 2408: 2263:{\displaystyle p} 1428:{\displaystyle i} 1044:{\displaystyle p} 355: 252:{\displaystyle p} 117:syndrome decoding 39:quantum computing 16:(Redirected from 9567: 9530: 9529: 9407:machine learning 9382:key distribution 9367:image processing 9357:error correction 9283: 9282: 9265: 9258: 9251: 9242: 9241: 9228: 9227: 8939:Quantum geometry 8934:Quantum dynamics 8791:Superdeterminism 8723:Rarita–Schwinger 8672:Matrix mechanics 8527:Bra–ket notation 8458: 8451: 8444: 8435: 8434: 8421: 8420: 8411: 8410: 8217: 8216: 8147:error correction 8076:computing models 8042:Relaxation times 7932:Quantum counting 7821: 7820: 7769:quantum capacity 7716:No-teleportation 7701:No-communication 7573: 7566: 7559: 7550: 7549: 7545: 7518: 7508: 7506:quant-ph/9810055 7487: 7480:Systolic freedom 7470: 7461: 7452: 7432: 7431: 7429: 7428: 7413: 7407: 7406: 7372: 7352: 7343: 7342: 7340: 7339: 7324: 7318: 7317: 7315: 7314: 7299: 7293: 7292: 7290: 7289: 7274: 7268: 7267: 7265: 7264: 7249: 7243: 7242: 7232: 7214: 7182: 7176: 7175: 7141: 7120: 7114: 7113: 7103: 7071: 7065: 7064: 7011: 7005: 7004: 7002: 7001: 6986: 6980: 6979: 6937: 6916: 6910: 6909: 6907: 6906: 6891: 6885: 6884: 6850: 6835:Science Bulletin 6826: 6820: 6819: 6793: 6778:Nature Photonics 6772: 6766: 6765: 6731: 6711: 6705: 6704: 6660: 6654: 6653: 6608: 6602: 6601: 6575: 6573:quant-ph/0502042 6555: 6549: 6548: 6522: 6520:quant-ph/9802018 6502: 6496: 6495: 6461: 6459:quant-ph/0410199 6441: 6435: 6434: 6408: 6388: 6382: 6381: 6379: 6367: 6361: 6360: 6334: 6314: 6308: 6307: 6287: 6281: 6280: 6254: 6252:quant-ph/0506023 6234: 6228: 6227: 6209: 6207:quant-ph/9608006 6200:(4): 1369–1387. 6189: 6183: 6182: 6129: 6120: 6119: 6077: 6056: 6047: 6046: 6020: 5999: 5993: 5992: 5966: 5946: 5935: 5934: 5908: 5906:quant-ph/0008040 5888: 5882: 5881: 5847: 5826: 5817: 5816: 5774: 5754: 5745: 5744: 5718: 5716:quant-ph/9809037 5709:(4): 2631–2634. 5698: 5689: 5688: 5646: 5622: 5616: 5615: 5608:Chuang, Isaac L. 5600: 5594: 5593: 5568:(6): 3266–3276. 5557: 5551: 5550: 5511: 5502: 5501: 5496: 5478: 5454: 5242:Bacon–Shor codes 5240:Two dimensional 5215:Daniel Gottesman 5211:stabilizer codes 5165:Raymond Laflamme 5120:is known as the 5119: 5117: 5116: 5111: 5109: 5108: 5095: 5093: 5092: 5087: 5085: 5084: 5067: 5065: 5064: 5059: 5057: 5056: 5055: 5038: 5036: 5035: 5030: 5025: 5024: 5023: 5010: 5009: 5008: 4970: 4969: 4960: 4959: 4943: 4941: 4940: 4935: 4933: 4932: 4919: 4917: 4916: 4911: 4909: 4908: 4895: 4893: 4892: 4887: 4885: 4884: 4875: 4874: 4861: 4859: 4858: 4853: 4851: 4850: 4822: 4820: 4819: 4814: 4812: 4811: 4806: 4805: 4797: 4786: 4784: 4783: 4778: 4776: 4775: 4767: 4764: 4763: 4758: 4757: 4749: 4738: 4736: 4735: 4730: 4728: 4727: 4722: 4721: 4713: 4699: 4697: 4696: 4691: 4686: 4685: 4677: 4674: 4673: 4668: 4667: 4659: 4633: 4631: 4630: 4625: 4623: 4622: 4614: 4601: 4599: 4598: 4593: 4591: 4590: 4575: 4573: 4572: 4567: 4565: 4564: 4547: 4545: 4544: 4539: 4537: 4518: 4501: 4485: 4480: 4471: 4450: 4436: 4420: 4415: 4406: 4382: 4365: 4349: 4344: 4335: 4314: 4300: 4284: 4279: 4270: 4248: 4246: 4245: 4240: 4231: 4226: 4217: 4205: 4200: 4191: 4176: 4174: 4173: 4168: 4159: 4154: 4145: 4133: 4128: 4119: 4099: 4098: 4094: 4077: 4075: 4074: 4069: 4058: 4046: 4044: 4043: 4038: 4030: 4018: 4016: 4015: 4010: 4005: 4004: 3996: 3983: 3981: 3980: 3975: 3964: 3953: 3952: 3951: 3941: 3932: 3926: 3925: 3918: 3904: 3898: 3890: 3889: 3888: 3878: 3860: 3859: 3855: 3830: 3828: 3827: 3822: 3801: 3799: 3798: 3793: 3775: 3773: 3772: 3767: 3739: 3737: 3736: 3731: 3729: 3722: 3721: 3666: 3665: 3610: 3609: 3549: 3547: 3546: 3541: 3539: 3538: 3522: 3520: 3519: 3514: 3512: 3511: 3495: 3493: 3492: 3487: 3485: 3484: 3468: 3466: 3465: 3460: 3458: 3457: 3441: 3439: 3438: 3433: 3428: 3427: 3412: 3411: 3396: 3395: 3380: 3379: 3357: 3355: 3354: 3349: 3337: 3335: 3334: 3329: 3321: 3300: 3298: 3297: 3292: 3281: 3267: 3247: 3233: 3213: 3199: 3191: 3189: 3188: 3183: 3174: 3166: 3165: 3164: 3154: 3143: 3141: 3140: 3135: 3124: 3110: 3090: 3076: 3056: 3042: 3034: 3032: 3031: 3026: 3017: 3009: 3008: 3007: 2997: 2985: 2983: 2982: 2977: 2972: 2971: 2962: 2957: 2956: 2941: 2940: 2931: 2926: 2925: 2910: 2902: 2890: 2888: 2887: 2882: 2874: 2869: 2868: 2850: 2845: 2844: 2826: 2809: 2807: 2806: 2801: 2766: 2764: 2763: 2758: 2756: 2755: 2752: 2739: 2737: 2736: 2731: 2723: 2711: 2709: 2708: 2703: 2701: 2700: 2697: 2682: 2680: 2679: 2674: 2666: 2661: 2656: 2651: 2646: 2645: 2630: 2625: 2620: 2615: 2610: 2609: 2594: 2586: 2574: 2572: 2571: 2566: 2558: 2553: 2552: 2534: 2529: 2528: 2510: 2496: 2494: 2493: 2488: 2483: 2478: 2476: 2462: 2448: 2431: 2419: 2417: 2416: 2411: 2409: 2404: 2402: 2388: 2374: 2357: 2345: 2343: 2342: 2337: 2329: 2317: 2315: 2314: 2309: 2301: 2269: 2267: 2266: 2261: 2249: 2247: 2246: 2241: 2236: 2211: 2209: 2208: 2203: 2201: 2179: 2177: 2176: 2171: 2166: 2165: 2150: 2149: 2128: 2127: 2097: 2096: 2069: 2058: 2057: 2045: 2028: 2007: 2005: 2004: 1999: 1997: 1996: 1980: 1978: 1977: 1972: 1949: 1935: 1920: 1919: 1889: 1888: 1857: 1855: 1854: 1849: 1847: 1846: 1816: 1815: 1787: 1785: 1784: 1779: 1777: 1776: 1760: 1758: 1757: 1752: 1741: 1727: 1713: 1712: 1703: 1702: 1697: 1696: 1686: 1685: 1669: 1667: 1666: 1661: 1656: 1638: 1636: 1635: 1630: 1625: 1601: 1599: 1598: 1593: 1591: 1590: 1571: 1569: 1568: 1563: 1558: 1557: 1548: 1547: 1534: 1533: 1524: 1523: 1513: 1508: 1490: 1489: 1477: 1476: 1455: 1454: 1449: 1448: 1434: 1432: 1431: 1426: 1410: 1408: 1407: 1402: 1400: 1399: 1383: 1381: 1380: 1375: 1373: 1372: 1356: 1354: 1353: 1348: 1346: 1339: 1322: 1314: 1297: 1285: 1284: 1268: 1251: 1243: 1226: 1214: 1213: 1197: 1180: 1172: 1155: 1143: 1142: 1126: 1109: 1101: 1084: 1072: 1071: 1050: 1048: 1047: 1042: 1026: 1024: 1023: 1018: 1013: 978: 976: 975: 970: 968: 967: 954: 952: 951: 946: 944: 943: 927: 925: 924: 919: 902: 901: 882: 880: 879: 874: 872: 871: 837: 835: 834: 829: 827: 826: 798: 796: 795: 790: 785: 764: 762: 761: 756: 751: 733: 731: 730: 725: 706:ancillary qubits 703: 701: 700: 695: 684: 683: 662: 661: 646: 628: 626: 625: 620: 602: 600: 599: 594: 577: 576: 575: 543: 541: 540: 535: 518: 517: 516: 479: 477: 476: 471: 453: 451: 450: 445: 437: 432: 431: 413: 408: 407: 389: 374: 372: 371: 366: 353: 316: 315: 302: 300: 299: 294: 282: 280: 279: 274: 272: 271: 258: 256: 255: 250: 238: 236: 235: 230: 228: 227: 214: 212: 211: 206: 153:quantum computer 67:error correction 21: 9575: 9574: 9570: 9569: 9568: 9566: 9565: 9564: 9545: 9544: 9543: 9538: 9514: 9451: 9362:finite automata 9274: 9269: 9239: 9234: 9216: 9202:Wigner's friend 9178: 9169:Quantum gravity 9130: 9116:Quantum sensing 9096:Quantum network 9076:Quantum machine 9046:Quantum imaging 9009:Quantum circuit 9004:Quantum channel 8963: 8909:Quantum biology 8895: 8871:Elitzur–Vaidman 8846:Davisson–Germer 8829: 8781:Hidden-variable 8771:de Broglie–Bohm 8748:Interpretations 8742: 8691: 8645: 8532:Complementarity 8510: 8467: 8462: 8432: 8427: 8399: 8349: 8338: 8311:Superconducting 8305: 8271: 8262:Neutral atom QC 8254:Ultracold atoms 8248: 8213:implementations 8212: 8206: 8146: 8139: 8106:Quantum circuit 8074: 8068: 8062: 8052: 8012: 8006: 7973: 7966: 7922:Hidden subgroup 7874: 7863:other protocols 7819: 7796:quantum network 7791:Quantum channel 7751: 7745: 7691:No-broadcasting 7681:Gottesman–Knill 7654: 7582: 7577: 7528: 7525: 7477: 7440: 7438:Further reading 7435: 7426: 7424: 7414: 7410: 7353: 7346: 7337: 7335: 7325: 7321: 7312: 7310: 7300: 7296: 7287: 7285: 7275: 7271: 7262: 7260: 7250: 7246: 7183: 7179: 7121: 7117: 7072: 7068: 7012: 7008: 6999: 6997: 6987: 6983: 6917: 6913: 6904: 6902: 6893: 6892: 6888: 6827: 6823: 6773: 6769: 6712: 6708: 6661: 6657: 6609: 6605: 6556: 6552: 6507:Phys. Rev. Lett 6503: 6499: 6452:(7029): 39–44. 6442: 6438: 6389: 6385: 6368: 6364: 6315: 6311: 6288: 6284: 6235: 6231: 6190: 6186: 6130: 6123: 6057: 6050: 6000: 5996: 5947: 5938: 5889: 5885: 5827: 5820: 5755: 5748: 5699: 5692: 5623: 5619: 5601: 5597: 5558: 5554: 5512: 5505: 5455: 5451: 5447: 5430: 5421: 5374:entangling gate 5344: 5231:N. J. A. Sloane 5142: 5104: 5103: 5101: 5098: 5097: 5080: 5079: 5077: 5074: 5073: 5051: 5050: 5046: 5044: 5041: 5040: 5019: 5018: 5014: 5004: 5003: 4999: 4965: 4964: 4955: 4954: 4949: 4946: 4945: 4928: 4927: 4925: 4922: 4921: 4904: 4903: 4901: 4898: 4897: 4880: 4879: 4870: 4869: 4867: 4864: 4863: 4846: 4845: 4843: 4840: 4839: 4837:quantum channel 4829: 4807: 4796: 4795: 4794: 4792: 4789: 4788: 4766: 4765: 4759: 4748: 4747: 4746: 4744: 4741: 4740: 4723: 4712: 4711: 4710: 4708: 4705: 4704: 4676: 4675: 4669: 4658: 4657: 4656: 4639: 4636: 4635: 4613: 4612: 4610: 4607: 4606: 4586: 4583: 4581: 4578: 4577: 4560: 4557: 4555: 4552: 4551: 4535: 4534: 4514: 4497: 4490: 4481: 4476: 4467: 4464: 4463: 4446: 4432: 4425: 4416: 4411: 4402: 4399: 4398: 4378: 4361: 4354: 4345: 4340: 4331: 4328: 4327: 4310: 4296: 4289: 4280: 4275: 4266: 4262: 4260: 4257: 4256: 4227: 4222: 4213: 4201: 4196: 4187: 4182: 4179: 4178: 4155: 4150: 4141: 4129: 4124: 4115: 4110: 4107: 4106: 4100: 4096: 4092: 4090: 4089: 4054: 4052: 4049: 4048: 4026: 4024: 4021: 4020: 3995: 3994: 3992: 3989: 3988: 3960: 3947: 3946: 3942: 3937: 3914: 3900: 3899: 3897: 3884: 3883: 3879: 3874: 3872: 3869: 3868: 3863:Written in the 3861: 3857: 3853: 3851: 3850: 3837: 3807: 3804: 3803: 3781: 3778: 3777: 3755: 3752: 3751: 3727: 3726: 3716: 3715: 3707: 3701: 3700: 3695: 3685: 3684: 3677: 3671: 3670: 3660: 3659: 3654: 3648: 3647: 3639: 3629: 3628: 3621: 3615: 3614: 3604: 3603: 3598: 3592: 3591: 3586: 3576: 3575: 3568: 3561: 3559: 3556: 3555: 3534: 3530: 3528: 3525: 3524: 3507: 3503: 3501: 3498: 3497: 3480: 3476: 3474: 3471: 3470: 3453: 3449: 3447: 3444: 3443: 3423: 3419: 3407: 3403: 3391: 3387: 3375: 3371: 3363: 3360: 3359: 3343: 3340: 3339: 3317: 3315: 3312: 3311: 3277: 3263: 3243: 3229: 3209: 3195: 3182: 3178: 3173: 3160: 3159: 3155: 3150: 3148: 3145: 3144: 3120: 3106: 3086: 3072: 3052: 3038: 3025: 3021: 3016: 3003: 3002: 2998: 2993: 2991: 2988: 2987: 2967: 2963: 2958: 2952: 2948: 2936: 2932: 2927: 2921: 2917: 2903: 2898: 2896: 2893: 2892: 2870: 2864: 2860: 2846: 2840: 2836: 2822: 2820: 2817: 2816: 2812:quantum channel 2795: 2792: 2791: 2784:Quantum circuit 2773: 2751: 2747: 2745: 2742: 2741: 2719: 2717: 2714: 2713: 2696: 2692: 2690: 2687: 2686: 2662: 2657: 2652: 2647: 2641: 2637: 2626: 2621: 2616: 2611: 2605: 2601: 2587: 2582: 2580: 2577: 2576: 2554: 2548: 2544: 2530: 2524: 2520: 2506: 2504: 2501: 2500: 2477: 2472: 2458: 2444: 2427: 2425: 2422: 2421: 2403: 2398: 2384: 2370: 2353: 2351: 2348: 2347: 2325: 2323: 2320: 2319: 2297: 2295: 2292: 2291: 2283:Quantum circuit 2276: 2255: 2252: 2251: 2232: 2221: 2218: 2217: 2191: 2189: 2186: 2185: 2161: 2157: 2145: 2141: 2123: 2119: 2092: 2088: 2062: 2053: 2049: 2038: 2021: 2013: 2010: 2009: 1992: 1988: 1986: 1983: 1982: 1942: 1928: 1915: 1911: 1884: 1880: 1863: 1860: 1859: 1842: 1838: 1811: 1807: 1793: 1790: 1789: 1772: 1768: 1766: 1763: 1762: 1734: 1720: 1708: 1707: 1698: 1692: 1691: 1690: 1681: 1677: 1675: 1672: 1671: 1670:and the output 1649: 1644: 1641: 1640: 1618: 1610: 1607: 1606: 1586: 1582: 1580: 1577: 1576: 1553: 1549: 1543: 1539: 1529: 1525: 1519: 1515: 1509: 1498: 1485: 1481: 1472: 1468: 1450: 1444: 1443: 1442: 1440: 1437: 1436: 1420: 1417: 1416: 1395: 1391: 1389: 1386: 1385: 1368: 1364: 1362: 1359: 1358: 1344: 1343: 1335: 1318: 1310: 1293: 1286: 1280: 1276: 1273: 1272: 1264: 1247: 1239: 1222: 1215: 1209: 1205: 1202: 1201: 1193: 1176: 1168: 1151: 1144: 1138: 1134: 1131: 1130: 1122: 1105: 1097: 1080: 1073: 1067: 1063: 1059: 1057: 1054: 1053: 1036: 1033: 1032: 1009: 998: 995: 994: 963: 962: 960: 957: 956: 939: 935: 933: 930: 929: 897: 893: 888: 885: 884: 867: 863: 843: 840: 839: 822: 818: 804: 801: 800: 778: 773: 770: 769: 744: 739: 736: 735: 713: 710: 709: 679: 675: 657: 653: 639: 634: 631: 630: 608: 605: 604: 571: 570: 566: 549: 546: 545: 512: 511: 507: 490: 487: 486: 459: 456: 455: 433: 427: 423: 409: 403: 399: 385: 383: 380: 379: 311: 310: 308: 305: 304: 288: 285: 284: 267: 266: 264: 261: 260: 244: 241: 240: 223: 222: 220: 217: 216: 194: 191: 190: 183:Quantum circuit 161: 75:repetition code 28: 23: 22: 15: 12: 11: 5: 9573: 9563: 9562: 9557: 9540: 9539: 9537: 9536: 9523: 9520: 9519: 9516: 9515: 9513: 9512: 9507: 9502: 9497: 9492: 9491: 9490: 9480: 9475: 9470: 9465: 9459: 9457: 9453: 9452: 9450: 9449: 9444: 9439: 9434: 9429: 9424: 9422:neural network 9419: 9414: 9409: 9404: 9399: 9394: 9389: 9384: 9379: 9374: 9369: 9364: 9359: 9354: 9349: 9344: 9343: 9342: 9332: 9327: 9322: 9317: 9312: 9307: 9302: 9297: 9291: 9289: 9280: 9276: 9275: 9268: 9267: 9260: 9253: 9245: 9236: 9235: 9233: 9232: 9221: 9218: 9217: 9215: 9214: 9209: 9204: 9199: 9198: 9197: 9186: 9184: 9180: 9179: 9177: 9176: 9171: 9166: 9165: 9164: 9154: 9149: 9147:Casimir effect 9144: 9138: 9136: 9132: 9131: 9129: 9128: 9123: 9118: 9113: 9108: 9106:Quantum optics 9103: 9098: 9093: 9088: 9083: 9078: 9073: 9068: 9063: 9058: 9053: 9048: 9043: 9038: 9033: 9028: 9027: 9026: 9016: 9011: 9006: 9001: 9000: 8999: 8989: 8984: 8979: 8973: 8971: 8965: 8964: 8962: 8961: 8956: 8951: 8946: 8941: 8936: 8931: 8926: 8921: 8916: 8911: 8905: 8903: 8897: 8896: 8894: 8893: 8888: 8883: 8881:Quantum eraser 8878: 8873: 8868: 8863: 8858: 8853: 8848: 8843: 8837: 8835: 8831: 8830: 8828: 8827: 8822: 8817: 8812: 8807: 8802: 8797: 8796: 8795: 8794: 8793: 8778: 8773: 8768: 8763: 8758: 8752: 8750: 8744: 8743: 8741: 8740: 8735: 8730: 8725: 8720: 8715: 8710: 8705: 8699: 8697: 8693: 8692: 8690: 8689: 8684: 8679: 8674: 8669: 8664: 8659: 8653: 8651: 8647: 8646: 8644: 8643: 8642: 8641: 8636: 8626: 8621: 8616: 8611: 8606: 8601: 8596: 8591: 8586: 8581: 8576: 8571: 8566: 8565: 8564: 8559: 8554: 8549: 8539: 8537:Density matrix 8534: 8529: 8524: 8518: 8516: 8512: 8511: 8509: 8508: 8503: 8498: 8493: 8492: 8491: 8481: 8475: 8473: 8469: 8468: 8461: 8460: 8453: 8446: 8438: 8429: 8428: 8426: 8425: 8415: 8404: 8401: 8400: 8398: 8397: 8395:many others... 8392: 8387: 8382: 8377: 8368: 8354: 8352: 8344: 8343: 8340: 8339: 8337: 8336: 8331: 8326: 8321: 8315: 8313: 8307: 8306: 8304: 8303: 8298: 8293: 8288: 8282: 8280: 8273: 8272: 8270: 8269: 8267:Trapped-ion QC 8264: 8258: 8256: 8250: 8249: 8247: 8246: 8241: 8236: 8231: 8225: 8223: 8221:Quantum optics 8214: 8208: 8207: 8205: 8204: 8199: 8198: 8197: 8190: 8185: 8180: 8175: 8170: 8165: 8160: 8151: 8149: 8141: 8140: 8138: 8137: 8132: 8127: 8126: 8125: 8115: 8114: 8113: 8103: 8102: 8101: 8091: 8086: 8080: 8078: 8070: 8069: 8067: 8066: 8065: 8064: 8060: 8054: 8050: 8039: 8038: 8037: 8027: 8025:Quantum volume 8022: 8016: 8014: 8008: 8007: 8005: 8004: 7999: 7994: 7989: 7984: 7978: 7976: 7968: 7967: 7965: 7964: 7959: 7954: 7949: 7944: 7939: 7934: 7929: 7924: 7919: 7914: 7909: 7904: 7902:Boson sampling 7899: 7894: 7888: 7886: 7880: 7879: 7876: 7875: 7873: 7872: 7867: 7866: 7865: 7860: 7855: 7845: 7840: 7835: 7829: 7827: 7818: 7817: 7812: 7811: 7810: 7800: 7799: 7798: 7788: 7783: 7778: 7773: 7772: 7771: 7766: 7755: 7753: 7747: 7746: 7744: 7743: 7738: 7736:Solovay–Kitaev 7733: 7728: 7723: 7718: 7713: 7708: 7703: 7698: 7693: 7688: 7683: 7678: 7673: 7668: 7662: 7660: 7656: 7655: 7653: 7652: 7651: 7650: 7640: 7639: 7638: 7628: 7623: 7618: 7613: 7612: 7611: 7601: 7596: 7590: 7588: 7584: 7583: 7576: 7575: 7568: 7561: 7553: 7547: 7546: 7524: 7523:External links 7521: 7520: 7519: 7499:(3): 325–332. 7488: 7475: 7471: 7462: 7453: 7439: 7436: 7434: 7433: 7408: 7344: 7319: 7294: 7269: 7244: 7177: 7115: 7066: 7006: 6981: 6911: 6886: 6821: 6767: 6706: 6655: 6603: 6550: 6497: 6436: 6399:(12): 123011. 6383: 6362: 6309: 6282: 6229: 6184: 6121: 6048: 5994: 5936: 5883: 5818: 5765:(12): 120501. 5746: 5690: 5617: 5595: 5552: 5515:Shor, Peter W. 5503: 5448: 5446: 5443: 5442: 5441: 5436: 5429: 5426: 5420: 5417: 5343: 5340: 5321:Dorit Aharonov 5317:Michael Ben-Or 5309: 5308: 5301: 5286:Min-Hsiu Hsieh 5275: 5257: 5238: 5235:additive codes 5213:discovered by 5207: 5176: 5169:fault-tolerant 5162: 5152: 5141: 5138: 5107: 5083: 5054: 5049: 5028: 5022: 5017: 5013: 5007: 5002: 4998: 4995: 4992: 4988: 4985: 4982: 4979: 4976: 4973: 4968: 4963: 4958: 4953: 4931: 4907: 4883: 4878: 4873: 4862:is a subspace 4849: 4831:In general, a 4828: 4825: 4810: 4803: 4800: 4773: 4770: 4762: 4755: 4752: 4726: 4719: 4716: 4689: 4683: 4680: 4672: 4665: 4662: 4655: 4652: 4649: 4646: 4643: 4620: 4617: 4589: 4585: 4563: 4559: 4533: 4530: 4527: 4524: 4521: 4517: 4513: 4510: 4507: 4504: 4500: 4496: 4493: 4491: 4489: 4484: 4479: 4475: 4470: 4466: 4465: 4462: 4459: 4456: 4453: 4449: 4445: 4442: 4439: 4435: 4431: 4428: 4426: 4424: 4419: 4414: 4410: 4405: 4401: 4400: 4397: 4394: 4391: 4388: 4385: 4381: 4377: 4374: 4371: 4368: 4364: 4360: 4357: 4355: 4353: 4348: 4343: 4339: 4334: 4330: 4329: 4326: 4323: 4320: 4317: 4313: 4309: 4306: 4303: 4299: 4295: 4292: 4290: 4288: 4283: 4278: 4274: 4269: 4265: 4264: 4251:coherent state 4238: 4235: 4230: 4225: 4221: 4216: 4212: 4209: 4204: 4199: 4195: 4190: 4186: 4166: 4163: 4158: 4153: 4149: 4144: 4140: 4137: 4132: 4127: 4123: 4118: 4114: 4088: 4085: 4067: 4064: 4061: 4057: 4036: 4033: 4029: 4008: 4002: 3999: 3973: 3970: 3967: 3963: 3959: 3956: 3950: 3945: 3940: 3935: 3929: 3924: 3921: 3917: 3913: 3910: 3907: 3903: 3896: 3893: 3887: 3882: 3877: 3849: 3846: 3836: 3833: 3820: 3817: 3814: 3811: 3791: 3788: 3785: 3765: 3762: 3759: 3725: 3720: 3714: 3711: 3708: 3706: 3703: 3702: 3699: 3696: 3694: 3691: 3690: 3688: 3683: 3680: 3678: 3676: 3673: 3672: 3669: 3664: 3658: 3655: 3653: 3650: 3649: 3646: 3643: 3640: 3638: 3635: 3634: 3632: 3627: 3624: 3622: 3620: 3617: 3616: 3613: 3608: 3602: 3599: 3597: 3594: 3593: 3590: 3587: 3585: 3582: 3581: 3579: 3574: 3571: 3569: 3567: 3564: 3563: 3552:Pauli matrices 3537: 3533: 3510: 3506: 3483: 3479: 3456: 3452: 3431: 3426: 3422: 3418: 3415: 3410: 3406: 3402: 3399: 3394: 3390: 3386: 3383: 3378: 3374: 3370: 3367: 3347: 3327: 3324: 3320: 3290: 3287: 3284: 3280: 3276: 3273: 3270: 3266: 3262: 3259: 3256: 3253: 3250: 3246: 3242: 3239: 3236: 3232: 3228: 3225: 3222: 3219: 3216: 3212: 3208: 3205: 3202: 3198: 3194: 3186: 3181: 3177: 3172: 3169: 3163: 3158: 3153: 3133: 3130: 3127: 3123: 3119: 3116: 3113: 3109: 3105: 3102: 3099: 3096: 3093: 3089: 3085: 3082: 3079: 3075: 3071: 3068: 3065: 3062: 3059: 3055: 3051: 3048: 3045: 3041: 3037: 3029: 3024: 3020: 3015: 3012: 3006: 3001: 2996: 2975: 2970: 2966: 2961: 2955: 2951: 2947: 2944: 2939: 2935: 2930: 2924: 2920: 2916: 2913: 2909: 2906: 2901: 2880: 2877: 2873: 2867: 2863: 2859: 2856: 2853: 2849: 2843: 2839: 2835: 2832: 2829: 2825: 2799: 2772: 2769: 2750: 2729: 2726: 2722: 2695: 2672: 2669: 2665: 2660: 2655: 2650: 2644: 2640: 2636: 2633: 2629: 2624: 2619: 2614: 2608: 2604: 2600: 2597: 2593: 2590: 2585: 2564: 2561: 2557: 2551: 2547: 2543: 2540: 2537: 2533: 2527: 2523: 2519: 2516: 2513: 2509: 2486: 2481: 2475: 2471: 2468: 2465: 2461: 2457: 2454: 2451: 2447: 2443: 2440: 2437: 2434: 2430: 2407: 2401: 2397: 2394: 2391: 2387: 2383: 2380: 2377: 2373: 2369: 2366: 2363: 2360: 2356: 2335: 2332: 2328: 2307: 2304: 2300: 2275: 2274:Sign flip code 2272: 2259: 2239: 2235: 2231: 2228: 2225: 2200: 2197: 2194: 2169: 2164: 2160: 2156: 2153: 2148: 2144: 2140: 2137: 2134: 2131: 2126: 2122: 2118: 2115: 2112: 2109: 2106: 2103: 2100: 2095: 2091: 2087: 2084: 2081: 2078: 2075: 2072: 2068: 2065: 2061: 2056: 2052: 2048: 2044: 2041: 2037: 2034: 2031: 2027: 2024: 2020: 2017: 1995: 1991: 1970: 1967: 1964: 1961: 1958: 1955: 1952: 1948: 1945: 1941: 1938: 1934: 1931: 1927: 1923: 1918: 1914: 1910: 1907: 1904: 1901: 1898: 1895: 1892: 1887: 1883: 1879: 1876: 1873: 1870: 1867: 1845: 1841: 1837: 1834: 1831: 1828: 1825: 1822: 1819: 1814: 1810: 1806: 1803: 1800: 1797: 1775: 1771: 1750: 1747: 1744: 1740: 1737: 1733: 1730: 1726: 1723: 1719: 1716: 1711: 1706: 1701: 1695: 1689: 1684: 1680: 1659: 1655: 1652: 1648: 1628: 1624: 1621: 1617: 1614: 1589: 1585: 1561: 1556: 1552: 1546: 1542: 1537: 1532: 1528: 1522: 1518: 1512: 1507: 1504: 1501: 1497: 1493: 1488: 1484: 1480: 1475: 1471: 1467: 1464: 1461: 1458: 1453: 1447: 1424: 1398: 1394: 1371: 1367: 1342: 1338: 1334: 1331: 1328: 1325: 1321: 1317: 1313: 1309: 1306: 1303: 1300: 1296: 1292: 1289: 1287: 1283: 1279: 1275: 1274: 1271: 1267: 1263: 1260: 1257: 1254: 1250: 1246: 1242: 1238: 1235: 1232: 1229: 1225: 1221: 1218: 1216: 1212: 1208: 1204: 1203: 1200: 1196: 1192: 1189: 1186: 1183: 1179: 1175: 1171: 1167: 1164: 1161: 1158: 1154: 1150: 1147: 1145: 1141: 1137: 1133: 1132: 1129: 1125: 1121: 1118: 1115: 1112: 1108: 1104: 1100: 1096: 1093: 1090: 1087: 1083: 1079: 1076: 1074: 1070: 1066: 1062: 1061: 1040: 1016: 1012: 1008: 1005: 1002: 966: 942: 938: 917: 914: 911: 908: 905: 900: 896: 892: 870: 866: 862: 859: 856: 853: 850: 847: 825: 821: 817: 814: 811: 808: 788: 784: 781: 777: 754: 750: 747: 743: 723: 720: 717: 693: 690: 687: 682: 678: 674: 671: 668: 665: 660: 656: 652: 649: 645: 642: 638: 618: 615: 612: 592: 589: 586: 583: 580: 574: 569: 565: 562: 559: 556: 553: 533: 530: 527: 524: 521: 515: 510: 506: 503: 500: 497: 494: 469: 466: 463: 443: 440: 436: 430: 426: 422: 419: 416: 412: 406: 402: 398: 395: 392: 388: 364: 361: 358: 352: 349: 346: 343: 340: 337: 334: 331: 328: 325: 322: 319: 314: 292: 270: 248: 226: 204: 201: 198: 160: 157: 126:Pauli matrices 69:often employs 26: 9: 6: 4: 3: 2: 9572: 9561: 9558: 9556: 9553: 9552: 9550: 9535: 9534: 9525: 9524: 9521: 9511: 9508: 9506: 9503: 9501: 9498: 9496: 9493: 9489: 9488:Plasma window 9486: 9485: 9484: 9481: 9479: 9476: 9474: 9471: 9469: 9466: 9464: 9461: 9460: 9458: 9454: 9448: 9447:teleportation 9445: 9443: 9440: 9438: 9435: 9433: 9430: 9428: 9425: 9423: 9420: 9418: 9415: 9413: 9410: 9408: 9405: 9403: 9400: 9398: 9395: 9393: 9390: 9388: 9385: 9383: 9380: 9378: 9375: 9373: 9370: 9368: 9365: 9363: 9360: 9358: 9355: 9353: 9350: 9348: 9345: 9341: 9338: 9337: 9336: 9333: 9331: 9328: 9326: 9323: 9321: 9318: 9316: 9313: 9311: 9308: 9306: 9303: 9301: 9298: 9296: 9293: 9292: 9290: 9288: 9284: 9281: 9277: 9273: 9266: 9261: 9259: 9254: 9252: 9247: 9246: 9243: 9231: 9223: 9222: 9219: 9213: 9210: 9208: 9205: 9203: 9200: 9196: 9193: 9192: 9191: 9188: 9187: 9185: 9181: 9175: 9172: 9170: 9167: 9163: 9160: 9159: 9158: 9155: 9153: 9150: 9148: 9145: 9143: 9140: 9139: 9137: 9133: 9127: 9124: 9122: 9119: 9117: 9114: 9112: 9109: 9107: 9104: 9102: 9099: 9097: 9094: 9092: 9089: 9087: 9084: 9082: 9079: 9077: 9074: 9072: 9069: 9067: 9066:Quantum logic 9064: 9062: 9059: 9057: 9054: 9052: 9049: 9047: 9044: 9042: 9039: 9037: 9034: 9032: 9029: 9025: 9022: 9021: 9020: 9017: 9015: 9012: 9010: 9007: 9005: 9002: 8998: 8995: 8994: 8993: 8990: 8988: 8985: 8983: 8980: 8978: 8975: 8974: 8972: 8970: 8966: 8960: 8957: 8955: 8952: 8950: 8947: 8945: 8942: 8940: 8937: 8935: 8932: 8930: 8927: 8925: 8922: 8920: 8919:Quantum chaos 8917: 8915: 8912: 8910: 8907: 8906: 8904: 8902: 8898: 8892: 8889: 8887: 8886:Stern–Gerlach 8884: 8882: 8879: 8877: 8874: 8872: 8869: 8867: 8864: 8862: 8859: 8857: 8854: 8852: 8849: 8847: 8844: 8842: 8839: 8838: 8836: 8832: 8826: 8823: 8821: 8820:Transactional 8818: 8816: 8813: 8811: 8810:Quantum logic 8808: 8806: 8803: 8801: 8798: 8792: 8789: 8788: 8787: 8784: 8783: 8782: 8779: 8777: 8774: 8772: 8769: 8767: 8764: 8762: 8759: 8757: 8754: 8753: 8751: 8749: 8745: 8739: 8736: 8734: 8731: 8729: 8726: 8724: 8721: 8719: 8716: 8714: 8711: 8709: 8706: 8704: 8701: 8700: 8698: 8694: 8688: 8685: 8683: 8680: 8678: 8675: 8673: 8670: 8668: 8665: 8663: 8660: 8658: 8655: 8654: 8652: 8648: 8640: 8637: 8635: 8632: 8631: 8630: 8629:Wave function 8627: 8625: 8622: 8620: 8617: 8615: 8612: 8610: 8607: 8605: 8604:Superposition 8602: 8600: 8599:Quantum state 8597: 8595: 8592: 8590: 8587: 8585: 8582: 8580: 8577: 8575: 8572: 8570: 8567: 8563: 8560: 8558: 8555: 8553: 8552:Excited state 8550: 8548: 8545: 8544: 8543: 8540: 8538: 8535: 8533: 8530: 8528: 8525: 8523: 8520: 8519: 8517: 8513: 8507: 8504: 8502: 8499: 8497: 8494: 8490: 8487: 8486: 8485: 8482: 8480: 8477: 8476: 8474: 8470: 8466: 8459: 8454: 8452: 8447: 8445: 8440: 8439: 8436: 8424: 8416: 8414: 8406: 8405: 8402: 8396: 8393: 8391: 8388: 8386: 8383: 8381: 8378: 8376: 8372: 8369: 8367: 8363: 8359: 8356: 8355: 8353: 8351: 8345: 8335: 8332: 8330: 8327: 8325: 8322: 8320: 8317: 8316: 8314: 8312: 8308: 8302: 8299: 8297: 8294: 8292: 8291:Spin qubit QC 8289: 8287: 8284: 8283: 8281: 8278: 8274: 8268: 8265: 8263: 8260: 8259: 8257: 8255: 8251: 8245: 8242: 8240: 8237: 8235: 8232: 8230: 8227: 8226: 8224: 8222: 8218: 8215: 8209: 8203: 8200: 8196: 8195: 8191: 8189: 8186: 8184: 8181: 8179: 8176: 8174: 8171: 8169: 8166: 8164: 8161: 8159: 8156: 8155: 8153: 8152: 8150: 8148: 8142: 8136: 8133: 8131: 8128: 8124: 8121: 8120: 8119: 8116: 8112: 8109: 8108: 8107: 8104: 8100: 8099:cluster state 8097: 8096: 8095: 8092: 8090: 8087: 8085: 8082: 8081: 8079: 8077: 8071: 8063: 8059: 8055: 8053: 8049: 8045: 8044: 8043: 8040: 8036: 8033: 8032: 8031: 8028: 8026: 8023: 8021: 8018: 8017: 8015: 8009: 8003: 8000: 7998: 7995: 7993: 7990: 7988: 7985: 7983: 7980: 7979: 7977: 7975: 7969: 7963: 7960: 7958: 7955: 7953: 7950: 7948: 7945: 7943: 7940: 7938: 7935: 7933: 7930: 7928: 7925: 7923: 7920: 7918: 7915: 7913: 7910: 7908: 7907:Deutsch–Jozsa 7905: 7903: 7900: 7898: 7895: 7893: 7890: 7889: 7887: 7885: 7881: 7871: 7868: 7864: 7861: 7859: 7856: 7854: 7851: 7850: 7849: 7846: 7844: 7843:Quantum money 7841: 7839: 7836: 7834: 7831: 7830: 7828: 7826: 7822: 7816: 7813: 7809: 7806: 7805: 7804: 7801: 7797: 7794: 7793: 7792: 7789: 7787: 7784: 7782: 7779: 7777: 7774: 7770: 7767: 7765: 7762: 7761: 7760: 7757: 7756: 7754: 7752:communication 7748: 7742: 7739: 7737: 7734: 7732: 7729: 7727: 7724: 7722: 7719: 7717: 7714: 7712: 7709: 7707: 7704: 7702: 7699: 7697: 7694: 7692: 7689: 7687: 7684: 7682: 7679: 7677: 7674: 7672: 7669: 7667: 7664: 7663: 7661: 7657: 7649: 7646: 7645: 7644: 7641: 7637: 7634: 7633: 7632: 7629: 7627: 7624: 7622: 7619: 7617: 7614: 7610: 7607: 7606: 7605: 7602: 7600: 7597: 7595: 7592: 7591: 7589: 7585: 7581: 7574: 7569: 7567: 7562: 7560: 7555: 7554: 7551: 7543: 7539: 7535: 7534:Quantum Light 7531: 7527: 7526: 7516: 7512: 7507: 7502: 7498: 7494: 7489: 7485: 7481: 7472: 7468: 7463: 7459: 7454: 7450: 7446: 7442: 7441: 7423: 7419: 7412: 7404: 7400: 7396: 7392: 7388: 7384: 7380: 7376: 7371: 7366: 7363:(2): 022428. 7362: 7358: 7351: 7349: 7334: 7330: 7323: 7309: 7305: 7298: 7284: 7280: 7273: 7259: 7258:Physics World 7255: 7248: 7240: 7236: 7231: 7226: 7222: 7218: 7213: 7208: 7204: 7200: 7196: 7192: 7188: 7181: 7173: 7169: 7165: 7161: 7157: 7153: 7149: 7145: 7140: 7135: 7131: 7127: 7124:operations". 7119: 7111: 7107: 7102: 7097: 7093: 7089: 7085: 7081: 7077: 7070: 7062: 7058: 7054: 7050: 7046: 7042: 7038: 7034: 7030: 7026: 7022: 7018: 7010: 6996: 6992: 6985: 6977: 6973: 6969: 6965: 6961: 6957: 6953: 6949: 6945: 6941: 6936: 6931: 6927: 6923: 6915: 6900: 6896: 6890: 6882: 6878: 6874: 6870: 6866: 6862: 6858: 6854: 6849: 6844: 6840: 6836: 6832: 6825: 6817: 6813: 6809: 6805: 6801: 6797: 6792: 6787: 6783: 6779: 6771: 6763: 6759: 6755: 6751: 6747: 6743: 6739: 6735: 6730: 6725: 6721: 6717: 6710: 6702: 6698: 6694: 6690: 6686: 6682: 6678: 6674: 6670: 6666: 6659: 6651: 6647: 6643: 6639: 6635: 6631: 6627: 6623: 6619: 6615: 6607: 6599: 6595: 6591: 6587: 6583: 6579: 6574: 6569: 6566:(5): 052332. 6565: 6561: 6554: 6546: 6542: 6538: 6534: 6530: 6526: 6521: 6516: 6512: 6508: 6501: 6493: 6489: 6485: 6481: 6477: 6473: 6469: 6465: 6460: 6455: 6451: 6447: 6440: 6432: 6428: 6424: 6420: 6416: 6412: 6407: 6402: 6398: 6394: 6387: 6378: 6373: 6366: 6358: 6354: 6350: 6346: 6342: 6338: 6333: 6328: 6325:(3): 032324. 6324: 6320: 6313: 6305: 6301: 6297: 6293: 6286: 6278: 6274: 6270: 6266: 6262: 6258: 6253: 6248: 6245:(1): 012340. 6244: 6240: 6233: 6225: 6221: 6217: 6213: 6208: 6203: 6199: 6195: 6188: 6180: 6176: 6172: 6168: 6164: 6160: 6156: 6152: 6148: 6144: 6140: 6136: 6128: 6126: 6117: 6113: 6109: 6105: 6101: 6097: 6093: 6089: 6085: 6081: 6076: 6071: 6067: 6063: 6055: 6053: 6044: 6040: 6036: 6032: 6028: 6024: 6019: 6014: 6011:(3): 032346. 6010: 6006: 5998: 5990: 5986: 5982: 5978: 5974: 5970: 5965: 5960: 5957:(3): 031006. 5956: 5952: 5945: 5943: 5941: 5932: 5928: 5924: 5920: 5916: 5912: 5907: 5902: 5899:(1): 012310. 5898: 5894: 5887: 5879: 5875: 5871: 5867: 5863: 5859: 5855: 5851: 5846: 5841: 5838:(4): 045014. 5837: 5833: 5825: 5823: 5814: 5810: 5806: 5802: 5798: 5794: 5790: 5786: 5782: 5778: 5773: 5768: 5764: 5760: 5753: 5751: 5742: 5738: 5734: 5730: 5726: 5722: 5717: 5712: 5708: 5704: 5697: 5695: 5686: 5682: 5678: 5674: 5670: 5666: 5662: 5658: 5654: 5650: 5645: 5640: 5637:(7): 076001. 5636: 5632: 5628: 5621: 5613: 5609: 5605: 5599: 5591: 5587: 5583: 5579: 5575: 5571: 5567: 5563: 5556: 5548: 5544: 5540: 5536: 5532: 5528: 5524: 5520: 5516: 5510: 5508: 5500: 5495: 5490: 5486: 5482: 5477: 5472: 5468: 5464: 5460: 5453: 5449: 5440: 5437: 5435: 5432: 5431: 5425: 5416: 5412: 5408: 5406: 5401: 5399: 5394: 5389: 5387: 5383: 5379: 5375: 5370: 5367: 5364: 5360: 5355: 5353: 5349: 5339: 5337: 5332: 5330: 5329:quantum gates 5326: 5322: 5318: 5314: 5306: 5302: 5299: 5295: 5291: 5287: 5283: 5279: 5276: 5273: 5269: 5265: 5261: 5260:Alexei Kitaev 5258: 5255: 5251: 5247: 5243: 5239: 5236: 5232: 5228: 5224: 5220: 5216: 5212: 5208: 5205: 5204:Andrew Steane 5201: 5197: 5193: 5189: 5185: 5181: 5177: 5174: 5170: 5166: 5163: 5160: 5156: 5155:Andrew Steane 5153: 5150: 5147: 5146: 5145: 5137: 5135: 5130: 5125: 5123: 5071: 5047: 5026: 5015: 5011: 5000: 4996: 4993: 4986: 4983: 4977: 4961: 4876: 4838: 4834: 4827:General codes 4824: 4808: 4798: 4768: 4760: 4750: 4724: 4714: 4701: 4678: 4670: 4660: 4653: 4650: 4644: 4641: 4615: 4605: 4587: 4584: 4561: 4558: 4548: 4531: 4525: 4522: 4519: 4511: 4505: 4502: 4494: 4492: 4482: 4477: 4473: 4460: 4454: 4451: 4443: 4437: 4429: 4427: 4417: 4412: 4408: 4395: 4389: 4386: 4383: 4375: 4369: 4366: 4358: 4356: 4346: 4341: 4337: 4324: 4318: 4315: 4307: 4301: 4293: 4291: 4281: 4276: 4272: 4254: 4252: 4228: 4223: 4219: 4210: 4202: 4197: 4193: 4156: 4151: 4147: 4138: 4130: 4125: 4121: 4104: 4095: 4084: 4081: 4080:photon number 4065: 4059: 4031: 4006: 3997: 3987: 3971: 3965: 3957: 3943: 3933: 3927: 3919: 3911: 3905: 3894: 3880: 3866: 3856: 3848:Binomial code 3845: 3842: 3835:Bosonic codes 3832: 3818: 3815: 3812: 3809: 3789: 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8515:Fundamentals 8479:Introduction 8319:Charge qubit 8244:KLM protocol 8193: 8144: 8057: 8047: 7741:Purification 7671:Eastin–Knill 7533: 7496: 7492: 7483: 7466: 7457: 7448: 7445:Daniel Lidar 7425:. 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8334:Transmon 8211:Physical 8011:Quantum 7912:Grover's 7686:Holevo's 7659:Theorems 7609:timeline 7599:NISQ era 7538:Archived 7333:Phys.org 7239:36813892 7164:35614250 7053:34608286 6995:phys.org 6968:33442044 6899:phys.org 6873:36654309 6816:55090423 6754:22297844 6701:32268350 6693:21617070 6642:15577904 6598:11679660 6545:11662810 6484:15744292 6171:27437573 6108:25043007 6043:51691343 5989:29518512 5931:18995200 5813:19929020 5805:24093235 5677:23787909 5610:(2000). 5428:See also 5352:transmon 5325:provided 4896:, where 4087:Cat code 2986:, where 2908:′ 2592:′ 2182:fidelity 2067:′ 2043:′ 2026:′ 1947:′ 1933:′ 1739:′ 1725:′ 1654:′ 1623:′ 1604:fidelity 783:′ 749:′ 644:′ 482:encoding 142:identity 87:decoding 83:encoding 9437:sensing 9417:network 9402:machine 9372:imaging 9320:circuit 9315:channel 9287:Quantum 9183:Related 9162:History 8901:Science 8733:Rydberg 8484:History 8348:Quantum 8286:Kane QC 8145:Quantum 8073:Quantum 8002:PostBQP 7972:Quantum 7957:Simon's 7750:Quantum 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