Quantum error correction

2268: 168: 66:. 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 975: 8408: 70:, 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 9200: 8398: 4535: 1344: 9502: 2769: 4247: 5413:

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.

1044: 3727: 5373:. 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 4530:{\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}}} 5412:

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.

1339:{\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}}} 2167: 6908:

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

2973: 5296:, 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. 473:

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".

2878: 2562: 1968: 1662: 441: 3722:{\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}}} 2278:

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".

4936: 5312:, 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— 1427: 2484: 2407: 3859: 362: 4687: 4236: 4164: 5938:

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".

4623:, 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 3292:

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

1845: 4094:, 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: 152:

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

2567: 616: 202: 5195:. 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. 5055: 2754: 2699: 1346:

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.

3283:{\displaystyle |1_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )} 3126:{\displaystyle |0_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )} 78:. 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. 5690:

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.

5351: 1591:, 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 6494:

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

5305: 5274: 1743:{\displaystyle \rho _{\operatorname {out} }\equiv {\mathcal {E}}_{\operatorname {corr} }({\mathcal {E}}(\vert \psi '\rangle \langle \psi '\vert ))} 9236: 8997: 7597: 7242: 5223: 133:). 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. 2807: 370: 5270: 8645: 7559: 5140:'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. 8289: 5336: 2162:{\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}.} 7344:

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.

1424:-th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel: 295: 4626: 8864: 8437: 4169: 4097: 8798: 4854: 156:, it is not possible to repeat a single qubit three times. To overcome this, a different method has to be used, such as the 8721: 8401: 7587: 5021:{\displaystyle ({\mathcal {R}}\circ {\mathcal {E}})(\rho )=\rho \quad \forall \rho =P_{\mathcal {C}}\rho P_{\mathcal {C}},} 2775:

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

9229: 8359: 1554:{\displaystyle {\mathcal {E}}_{\operatorname {corr} }(\rho )=P_{0}\rho P_{0}+\sum _{i=1}^{3}X_{i}P_{i}\rho \,P_{i}X_{i}.} 6181:

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".

85:. This theorem seems to present an obstacle to formulating a theory of quantum error correction. But it is possible to 3966:{\displaystyle |0_{\rm {L}}\rangle ={\frac {|0\rangle +|4\rangle }{\sqrt {2}}},\quad |1_{\rm {L}}\rangle =|2\rangle ,} 8467: 8430: 8411: 8299: 7552: 5396:

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

3829: 2239:. Note that this is consistent with the working assumption that was made while deriving the protocol (of 595: 181: 118: 9528: 9483: 9283: 9203: 8965: 8773: 8744: 8151: 8023: 7920: 7796: 7724: 7631: 5335:

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;

5339:. Subsequently, demonstrations have been made with linear optics, trapped ions, and superconducting ( 5301: 700: 43: 5088: 5064: 4912: 4888: 4830: 4695: 3302: 2704: 947: 251: 207: 9456: 9451: 9380: 9355: 9340: 9335: 9054: 9034: 9024: 8970: 8545: 8106: 8082: 8045: 7836: 7769: 7704: 7689: 7659: 7582: 5386: 5292:

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

4039: 3979: 875: 830: 228:. Let us moreover assume that this channel either flips the state of the qubit, with probability 9163: 8711: 4597: 4011: 2310: 2282: 693:. This mapping can be realized for example using two CNOT gates, entangling the system with two 9473: 9395: 9074: 8849: 8829: 8754: 8622: 8354: 8118: 8111: 7858: 5164:

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

1597: 985: 791: 130: 126: 125:, so even if the error due to the noise was arbitrary, it can be expressed as a combination of 8665: 6979: 2208: 686:{\displaystyle \vert \psi '\rangle =\alpha _{0}\vert 000\rangle +\alpha _{1}\vert 111\rangle } 9446: 9420: 9245: 9099: 8612: 8592: 8274: 7900: 7826: 7791: 5058: 788:

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".

2665:{\displaystyle |\psi '\rangle =\alpha _{0}|{+}{+}{+}\rangle +\alpha _{1}|{-}{-}{-}\rangle .} 585:{\displaystyle \vert 1\rangle \rightarrow \vert 1_{\rm {L}}\rangle \equiv \vert 111\rangle } 526:{\displaystyle \vert 0\rangle \rightarrow \vert 0_{\rm {L}}\rangle \equiv \vert 000\rangle } 9385: 9308: 9130: 9059: 9004: 8734: 8557: 8515: 8064: 7813: 7664: 7499: 7480:

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

7363: 7187: 7132: 7076: 7013: 6928: 6841: 6784: 6722: 6661: 6610: 6566: 6513: 6452: 6399: 6325: 6280: 6245: 6131: 6068: 6011: 5957: 5899: 5838: 5765: 5709: 5637: 5558: 5515: 5469: 5286: 5282: 3515: 3488: 3461: 3434: 2176: 1567: 1376: 1349: 920: 275: 161: 7407:"What's the difference between error suppression, error mitigation, and error correction?" 6359:

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.

9415: 9303: 9273: 9268: 9185: 9094: 9064: 8992: 8955: 8950: 8932: 8897: 8887: 8602: 8567: 8550: 8453: 8363: 8008: 7915: 7872: 7803: 7719: 7699: 7654: 7614: 7592: 7391: 7379: 7223: 7205: 7160: 7148: 7098: 7049: 7037: 7029: 6980:"Foundational step shows quantum computers can be better than the sum of their parts" 6964: 6952: 6944: 6869: 6857: 6738: 6677: 6626: 6468: 6415: 6341: 6265: 6155: 6147: 6092: 6084: 5854: 5789: 5781: 5729: 5673: 5661: 5653: 5574: 5531: 5207: 5184: 4592: 3974: 164:

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

6804: 6689: 6586: 6533: 6031: 5977: 5919: 5801: 9320: 8912: 8907: 8764: 8660: 8030: 7980: 7757: 7468: 7371: 7213: 7195: 7140: 7084: 7021: 6936: 6849: 6792: 6750: 6730: 6669: 6618: 6574: 6521: 6480: 6460: 6407: 6333: 6288: 6253: 6212: 6200: 6139: 6076: 6019: 5965: 5907: 5866: 5846: 5777: 5773: 5717: 5645: 5615: 5566: 5523: 5477: 5203: 5153: 2968:{\displaystyle |\psi '\rangle =\alpha _{0}|0_{S}\rangle +\alpha _{1}|1_{S}\rangle } 141: 55: 27: 6638: 6167: 6104: 9390: 9375: 9345: 9293: 9288: 9142: 9069: 9049: 9019: 8982: 8977: 8882: 8706: 8156: 8094: 7784: 7779: 5596: 5219: 5199: 4825: 2800: 2772: 2271: 171: 63: 7375: 6525: 944:. Note that a further assumption about the channel is made here: we assume that 178:

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

1997:

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

9522: 9461: 9360: 9039: 8892: 8783: 8617: 8587: 8540: 8087: 7905: 7831: 7383: 7209: 7033: 6948: 6796: 6419: 6345: 6151: 6088: 5858: 5785: 5721: 5657: 5570: 5503: 5355:

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

5248: 5192: 5168: 5143: 4068: 1373:

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

110: 39: 6673: 5448:"Bosonic quantum error correction codes in superconducting quantum circuits" 9441: 8922: 8535: 8530: 8307: 8232: 7433: 7227: 7152: 7089: 7064: 7041: 6956: 6861: 6742: 6681: 6630: 6472: 6159: 6096: 5793: 5665: 5393: 5317: 5293: 5176: 4909:

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

6433:

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

5879: 5578: 5535: 2873:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 2557:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 436:{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle } 9278: 8960: 8317: 8171: 7709: 7494: 6561: 6508: 6447: 6240: 6195: 5894: 5704: 5374: 5172: 5147: 35: 9214: 6822:(2021). "Testing a quantum error-correcting code on various platforms". 6734: 6622: 6464: 6143: 6080: 144:, which would prevent it from being used to convey quantum information. 8378: 8312: 8176: 5427: 5366: 5256: 5252: 5215: 5211: 5188: 5137: 3853: 94: 6204: 5506:(1995). "Scheme for reducing decoherence in quantum computer memory". 8814: 8510: 8161: 5347: 5266: 4091: 8422: 7176:"Suppressing quantum errors by scaling a surface code logical qubit" 5179:, led to the construction of an important class of codes called the 8346: 8322: 8181: 8146: 7358: 7127: 6923: 6836: 6819: 6365: 6006: 5952: 5745: 5464: 5340: 5180: 6779: 6717: 6394: 6320: 6063: 5833: 5760: 5632: 8373: 7990: 7530: 7268:"Google's quantum computer reached an error-correcting milestone" 6599: 6907: 5198:

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

5146:

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

1963:{\displaystyle \,\vert \psi '\rangle \langle \psi '\vert +(...)} 42:. Quantum error correction is theorised as essential to achieve 8350: 7846: 6305: 5549:

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

2479:{\displaystyle |+\rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}.} 7293:"Microsoft And Quantinuum Improve Quantum Error Rates By 800x" 6651: 2402:{\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}} 7619: 7456:

Quantum Error Correction and Fault Tolerant Quantum Computing

7111: 5817: 5614:

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

5324: 2768: 90: 7444: 5937: 4692:

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

93:

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

8368: 7841: 7774: 7463:

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

6884:"Error-protected quantum bits entangled for the first time" 6047: 5990: 5408:

Quantum error-correction without encoding and parity-checks

357:{\displaystyle {\mathcal {E}}(\rho )=(1-p)\rho +p\ X\rho X} 160:

first proposed by Asher Peres in 1985. This technique uses

7432: 6120: 4682:{\displaystyle \exp(i\pi {\hat {a}}^{\dagger }{\hat {a}})} 7985: 7970: 6180: 4231:{\displaystyle \{|0_{L}^{-}\rangle ,|1_{L}^{-}\rangle \}} 4159:{\displaystyle \{|0_{L}^{+}\rangle ,|1_{L}^{+}\rangle \}} 6379: 5133:

Over time, researchers have come up with several codes:

2335:

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

6358: 5350:

encoded in a superconducting resonator, and employed a

81:

Copying quantum information is not possible due to the

7002: 6763: 4878:{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}} 4565:

only occupy even photon number states and states with

3678: 3622: 3569: 117:

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

5091: 5067: 5034: 4939: 4915: 4891: 4857: 4833: 4782: 4734: 4698: 4629: 4600: 4571: 4545: 4250: 4172: 4100: 4042: 4014: 3982: 3862: 3797: 3771: 3745: 3549: 3518: 3491: 3464: 3437: 3353: 3333: 3305: 3138: 2981: 2886: 2810: 2785: 2735: 2707: 2680: 2570: 2494: 2415: 2341: 2313: 2285: 2245: 2211: 2205:

error correction is greater than the one without for

2179: 2003: 1976: 1853: 1783: 1756: 1665: 1634: 1600: 1570: 1430: 1410: 1379: 1352: 1047: 1026: 988: 950: 923: 878: 833: 794: 763: 729: 703: 624: 598: 539: 480: 449: 373: 298: 278: 254: 234: 210: 184: 6546: 5689: 1016:, the error correction scheme improves the fidelity. 917:, and all three qubits are flipped with probability 7812: 7065:"Real-Time Error Correction for Quantum Computing" 6702: 6493: 5101: 5077: 5049: 5020: 4925: 4901: 4877: 4843: 4804: 4768: 4720: 4681: 4615: 4583: 4557: 4529: 4230: 4158: 4059: 4028: 4000: 3965: 3812: 3783: 3757: 3721: 3539:are complex constants, I is the identity, and the 3531: 3504: 3477: 3450: 3423: 3339: 3319: 3282: 3125: 2967: 2872: 2791: 2748: 2721: 2693: 2664: 2556: 2478: 2401: 2327: 2299: 2251: 2231: 2193: 2161: 1989: 1962: 1839: 1769: 1742: 1651: 1620: 1583: 1553: 1416: 1392: 1365: 1338: 1032: 1008: 960: 936: 909: 864: 819: 780: 746: 715: 685: 610: 584: 525: 461: 435: 356: 284: 264: 240: 220: 196: 62:. The simplest albeit inefficient approach is the 7173: 6285:Quantum Communication, Computing, and Measurement 5613: 4238:, where each of the states is a superposition of 2880:will be transformed into the product of 9 qubits 754:is what is now passed through the noisy channel. 9520: 7462: 6817: 5233:are a family of codes parameterized by integers 2201:. A little algebra then shows that the fidelity 7567: 7343: 6281:"Quantum Error Correction with Imperfect Gates" 4769:{\displaystyle {\hat {a}}^{\dagger }{\hat {a}}} 140:of this logical qubit with other qubits in the 6043: 6041: 4689:using a dispersively coupled ancillary qubit. 97:first discovered this method of formulating a 9230: 8438: 7553: 7479: 7290: 5591: 3424:{\displaystyle U=c_{0}I+c_{1}X+c_{2}Y+c_{3}Z} 827:, a single qubit is flipped with probability 7529:from the original on 2021-12-22 – via 7453: 4517: 4497: 4476: 4446: 4429: 4411: 4381: 4361: 4340: 4310: 4293: 4275: 4225: 4222: 4196: 4173: 4153: 4150: 4124: 4101: 4051: 4023: 3957: 3943: 3911: 3897: 3880: 3314: 3274: 3260: 3240: 3226: 3206: 3192: 3156: 3117: 3103: 3083: 3069: 3049: 3035: 2999: 2962: 2931: 2900: 2867: 2843: 2819: 2716: 2656: 2620: 2584: 2551: 2527: 2503: 2455: 2441: 2424: 2381: 2367: 2350: 2322: 2294: 2059: 2048: 2035: 2024: 1990:{\displaystyle \rho _{\operatorname {out} }} 1939: 1928: 1925: 1914: 1770:{\displaystyle \rho _{\operatorname {out} }} 1731: 1720: 1717: 1706: 1646: 1635: 1318: 1315: 1293: 1290: 1247: 1244: 1222: 1219: 1176: 1173: 1151: 1148: 1105: 1102: 1080: 1077: 775: 764: 741: 730: 710: 704: 680: 674: 658: 652: 636: 625: 605: 599: 579: 573: 567: 552: 546: 540: 520: 514: 508: 493: 487: 481: 430: 406: 382: 191: 185: 6038: 5601:Quantum Computation and Quantum Information 5369:has first been realized using 10 ions in a 5330: 469:. We can however improve on this number by 9501: 9237: 9223: 8445: 8431: 7560: 7546: 7445:La Guardia, Giuliano Gadioli, ed. (2020). 5498: 5496: 5367:topological quantum error-correction codes 5279:entanglement-assisted stabilizer formalism 5167:A generalisation of the technique used by 872:, two qubits are flipped with probability 9244: 7493: 7357: 7217: 7199: 7126: 7088: 6922: 6835: 6778: 6716: 6560: 6507: 6446: 6393: 6364: 6319: 6239: 6194: 6062: 6005: 5951: 5893: 5832: 5759: 5703: 5631: 5481: 5463: 3856:basis, the simplest binomial encoding is 1913: 1527: 89:the (logical) information of one logical 7240: 5616:"Quantum error correction for beginners" 2767: 2266: 973: 248:, or leaves it unchanged. The action of 166: 8078:Continuous-variable quantum information 7525:. University of Sheffield. 2018-09-28. 7404: 7265: 6977: 6183:IEEE Transactions on Information Theory 5493: 5289:shared between a sender and a receiver. 9521: 7519:"Topological Quantum Error Correction" 6278: 5365:between two logical qubits encoded in 1970:, where the dots denote components of 113:) flip, or both (corresponding to the 9218: 8452: 8426: 7541: 7339: 7337: 7315: 6432: 6225: 6116: 6114: 5933: 5931: 5929: 5548: 5445: 5263:are the basis for various code types. 4805:{\displaystyle {\hat {a}}^{\dagger }} 1840:{\displaystyle (1-p)^{3}+3p(1-p)^{2}} 7062: 5813: 5811: 5741: 5739: 5685: 5683: 5502: 5245:qubits arranged in a square lattice. 1652:{\displaystyle \vert \psi '\rangle } 781:{\displaystyle \vert \psi '\rangle } 747:{\displaystyle \vert \psi '\rangle } 4008:the corresponding error states are 611:{\displaystyle \vert \psi \rangle } 197:{\displaystyle \vert \psi \rangle } 13: 7473:Mathematics of quantum computation 7426: 7334: 7291:Smith-Goodson, Paul (2024-04-18). 6111: 5926: 5316:that the error rate of individual 5094: 5070: 5041: 5009: 4994: 4979: 4955: 4945: 4918: 4894: 4870: 4860: 4836: 3937: 3874: 3150: 2993: 1698: 1682: 1434: 1020:Let us assume for simplicity that 953: 561: 502: 301: 257: 213: 14: 9545: 7511: 5808: 5736: 5680: 5337:nuclear magnetic resonance qubits 5259:, and the more general idea of a 2262: 26:) is a set of techniques used in 9500: 9199: 9198: 8407: 8406: 8397: 8396: 7405:Steffen, Matthias (2022-10-20). 7241:Boerkamp, Martijn (2023-03-20). 7174:Google Quantum AI (2023-02-22). 5446:Cai, Weizhou; Ma, Yuwei (2021). 5281:as an extension of the standard 5050:{\displaystyle P_{\mathcal {C}}} 4815: 3836: 3823: 2749:{\displaystyle E_{\text{phase}}} 2694:{\displaystyle E_{\text{phase}}} 2488:The original state of the qubit 147: 44:fault tolerant quantum computing 9534:Fault-tolerant computer systems 7398: 7309: 7284: 7259: 7234: 7167: 7105: 7056: 6996: 6971: 6901: 6876: 6811: 6757: 6696: 6645: 6593: 6540: 6487: 6426: 6373: 6352: 6299: 6272: 6219: 6174: 5984: 4978: 3925: 3791:, a sign flip error occurs. If 2564:will be changed into the state 716:{\displaystyle \vert 0\rangle } 104:In classical error correction, 9148:Relativistic quantum mechanics 6978:Bedford, Bailey (2021-10-04). 6412:10.1088/1367-2630/14/12/123011 6287:. Springer. pp. 181–188. 5873: 5778:10.1103/physrevlett.111.120501 5620:Reports on Progress in Physics 5607: 5585: 5542: 5439: 5423:Error detection and correction 5102:{\displaystyle {\mathcal {R}}} 5078:{\displaystyle {\mathcal {C}}} 4969: 4963: 4960: 4940: 4926:{\displaystyle {\mathcal {R}}} 4902:{\displaystyle {\mathcal {H}}} 4844:{\displaystyle {\mathcal {E}}} 4790: 4760: 4742: 4721:{\displaystyle {\hat {a}}^{2}} 4706: 4676: 4670: 4652: 4636: 4607: 4504: 4487: 4457: 4436: 4422: 4392: 4368: 4351: 4321: 4300: 4286: 4256: 4203: 4177: 4131: 4105: 4044: 4016: 3989: 3950: 3927: 3904: 3890: 3864: 3765:, a bit flip error occurs. If 3320:{\displaystyle |\psi \rangle } 3307: 3277: 3267: 3253: 3249: 3243: 3233: 3219: 3215: 3209: 3199: 3185: 3181: 3140: 3120: 3110: 3096: 3092: 3086: 3076: 3062: 3058: 3052: 3042: 3028: 3024: 2983: 2948: 2917: 2888: 2860: 2836: 2812: 2722:{\displaystyle |\psi \rangle } 2709: 2637: 2601: 2572: 2544: 2520: 2496: 2458: 2448: 2434: 2430: 2417: 2384: 2374: 2360: 2356: 2343: 2315: 2287: 2109: 2096: 2078: 2065: 2018: 2007: 1957: 1945: 1910: 1901: 1888: 1870: 1857: 1854: 1828: 1815: 1797: 1784: 1737: 1734: 1703: 1693: 1615: 1604: 1451: 1445: 1325: 1308: 1300: 1283: 1254: 1237: 1229: 1212: 1183: 1166: 1158: 1141: 1112: 1095: 1087: 1070: 961:{\displaystyle {\mathcal {E}}} 904: 892: 853: 840: 808: 795: 549: 490: 423: 399: 375: 330: 318: 312: 306: 265:{\displaystyle {\mathcal {E}}} 221:{\displaystyle {\mathcal {E}}} 1: 9126:Quantum statistical mechanics 8903:Quantum differential calculus 8825:Delayed-choice quantum eraser 8608:Symmetry in quantum mechanics 8073:Adiabatic quantum computation 7440:. Cambridge University Press. 7266:Conover, Emily (2023-02-22). 6279:Kitaev, Alexei (1997-07-31). 5851:10.1088/1367-2630/16/4/045014 5650:10.1088/0034-4885/76/7/076001 5603:. Cambridge University Press. 5433: 5183:, named for their inventors: 3347:can be described in the form 99:quantum error correcting code 8124:Topological quantum computer 5380:In 2022, researchers at the 5371:trapped-ion quantum computer 5261:topological quantum computer 5255:, introduced in 1997 as the 2759: 1400:is observed, then the Pauli 292:can therefore be written as 16:Process in quantum computing 7: 8928:Quantum stochastic calculus 8918:Quantum measurement problem 8840:Mach–Zehnder interferometer 8402:Quantum information science 7569:Quantum information science 7436:and Todd Brun, ed. (2013). 7376:10.1103/physreva.105.022428 7063:Ball, Philip (2021-12-23). 6526:10.1103/PhysRevLett.81.2152 5416: 4075: 4060:{\displaystyle |1\rangle ,} 4001:{\displaystyle {\hat {a}},} 3830:quantum harmonic oscillator 3739:, then no error occurs. If 910:{\displaystyle 3p^{2}(1-p)} 865:{\displaystyle 3p(1-p)^{2}} 10: 9550: 9484:Thermoacoustic heat engine 7797:quantum gate teleportation 7201:10.1038/s41586-022-05434-1 7145:10.1038/s41586-022-04721-1 7026:10.1038/s41586-021-03928-y 6941:10.1038/s41586-020-03079-6 6854:10.1016/j.scib.2020.07.033 6579:10.1103/PhysRevA.71.052332 6338:10.1103/PhysRevA.86.032324 6258:10.1103/PhysRevA.73.012340 6024:10.1103/PhysRevA.97.032346 5912:10.1103/PhysRevA.64.012310 5483:10.1016/j.fmre.2020.12.006 4616:{\displaystyle {\hat {a}}} 4029:{\displaystyle |3\rangle } 2409:may have its sign flip to 2328:{\displaystyle |1\rangle } 2300:{\displaystyle |0\rangle } 618:is encoded into the state 9496: 9469:Immersive virtual reality 9429: 9259: 9252: 9194: 9156: 9108: 8988:Quantum complexity theory 8966:Quantum cellular automata 8941: 8873: 8807: 8720: 8684: 8671:Path integral formulation 8638: 8503: 8460: 8392: 8335: 8298: 8264: 8241: 8208: 8199: 8132: 8061: 7999: 7959: 7926:Quantum Fourier transform 7871: 7822:Post-quantum cryptography 7765:Entanglement distillation 7738: 7647: 7575: 7316:Yirka, Bob (2024-04-05). 6293:10.1007/978-1-4615-5923-8 5970:10.1103/PhysRevX.6.031006 5528:10.1103/PhysRevA.52.R2493 5302:quantum threshold theorem 5253:topological quantum codes 5128: 1750:. Being the output state 1621:{\displaystyle F(\psi ')} 1009:{\displaystyle p\leq 1/2} 820:{\displaystyle (1-p)^{3}} 697:initialized in the state 158:three-qubit bit flip code 9452:Digital scent technology 9055:Quantum machine learning 9035:Quantum key distribution 9025:Quantum image processing 9015:Quantum error correction 8865:Wheeler's delayed choice 8412:Quantum mechanics topics 8107:Quantum machine learning 8083:One-way quantum computer 7936:Quantum phase estimation 7837:Quantum key distribution 7770:Monogamy of entanglement 7438:Quantum Error Correction 6797:10.1038/nphoton.2010.168 5722:10.1103/PhysRevA.59.2631 5571:10.1103/PhysRevA.32.3266 5331:Experimental realization 5222:; these are also called 2232:{\displaystyle p<1/2} 204:through a noisy channel 20:Quantum error correction 8971:Quantum finite automata 8019:Randomized benchmarking 7881:Amplitude amplification 7458:. Taylor & Francis. 6674:10.1126/science.1203329 5748:Physical Review Letters 5382:University of Innsbruck 5177:classical Hamming code 1404:gate is applied to the 9474:Magnetic refrigeration 9075:Quantum neural network 8119:Quantum Turing machine 8112:quantum neural network 7859:Quantum secret sharing 7090:10.1103/Physics.14.184 6382:New Journal of Physics 5821:New Journal of Physics 5348:Schrodinger-cat states 5103: 5079: 5051: 5022: 4927: 4903: 4879: 4845: 4806: 4770: 4722: 4683: 4617: 4585: 4559: 4531: 4232: 4160: 4092:Schrödinger cat states 4061: 4030: 4002: 3967: 3814: 3785: 3759: 3723: 3533: 3506: 3479: 3452: 3425: 3341: 3321: 3284: 3127: 2969: 2874: 2793: 2776: 2750: 2723: 2695: 2666: 2558: 2480: 2403: 2329: 2301: 2275: 2274:of the phase flip code 2253: 2233: 2195: 2163: 1991: 1964: 1841: 1771: 1744: 1653: 1622: 1585: 1555: 1503: 1418: 1394: 1367: 1340: 1034: 1017: 1010: 962: 938: 911: 866: 821: 782: 748: 717: 687: 612: 586: 527: 463: 437: 358: 286: 266: 242: 222: 198: 175: 9447:Cloak of invisibility 9246:Emerging technologies 9100:Quantum teleportation 8628:Wave–particle duality 8191:Entanglement-assisted 8152:quantum convolutional 7827:Quantum coin flipping 7792:Quantum teleportation 7753:entanglement-assisted 7583:DiVincenzo's criteria 7454:Frank Gaitan (2008). 5277:also constructed the 5104: 5080: 5059:orthogonal projection 5052: 5023: 4928: 4904: 4880: 4846: 4807: 4771: 4723: 4684: 4618: 4586: 4560: 4532: 4233: 4161: 4062: 4031: 4003: 3968: 3815: 3786: 3760: 3724: 3534: 3532:{\displaystyle c_{3}} 3507: 3505:{\displaystyle c_{2}} 3480: 3478:{\displaystyle c_{1}} 3453: 3451:{\displaystyle c_{0}} 3426: 3342: 3322: 3285: 3128: 2970: 2875: 2794: 2771: 2751: 2724: 2696: 2667: 2559: 2481: 2404: 2330: 2302: 2270: 2259:being small enough). 2254: 2234: 2196: 2194:{\displaystyle {1-p}} 2164: 1992: 1965: 1847:, we can write it as 1842: 1772: 1745: 1654: 1623: 1586: 1584:{\displaystyle P_{3}} 1556: 1483: 1419: 1395: 1393:{\displaystyle P_{i}} 1368: 1366:{\displaystyle P_{0}} 1341: 1035: 1011: 978:Comparison of output 977: 963: 939: 937:{\displaystyle p^{3}} 912: 867: 822: 783: 749: 718: 688: 613: 587: 528: 464: 438: 359: 287: 285:{\displaystyle \rho } 267: 243: 223: 199: 170: 138:quantum superposition 9131:Quantum field theory 9060:Quantum metamaterial 9005:Quantum cryptography 8735:Consistent histories 8002:processor benchmarks 7931:Quantum optimization 7814:Quantum cryptography 7625:physical vs. logical 7471:and quantum codes". 5452:Fundamental Research 5287:quantum entanglement 5283:stabilizer formalism 5111:correction operation 5089: 5065: 5032: 4937: 4913: 4889: 4855: 4831: 4780: 4776:, photon-gain error 4732: 4696: 4627: 4598: 4584:{\displaystyle ^{-}} 4569: 4558:{\displaystyle ^{+}} 4543: 4248: 4170: 4098: 4040: 4012: 3980: 3860: 3813:{\displaystyle U=iY} 3795: 3769: 3743: 3547: 3516: 3489: 3462: 3435: 3351: 3331: 3303: 3136: 2979: 2884: 2808: 2783: 2733: 2705: 2678: 2568: 2492: 2413: 2339: 2311: 2283: 2243: 2209: 2177: 2001: 1974: 1851: 1781: 1754: 1663: 1632: 1598: 1568: 1428: 1408: 1377: 1350: 1045: 1024: 986: 948: 921: 876: 831: 792: 761: 757:The channel acts on 727: 723:. The encoded state 701: 622: 596: 537: 478: 447: 371: 296: 276: 252: 232: 208: 182: 174:of the bit flip code 9479:Phased-array optics 9437:Acoustic levitation 9116:Quantum fluctuation 9085:Quantum programming 9045:Quantum logic gates 9030:Quantum information 9010:Quantum electronics 8485:Classical mechanics 7715:Quantum speed limit 7610:Quantum programming 7605:Quantum information 7504:1998quant.ph.10055F 7482:Found. Comput. Math 7368:2022PhRvA.105b2428A 7192:2023Natur.614..676G 7137:2022Natur.605..675P 7081:2021PhyOJ..14..184B 7018:2021Natur.598..281E 6933:2021Natur.589..220E 6846:2021SciBu..66...29G 6789:2010NaPho...4..700L 6735:10.1038/nature10786 6727:2012Natur.482..382R 6666:2011Sci...332.1059S 6660:(6033): 1059–1061. 6623:10.1038/nature03074 6615:2004Natur.432..602C 6571:2005PhRvA..71e2332P 6518:1998PhRvL..81.2152C 6465:10.1038/nature03350 6457:2005Natur.434...39K 6404:2012NJPh...14l3011H 6330:2012PhRvA..86c2324F 6250:2006PhRvA..73a2340B 6144:10.1038/nature18949 6136:2016Natur.536..441O 6081:10.1038/nature13436 6073:2014Natur.511..444S 6016:2018PhRvA..97c2346A 5962:2016PhRvX...6c1006M 5904:2001PhRvA..64a2310G 5843:2014NJPh...16d5014M 5770:2013PhRvL.111l0501L 5714:1999PhRvA..59.2631C 5642:2013RPPh...76g6001D 5593:Nielsen, Michael A. 5563:1985PhRvA..32.3266P 5520:1995PhRvA..52.2493S 5474:2021FunRe...1...50C 5118:non-degenerate code 4728:, dephasing noise 4475: 4410: 4339: 4274: 4221: 4195: 4149: 4123: 3784:{\displaystyle U=Z} 3758:{\displaystyle U=X} 462:{\displaystyle 1-p} 272:on a general input 123:quantum measurement 34:from errors due to 32:quantum information 9169:in popular culture 8951:Quantum algorithms 8799:Von Neumann–Wigner 8779:Objective collapse 8490:Old quantum theory 8364:Forest/Rigetti QCS 8100:quantum logic gate 7886:Bernstein–Vazirani 7873:Quantum algorithms 7748:Classical capacity 7632:Quantum processors 7615:Quantum simulation 7449:. Springer Nature. 5514:(4): R2493–R2496. 5352:quantum controller 5285:that incorporates 5099: 5075: 5047: 5018: 4923: 4899: 4875: 4841: 4802: 4766: 4718: 4679: 4613: 4581: 4555: 4527: 4525: 4461: 4396: 4325: 4260: 4228: 4207: 4181: 4156: 4135: 4109: 4057: 4026: 3998: 3963: 3810: 3781: 3755: 3719: 3717: 3706: 3650: 3594: 3529: 3502: 3475: 3448: 3421: 3337: 3317: 3280: 3123: 2965: 2870: 2789: 2777: 2746: 2719: 2691: 2662: 2554: 2476: 2399: 2325: 2297: 2276: 2249: 2229: 2191: 2159: 1987: 1960: 1837: 1767: 1740: 1649: 1628:between the input 1618: 1581: 1551: 1414: 1390: 1363: 1336: 1334: 1030: 1018: 1006: 958: 934: 907: 862: 817: 778: 744: 713: 683: 608: 592:. The input state 582: 523: 459: 433: 354: 282: 262: 238: 218: 194: 176: 154:no-cloning theorem 83:no-cloning theorem 9529:Quantum computing 9516: 9515: 9492: 9491: 9299:complexity theory 9284:cellular automata 9212: 9211: 9186:Quantum mysticism 9164:Schrödinger's cat 9095:Quantum simulator 9065:Quantum metrology 8993:Quantum computing 8956:Quantum amplifier 8933:Quantum spacetime 8898:Quantum cosmology 8888:Quantum chemistry 8603:Scattering theory 8551:Zero-point energy 8546:Degenerate levels 8454:Quantum mechanics 8420: 8419: 8331: 8330: 8228:Linear optical QC 8009:Quantum supremacy 7963:complexity theory 7916:Quantum annealing 7867: 7866: 7804:Superdense coding 7593:Quantum computing 7411:IBM Research Blog 7346:Physical Review A 7186:(7949): 676–681. 7121:(7911): 675–680. 7012:(7880): 281–286. 6917:(7841): 220–224. 6711:(7385): 382–385. 6609:(7017): 602–605. 6502:(10): 2152–2155. 6308:Physical Review A 6228:Physical Review A 6205:10.1109/18.681315 6130:(7617): 441–445. 6057:(7510): 444–448. 5994:Physical Review A 5940:Physical Review X 5882:Physical Review A 5692:Physical Review A 5551:Physical Review A 5508:Physical Review A 5208:Robert Calderbank 5185:Robert Calderbank 5171:, to develop the 4793: 4763: 4745: 4709: 4673: 4655: 4610: 4593:lowering operator 3992: 3975:lowering operator 3920: 3919: 3340:{\displaystyle U} 3179: 3176: 3022: 3019: 2792:{\displaystyle E} 2743: 2688: 2471: 2397: 2252:{\displaystyle p} 1417:{\displaystyle i} 1033:{\displaystyle p} 344: 241:{\displaystyle p} 106:syndrome decoding 28:quantum computing 9541: 9504: 9503: 9381:machine learning 9356:key distribution 9341:image processing 9331:error correction 9257: 9256: 9239: 9232: 9225: 9216: 9215: 9202: 9201: 8913:Quantum geometry 8908:Quantum dynamics 8765:Superdeterminism 8661:Matrix mechanics 8516:Bra–ket notation 8447: 8440: 8433: 8424: 8423: 8410: 8409: 8400: 8399: 8206: 8205: 8136:error correction 8065:computing models 8031:Relaxation times 7921:Quantum counting 7810: 7809: 7758:quantum capacity 7705:No-teleportation 7690:No-communication 7562: 7555: 7548: 7539: 7538: 7534: 7507: 7497: 7495:quant-ph/9810055 7476: 7469:Systolic freedom 7459: 7450: 7441: 7421: 7420: 7418: 7417: 7402: 7396: 7395: 7361: 7341: 7332: 7331: 7329: 7328: 7313: 7307: 7306: 7304: 7303: 7288: 7282: 7281: 7279: 7278: 7263: 7257: 7256: 7254: 7253: 7238: 7232: 7231: 7221: 7203: 7171: 7165: 7164: 7130: 7109: 7103: 7102: 7092: 7060: 7054: 7053: 7000: 6994: 6993: 6991: 6990: 6975: 6969: 6968: 6926: 6905: 6899: 6898: 6896: 6895: 6880: 6874: 6873: 6839: 6824:Science Bulletin 6815: 6809: 6808: 6782: 6767:Nature Photonics 6761: 6755: 6754: 6720: 6700: 6694: 6693: 6649: 6643: 6642: 6597: 6591: 6590: 6564: 6562:quant-ph/0502042 6544: 6538: 6537: 6511: 6509:quant-ph/9802018 6491: 6485: 6484: 6450: 6448:quant-ph/0410199 6430: 6424: 6423: 6397: 6377: 6371: 6370: 6368: 6356: 6350: 6349: 6323: 6303: 6297: 6296: 6276: 6270: 6269: 6243: 6241:quant-ph/0506023 6223: 6217: 6216: 6198: 6196:quant-ph/9608006 6189:(4): 1369–1387. 6178: 6172: 6171: 6118: 6109: 6108: 6066: 6045: 6036: 6035: 6009: 5988: 5982: 5981: 5955: 5935: 5924: 5923: 5897: 5895:quant-ph/0008040 5877: 5871: 5870: 5836: 5815: 5806: 5805: 5763: 5743: 5734: 5733: 5707: 5705:quant-ph/9809037 5698:(4): 2631–2634. 5687: 5678: 5677: 5635: 5611: 5605: 5604: 5597:Chuang, Isaac L. 5589: 5583: 5582: 5557:(6): 3266–3276. 5546: 5540: 5539: 5500: 5491: 5490: 5485: 5467: 5443: 5231:Bacon–Shor codes 5229:Two dimensional 5204:Daniel Gottesman 5200:stabilizer codes 5154:Raymond Laflamme 5109:is known as the 5108: 5106: 5105: 5100: 5098: 5097: 5084: 5082: 5081: 5076: 5074: 5073: 5056: 5054: 5053: 5048: 5046: 5045: 5044: 5027: 5025: 5024: 5019: 5014: 5013: 5012: 4999: 4998: 4997: 4959: 4958: 4949: 4948: 4932: 4930: 4929: 4924: 4922: 4921: 4908: 4906: 4905: 4900: 4898: 4897: 4884: 4882: 4881: 4876: 4874: 4873: 4864: 4863: 4850: 4848: 4847: 4842: 4840: 4839: 4811: 4809: 4808: 4803: 4801: 4800: 4795: 4794: 4786: 4775: 4773: 4772: 4767: 4765: 4764: 4756: 4753: 4752: 4747: 4746: 4738: 4727: 4725: 4724: 4719: 4717: 4716: 4711: 4710: 4702: 4688: 4686: 4685: 4680: 4675: 4674: 4666: 4663: 4662: 4657: 4656: 4648: 4622: 4620: 4619: 4614: 4612: 4611: 4603: 4590: 4588: 4587: 4582: 4580: 4579: 4564: 4562: 4561: 4556: 4554: 4553: 4536: 4534: 4533: 4528: 4526: 4507: 4490: 4474: 4469: 4460: 4439: 4425: 4409: 4404: 4395: 4371: 4354: 4338: 4333: 4324: 4303: 4289: 4273: 4268: 4259: 4237: 4235: 4234: 4229: 4220: 4215: 4206: 4194: 4189: 4180: 4165: 4163: 4162: 4157: 4148: 4143: 4134: 4122: 4117: 4108: 4088: 4087: 4083: 4066: 4064: 4063: 4058: 4047: 4035: 4033: 4032: 4027: 4019: 4007: 4005: 4004: 3999: 3994: 3993: 3985: 3972: 3970: 3969: 3964: 3953: 3942: 3941: 3940: 3930: 3921: 3915: 3914: 3907: 3893: 3887: 3879: 3878: 3877: 3867: 3849: 3848: 3844: 3819: 3817: 3816: 3811: 3790: 3788: 3787: 3782: 3764: 3762: 3761: 3756: 3728: 3726: 3725: 3720: 3718: 3711: 3710: 3655: 3654: 3599: 3598: 3538: 3536: 3535: 3530: 3528: 3527: 3511: 3509: 3508: 3503: 3501: 3500: 3484: 3482: 3481: 3476: 3474: 3473: 3457: 3455: 3454: 3449: 3447: 3446: 3430: 3428: 3427: 3422: 3417: 3416: 3401: 3400: 3385: 3384: 3369: 3368: 3346: 3344: 3343: 3338: 3326: 3324: 3323: 3318: 3310: 3289: 3287: 3286: 3281: 3270: 3256: 3236: 3222: 3202: 3188: 3180: 3178: 3177: 3172: 3163: 3155: 3154: 3153: 3143: 3132: 3130: 3129: 3124: 3113: 3099: 3079: 3065: 3045: 3031: 3023: 3021: 3020: 3015: 3006: 2998: 2997: 2996: 2986: 2974: 2972: 2971: 2966: 2961: 2960: 2951: 2946: 2945: 2930: 2929: 2920: 2915: 2914: 2899: 2891: 2879: 2877: 2876: 2871: 2863: 2858: 2857: 2839: 2834: 2833: 2815: 2798: 2796: 2795: 2790: 2755: 2753: 2752: 2747: 2745: 2744: 2741: 2728: 2726: 2725: 2720: 2712: 2700: 2698: 2697: 2692: 2690: 2689: 2686: 2671: 2669: 2668: 2663: 2655: 2650: 2645: 2640: 2635: 2634: 2619: 2614: 2609: 2604: 2599: 2598: 2583: 2575: 2563: 2561: 2560: 2555: 2547: 2542: 2541: 2523: 2518: 2517: 2499: 2485: 2483: 2482: 2477: 2472: 2467: 2465: 2451: 2437: 2420: 2408: 2406: 2405: 2400: 2398: 2393: 2391: 2377: 2363: 2346: 2334: 2332: 2331: 2326: 2318: 2306: 2304: 2303: 2298: 2290: 2258: 2256: 2255: 2250: 2238: 2236: 2235: 2230: 2225: 2200: 2198: 2197: 2192: 2190: 2168: 2166: 2165: 2160: 2155: 2154: 2139: 2138: 2117: 2116: 2086: 2085: 2058: 2047: 2046: 2034: 2017: 1996: 1994: 1993: 1988: 1986: 1985: 1969: 1967: 1966: 1961: 1938: 1924: 1909: 1908: 1878: 1877: 1846: 1844: 1843: 1838: 1836: 1835: 1805: 1804: 1776: 1774: 1773: 1768: 1766: 1765: 1749: 1747: 1746: 1741: 1730: 1716: 1702: 1701: 1692: 1691: 1686: 1685: 1675: 1674: 1658: 1656: 1655: 1650: 1645: 1627: 1625: 1624: 1619: 1614: 1590: 1588: 1587: 1582: 1580: 1579: 1560: 1558: 1557: 1552: 1547: 1546: 1537: 1536: 1523: 1522: 1513: 1512: 1502: 1497: 1479: 1478: 1466: 1465: 1444: 1443: 1438: 1437: 1423: 1421: 1420: 1415: 1399: 1397: 1396: 1391: 1389: 1388: 1372: 1370: 1369: 1364: 1362: 1361: 1345: 1343: 1342: 1337: 1335: 1328: 1311: 1303: 1286: 1274: 1273: 1257: 1240: 1232: 1215: 1203: 1202: 1186: 1169: 1161: 1144: 1132: 1131: 1115: 1098: 1090: 1073: 1061: 1060: 1039: 1037: 1036: 1031: 1015: 1013: 1012: 1007: 1002: 967: 965: 964: 959: 957: 956: 943: 941: 940: 935: 933: 932: 916: 914: 913: 908: 891: 890: 871: 869: 868: 863: 861: 860: 826: 824: 823: 818: 816: 815: 787: 785: 784: 779: 774: 753: 751: 750: 745: 740: 722: 720: 719: 714: 695:ancillary qubits 692: 690: 689: 684: 673: 672: 651: 650: 635: 617: 615: 614: 609: 591: 589: 588: 583: 566: 565: 564: 532: 530: 529: 524: 507: 506: 505: 468: 466: 465: 460: 442: 440: 439: 434: 426: 421: 420: 402: 397: 396: 378: 363: 361: 360: 355: 342: 305: 304: 291: 289: 288: 283: 271: 269: 268: 263: 261: 260: 247: 245: 244: 239: 227: 225: 224: 219: 217: 216: 203: 201: 200: 195: 142:quantum computer 56:error correction 9549: 9548: 9544: 9543: 9542: 9540: 9539: 9538: 9519: 9518: 9517: 9512: 9488: 9425: 9336:finite automata 9248: 9243: 9213: 9208: 9190: 9176:Wigner's friend 9152: 9143:Quantum gravity 9104: 9090:Quantum sensing 9070:Quantum network 9050:Quantum machine 9020:Quantum imaging 8983:Quantum circuit 8978:Quantum channel 8937: 8883:Quantum biology 8869: 8845:Elitzur–Vaidman 8820:Davisson–Germer 8803: 8755:Hidden-variable 8745:de Broglie–Bohm 8722:Interpretations 8716: 8680: 8634: 8521:Complementarity 8499: 8456: 8451: 8421: 8416: 8388: 8338: 8327: 8300:Superconducting 8294: 8260: 8251:Neutral atom QC 8243:Ultracold atoms 8237: 8202:implementations 8201: 8195: 8135: 8128: 8095:Quantum circuit 8063: 8057: 8051: 8041: 8001: 7995: 7962: 7955: 7911:Hidden subgroup 7863: 7852:other protocols 7808: 7785:quantum network 7780:Quantum channel 7740: 7734: 7680:No-broadcasting 7670:Gottesman–Knill 7643: 7571: 7566: 7517: 7514: 7466: 7429: 7427:Further reading 7424: 7415: 7413: 7403: 7399: 7342: 7335: 7326: 7324: 7314: 7310: 7301: 7299: 7289: 7285: 7276: 7274: 7264: 7260: 7251: 7249: 7239: 7235: 7172: 7168: 7110: 7106: 7061: 7057: 7001: 6997: 6988: 6986: 6976: 6972: 6906: 6902: 6893: 6891: 6882: 6881: 6877: 6816: 6812: 6762: 6758: 6701: 6697: 6650: 6646: 6598: 6594: 6545: 6541: 6496:Phys. Rev. Lett 6492: 6488: 6441:(7029): 39–44. 6431: 6427: 6378: 6374: 6357: 6353: 6304: 6300: 6277: 6273: 6224: 6220: 6179: 6175: 6119: 6112: 6046: 6039: 5989: 5985: 5936: 5927: 5878: 5874: 5816: 5809: 5744: 5737: 5688: 5681: 5612: 5608: 5590: 5586: 5547: 5543: 5501: 5494: 5444: 5440: 5436: 5419: 5410: 5363:entangling gate 5333: 5220:N. J. A. Sloane 5131: 5093: 5092: 5090: 5087: 5086: 5069: 5068: 5066: 5063: 5062: 5040: 5039: 5035: 5033: 5030: 5029: 5008: 5007: 5003: 4993: 4992: 4988: 4954: 4953: 4944: 4943: 4938: 4935: 4934: 4917: 4916: 4914: 4911: 4910: 4893: 4892: 4890: 4887: 4886: 4869: 4868: 4859: 4858: 4856: 4853: 4852: 4835: 4834: 4832: 4829: 4828: 4826:quantum channel 4818: 4796: 4785: 4784: 4783: 4781: 4778: 4777: 4755: 4754: 4748: 4737: 4736: 4735: 4733: 4730: 4729: 4712: 4701: 4700: 4699: 4697: 4694: 4693: 4665: 4664: 4658: 4647: 4646: 4645: 4628: 4625: 4624: 4602: 4601: 4599: 4596: 4595: 4575: 4572: 4570: 4567: 4566: 4549: 4546: 4544: 4541: 4540: 4524: 4523: 4503: 4486: 4479: 4470: 4465: 4456: 4453: 4452: 4435: 4421: 4414: 4405: 4400: 4391: 4388: 4387: 4367: 4350: 4343: 4334: 4329: 4320: 4317: 4316: 4299: 4285: 4278: 4269: 4264: 4255: 4251: 4249: 4246: 4245: 4216: 4211: 4202: 4190: 4185: 4176: 4171: 4168: 4167: 4144: 4139: 4130: 4118: 4113: 4104: 4099: 4096: 4095: 4089: 4085: 4081: 4079: 4078: 4043: 4041: 4038: 4037: 4015: 4013: 4010: 4009: 3984: 3983: 3981: 3978: 3977: 3949: 3936: 3935: 3931: 3926: 3903: 3889: 3888: 3886: 3873: 3872: 3868: 3863: 3861: 3858: 3857: 3852:Written in the 3850: 3846: 3842: 3840: 3839: 3826: 3796: 3793: 3792: 3770: 3767: 3766: 3744: 3741: 3740: 3716: 3715: 3705: 3704: 3696: 3690: 3689: 3684: 3674: 3673: 3666: 3660: 3659: 3649: 3648: 3643: 3637: 3636: 3628: 3618: 3617: 3610: 3604: 3603: 3593: 3592: 3587: 3581: 3580: 3575: 3565: 3564: 3557: 3550: 3548: 3545: 3544: 3523: 3519: 3517: 3514: 3513: 3496: 3492: 3490: 3487: 3486: 3469: 3465: 3463: 3460: 3459: 3442: 3438: 3436: 3433: 3432: 3412: 3408: 3396: 3392: 3380: 3376: 3364: 3360: 3352: 3349: 3348: 3332: 3329: 3328: 3306: 3304: 3301: 3300: 3266: 3252: 3232: 3218: 3198: 3184: 3171: 3167: 3162: 3149: 3148: 3144: 3139: 3137: 3134: 3133: 3109: 3095: 3075: 3061: 3041: 3027: 3014: 3010: 3005: 2992: 2991: 2987: 2982: 2980: 2977: 2976: 2956: 2952: 2947: 2941: 2937: 2925: 2921: 2916: 2910: 2906: 2892: 2887: 2885: 2882: 2881: 2859: 2853: 2849: 2835: 2829: 2825: 2811: 2809: 2806: 2805: 2801:quantum channel 2784: 2781: 2780: 2773:Quantum circuit 2762: 2740: 2736: 2734: 2731: 2730: 2708: 2706: 2703: 2702: 2685: 2681: 2679: 2676: 2675: 2651: 2646: 2641: 2636: 2630: 2626: 2615: 2610: 2605: 2600: 2594: 2590: 2576: 2571: 2569: 2566: 2565: 2543: 2537: 2533: 2519: 2513: 2509: 2495: 2493: 2490: 2489: 2466: 2461: 2447: 2433: 2416: 2414: 2411: 2410: 2392: 2387: 2373: 2359: 2342: 2340: 2337: 2336: 2314: 2312: 2309: 2308: 2286: 2284: 2281: 2280: 2272:Quantum circuit 2265: 2244: 2241: 2240: 2221: 2210: 2207: 2206: 2180: 2178: 2175: 2174: 2150: 2146: 2134: 2130: 2112: 2108: 2081: 2077: 2051: 2042: 2038: 2027: 2010: 2002: 1999: 1998: 1981: 1977: 1975: 1972: 1971: 1931: 1917: 1904: 1900: 1873: 1869: 1852: 1849: 1848: 1831: 1827: 1800: 1796: 1782: 1779: 1778: 1761: 1757: 1755: 1752: 1751: 1723: 1709: 1697: 1696: 1687: 1681: 1680: 1679: 1670: 1666: 1664: 1661: 1660: 1659:and the output 1638: 1633: 1630: 1629: 1607: 1599: 1596: 1595: 1575: 1571: 1569: 1566: 1565: 1542: 1538: 1532: 1528: 1518: 1514: 1508: 1504: 1498: 1487: 1474: 1470: 1461: 1457: 1439: 1433: 1432: 1431: 1429: 1426: 1425: 1409: 1406: 1405: 1384: 1380: 1378: 1375: 1374: 1357: 1353: 1351: 1348: 1347: 1333: 1332: 1324: 1307: 1299: 1282: 1275: 1269: 1265: 1262: 1261: 1253: 1236: 1228: 1211: 1204: 1198: 1194: 1191: 1190: 1182: 1165: 1157: 1140: 1133: 1127: 1123: 1120: 1119: 1111: 1094: 1086: 1069: 1062: 1056: 1052: 1048: 1046: 1043: 1042: 1025: 1022: 1021: 998: 987: 984: 983: 952: 951: 949: 946: 945: 928: 924: 922: 919: 918: 886: 882: 877: 874: 873: 856: 852: 832: 829: 828: 811: 807: 793: 790: 789: 767: 762: 759: 758: 733: 728: 725: 724: 702: 699: 698: 668: 664: 646: 642: 628: 623: 620: 619: 597: 594: 593: 560: 559: 555: 538: 535: 534: 501: 500: 496: 479: 476: 475: 448: 445: 444: 422: 416: 412: 398: 392: 388: 374: 372: 369: 368: 300: 299: 297: 294: 293: 277: 274: 273: 256: 255: 253: 250: 249: 233: 230: 229: 212: 211: 209: 206: 205: 183: 180: 179: 172:Quantum circuit 150: 64:repetition code 17: 12: 11: 5: 9547: 9537: 9536: 9531: 9514: 9513: 9511: 9510: 9497: 9494: 9493: 9490: 9489: 9487: 9486: 9481: 9476: 9471: 9466: 9465: 9464: 9454: 9449: 9444: 9439: 9433: 9431: 9427: 9426: 9424: 9423: 9418: 9413: 9408: 9403: 9398: 9396:neural network 9393: 9388: 9383: 9378: 9373: 9368: 9363: 9358: 9353: 9348: 9343: 9338: 9333: 9328: 9323: 9318: 9317: 9316: 9306: 9301: 9296: 9291: 9286: 9281: 9276: 9271: 9265: 9263: 9254: 9250: 9249: 9242: 9241: 9234: 9227: 9219: 9210: 9209: 9207: 9206: 9195: 9192: 9191: 9189: 9188: 9183: 9178: 9173: 9172: 9171: 9160: 9158: 9154: 9153: 9151: 9150: 9145: 9140: 9139: 9138: 9128: 9123: 9121:Casimir effect 9118: 9112: 9110: 9106: 9105: 9103: 9102: 9097: 9092: 9087: 9082: 9080:Quantum optics 9077: 9072: 9067: 9062: 9057: 9052: 9047: 9042: 9037: 9032: 9027: 9022: 9017: 9012: 9007: 9002: 9001: 9000: 8990: 8985: 8980: 8975: 8974: 8973: 8963: 8958: 8953: 8947: 8945: 8939: 8938: 8936: 8935: 8930: 8925: 8920: 8915: 8910: 8905: 8900: 8895: 8890: 8885: 8879: 8877: 8871: 8870: 8868: 8867: 8862: 8857: 8855:Quantum eraser 8852: 8847: 8842: 8837: 8832: 8827: 8822: 8817: 8811: 8809: 8805: 8804: 8802: 8801: 8796: 8791: 8786: 8781: 8776: 8771: 8770: 8769: 8768: 8767: 8752: 8747: 8742: 8737: 8732: 8726: 8724: 8718: 8717: 8715: 8714: 8709: 8704: 8699: 8694: 8688: 8686: 8682: 8681: 8679: 8678: 8673: 8668: 8663: 8658: 8653: 8648: 8642: 8640: 8636: 8635: 8633: 8632: 8631: 8630: 8625: 8615: 8610: 8605: 8600: 8595: 8590: 8585: 8580: 8575: 8570: 8565: 8560: 8555: 8554: 8553: 8548: 8543: 8538: 8528: 8526:Density matrix 8523: 8518: 8513: 8507: 8505: 8501: 8500: 8498: 8497: 8492: 8487: 8482: 8481: 8480: 8470: 8464: 8462: 8458: 8457: 8450: 8449: 8442: 8435: 8427: 8418: 8417: 8415: 8414: 8404: 8393: 8390: 8389: 8387: 8386: 8384:many others... 8381: 8376: 8371: 8366: 8357: 8343: 8341: 8333: 8332: 8329: 8328: 8326: 8325: 8320: 8315: 8310: 8304: 8302: 8296: 8295: 8293: 8292: 8287: 8282: 8277: 8271: 8269: 8262: 8261: 8259: 8258: 8256:Trapped-ion QC 8253: 8247: 8245: 8239: 8238: 8236: 8235: 8230: 8225: 8220: 8214: 8212: 8210:Quantum optics 8203: 8197: 8196: 8194: 8193: 8188: 8187: 8186: 8179: 8174: 8169: 8164: 8159: 8154: 8149: 8140: 8138: 8130: 8129: 8127: 8126: 8121: 8116: 8115: 8114: 8104: 8103: 8102: 8092: 8091: 8090: 8080: 8075: 8069: 8067: 8059: 8058: 8056: 8055: 8054: 8053: 8049: 8043: 8039: 8028: 8027: 8026: 8016: 8014:Quantum volume 8011: 8005: 8003: 7997: 7996: 7994: 7993: 7988: 7983: 7978: 7973: 7967: 7965: 7957: 7956: 7954: 7953: 7948: 7943: 7938: 7933: 7928: 7923: 7918: 7913: 7908: 7903: 7898: 7893: 7891:Boson sampling 7888: 7883: 7877: 7875: 7869: 7868: 7865: 7864: 7862: 7861: 7856: 7855: 7854: 7849: 7844: 7834: 7829: 7824: 7818: 7816: 7807: 7806: 7801: 7800: 7799: 7789: 7788: 7787: 7777: 7772: 7767: 7762: 7761: 7760: 7755: 7744: 7742: 7736: 7735: 7733: 7732: 7727: 7725:Solovay–Kitaev 7722: 7717: 7712: 7707: 7702: 7697: 7692: 7687: 7682: 7677: 7672: 7667: 7662: 7657: 7651: 7649: 7645: 7644: 7642: 7641: 7640: 7639: 7629: 7628: 7627: 7617: 7612: 7607: 7602: 7601: 7600: 7590: 7585: 7579: 7577: 7573: 7572: 7565: 7564: 7557: 7550: 7542: 7536: 7535: 7513: 7512:External links 7510: 7509: 7508: 7488:(3): 325–332. 7477: 7464: 7460: 7451: 7442: 7428: 7425: 7423: 7422: 7397: 7333: 7308: 7283: 7258: 7233: 7166: 7104: 7055: 6995: 6970: 6900: 6875: 6810: 6756: 6695: 6644: 6592: 6539: 6486: 6425: 6388:(12): 123011. 6372: 6351: 6298: 6271: 6218: 6173: 6110: 6037: 5983: 5925: 5872: 5807: 5754:(12): 120501. 5735: 5679: 5606: 5584: 5541: 5504:Shor, Peter W. 5492: 5437: 5435: 5432: 5431: 5430: 5425: 5418: 5415: 5409: 5406: 5332: 5329: 5310:Dorit Aharonov 5306:Michael Ben-Or 5298: 5297: 5290: 5275:Min-Hsiu Hsieh 5264: 5246: 5227: 5224:additive codes 5202:discovered by 5196: 5165: 5158:fault-tolerant 5151: 5141: 5130: 5127: 5096: 5072: 5043: 5038: 5017: 5011: 5006: 5002: 4996: 4991: 4987: 4984: 4981: 4977: 4974: 4971: 4968: 4965: 4962: 4957: 4952: 4947: 4942: 4920: 4896: 4872: 4867: 4862: 4851:is a subspace 4838: 4820:In general, a 4817: 4814: 4799: 4792: 4789: 4762: 4759: 4751: 4744: 4741: 4715: 4708: 4705: 4678: 4672: 4669: 4661: 4654: 4651: 4644: 4641: 4638: 4635: 4632: 4609: 4606: 4578: 4574: 4552: 4548: 4522: 4519: 4516: 4513: 4510: 4506: 4502: 4499: 4496: 4493: 4489: 4485: 4482: 4480: 4478: 4473: 4468: 4464: 4459: 4455: 4454: 4451: 4448: 4445: 4442: 4438: 4434: 4431: 4428: 4424: 4420: 4417: 4415: 4413: 4408: 4403: 4399: 4394: 4390: 4389: 4386: 4383: 4380: 4377: 4374: 4370: 4366: 4363: 4360: 4357: 4353: 4349: 4346: 4344: 4342: 4337: 4332: 4328: 4323: 4319: 4318: 4315: 4312: 4309: 4306: 4302: 4298: 4295: 4292: 4288: 4284: 4281: 4279: 4277: 4272: 4267: 4263: 4258: 4254: 4253: 4240:coherent state 4227: 4224: 4219: 4214: 4210: 4205: 4201: 4198: 4193: 4188: 4184: 4179: 4175: 4155: 4152: 4147: 4142: 4138: 4133: 4129: 4126: 4121: 4116: 4112: 4107: 4103: 4077: 4074: 4056: 4053: 4050: 4046: 4025: 4022: 4018: 3997: 3991: 3988: 3962: 3959: 3956: 3952: 3948: 3945: 3939: 3934: 3929: 3924: 3918: 3913: 3910: 3906: 3902: 3899: 3896: 3892: 3885: 3882: 3876: 3871: 3866: 3838: 3835: 3825: 3822: 3809: 3806: 3803: 3800: 3780: 3777: 3774: 3754: 3751: 3748: 3714: 3709: 3703: 3700: 3697: 3695: 3692: 3691: 3688: 3685: 3683: 3680: 3679: 3677: 3672: 3669: 3667: 3665: 3662: 3661: 3658: 3653: 3647: 3644: 3642: 3639: 3638: 3635: 3632: 3629: 3627: 3624: 3623: 3621: 3616: 3613: 3611: 3609: 3606: 3605: 3602: 3597: 3591: 3588: 3586: 3583: 3582: 3579: 3576: 3574: 3571: 3570: 3568: 3563: 3560: 3558: 3556: 3553: 3552: 3541:Pauli matrices 3526: 3522: 3499: 3495: 3472: 3468: 3445: 3441: 3420: 3415: 3411: 3407: 3404: 3399: 3395: 3391: 3388: 3383: 3379: 3375: 3372: 3367: 3363: 3359: 3356: 3336: 3316: 3313: 3309: 3279: 3276: 3273: 3269: 3265: 3262: 3259: 3255: 3251: 3248: 3245: 3242: 3239: 3235: 3231: 3228: 3225: 3221: 3217: 3214: 3211: 3208: 3205: 3201: 3197: 3194: 3191: 3187: 3183: 3175: 3170: 3166: 3161: 3158: 3152: 3147: 3142: 3122: 3119: 3116: 3112: 3108: 3105: 3102: 3098: 3094: 3091: 3088: 3085: 3082: 3078: 3074: 3071: 3068: 3064: 3060: 3057: 3054: 3051: 3048: 3044: 3040: 3037: 3034: 3030: 3026: 3018: 3013: 3009: 3004: 3001: 2995: 2990: 2985: 2964: 2959: 2955: 2950: 2944: 2940: 2936: 2933: 2928: 2924: 2919: 2913: 2909: 2905: 2902: 2898: 2895: 2890: 2869: 2866: 2862: 2856: 2852: 2848: 2845: 2842: 2838: 2832: 2828: 2824: 2821: 2818: 2814: 2788: 2761: 2758: 2739: 2718: 2715: 2711: 2684: 2661: 2658: 2654: 2649: 2644: 2639: 2633: 2629: 2625: 2622: 2618: 2613: 2608: 2603: 2597: 2593: 2589: 2586: 2582: 2579: 2574: 2553: 2550: 2546: 2540: 2536: 2532: 2529: 2526: 2522: 2516: 2512: 2508: 2505: 2502: 2498: 2475: 2470: 2464: 2460: 2457: 2454: 2450: 2446: 2443: 2440: 2436: 2432: 2429: 2426: 2423: 2419: 2396: 2390: 2386: 2383: 2380: 2376: 2372: 2369: 2366: 2362: 2358: 2355: 2352: 2349: 2345: 2324: 2321: 2317: 2296: 2293: 2289: 2264: 2263:Sign flip code 2261: 2248: 2228: 2224: 2220: 2217: 2214: 2189: 2186: 2183: 2158: 2153: 2149: 2145: 2142: 2137: 2133: 2129: 2126: 2123: 2120: 2115: 2111: 2107: 2104: 2101: 2098: 2095: 2092: 2089: 2084: 2080: 2076: 2073: 2070: 2067: 2064: 2061: 2057: 2054: 2050: 2045: 2041: 2037: 2033: 2030: 2026: 2023: 2020: 2016: 2013: 2009: 2006: 1984: 1980: 1959: 1956: 1953: 1950: 1947: 1944: 1941: 1937: 1934: 1930: 1927: 1923: 1920: 1916: 1912: 1907: 1903: 1899: 1896: 1893: 1890: 1887: 1884: 1881: 1876: 1872: 1868: 1865: 1862: 1859: 1856: 1834: 1830: 1826: 1823: 1820: 1817: 1814: 1811: 1808: 1803: 1799: 1795: 1792: 1789: 1786: 1764: 1760: 1739: 1736: 1733: 1729: 1726: 1722: 1719: 1715: 1712: 1708: 1705: 1700: 1695: 1690: 1684: 1678: 1673: 1669: 1648: 1644: 1641: 1637: 1617: 1613: 1610: 1606: 1603: 1578: 1574: 1550: 1545: 1541: 1535: 1531: 1526: 1521: 1517: 1511: 1507: 1501: 1496: 1493: 1490: 1486: 1482: 1477: 1473: 1469: 1464: 1460: 1456: 1453: 1450: 1447: 1442: 1436: 1413: 1387: 1383: 1360: 1356: 1331: 1327: 1323: 1320: 1317: 1314: 1310: 1306: 1302: 1298: 1295: 1292: 1289: 1285: 1281: 1278: 1276: 1272: 1268: 1264: 1263: 1260: 1256: 1252: 1249: 1246: 1243: 1239: 1235: 1231: 1227: 1224: 1221: 1218: 1214: 1210: 1207: 1205: 1201: 1197: 1193: 1192: 1189: 1185: 1181: 1178: 1175: 1172: 1168: 1164: 1160: 1156: 1153: 1150: 1147: 1143: 1139: 1136: 1134: 1130: 1126: 1122: 1121: 1118: 1114: 1110: 1107: 1104: 1101: 1097: 1093: 1089: 1085: 1082: 1079: 1076: 1072: 1068: 1065: 1063: 1059: 1055: 1051: 1050: 1029: 1005: 1001: 997: 994: 991: 955: 931: 927: 906: 903: 900: 897: 894: 889: 885: 881: 859: 855: 851: 848: 845: 842: 839: 836: 814: 810: 806: 803: 800: 797: 777: 773: 770: 766: 743: 739: 736: 732: 712: 709: 706: 682: 679: 676: 671: 667: 663: 660: 657: 654: 649: 645: 641: 638: 634: 631: 627: 607: 604: 601: 581: 578: 575: 572: 569: 563: 558: 554: 551: 548: 545: 542: 522: 519: 516: 513: 510: 504: 499: 495: 492: 489: 486: 483: 458: 455: 452: 432: 429: 425: 419: 415: 411: 408: 405: 401: 395: 391: 387: 384: 381: 377: 353: 350: 347: 341: 338: 335: 332: 329: 326: 323: 320: 317: 314: 311: 308: 303: 281: 259: 237: 215: 193: 190: 187: 149: 146: 115:Pauli matrices 58:often employs 15: 9: 6: 4: 3: 2: 9546: 9535: 9532: 9530: 9527: 9526: 9524: 9509: 9508: 9499: 9498: 9495: 9485: 9482: 9480: 9477: 9475: 9472: 9470: 9467: 9463: 9462:Plasma window 9460: 9459: 9458: 9455: 9453: 9450: 9448: 9445: 9443: 9440: 9438: 9435: 9434: 9432: 9428: 9422: 9421:teleportation 9419: 9417: 9414: 9412: 9409: 9407: 9404: 9402: 9399: 9397: 9394: 9392: 9389: 9387: 9384: 9382: 9379: 9377: 9374: 9372: 9369: 9367: 9364: 9362: 9359: 9357: 9354: 9352: 9349: 9347: 9344: 9342: 9339: 9337: 9334: 9332: 9329: 9327: 9324: 9322: 9319: 9315: 9312: 9311: 9310: 9307: 9305: 9302: 9300: 9297: 9295: 9292: 9290: 9287: 9285: 9282: 9280: 9277: 9275: 9272: 9270: 9267: 9266: 9264: 9262: 9258: 9255: 9251: 9247: 9240: 9235: 9233: 9228: 9226: 9221: 9220: 9217: 9205: 9197: 9196: 9193: 9187: 9184: 9182: 9179: 9177: 9174: 9170: 9167: 9166: 9165: 9162: 9161: 9159: 9155: 9149: 9146: 9144: 9141: 9137: 9134: 9133: 9132: 9129: 9127: 9124: 9122: 9119: 9117: 9114: 9113: 9111: 9107: 9101: 9098: 9096: 9093: 9091: 9088: 9086: 9083: 9081: 9078: 9076: 9073: 9071: 9068: 9066: 9063: 9061: 9058: 9056: 9053: 9051: 9048: 9046: 9043: 9041: 9040:Quantum logic 9038: 9036: 9033: 9031: 9028: 9026: 9023: 9021: 9018: 9016: 9013: 9011: 9008: 9006: 9003: 8999: 8996: 8995: 8994: 8991: 8989: 8986: 8984: 8981: 8979: 8976: 8972: 8969: 8968: 8967: 8964: 8962: 8959: 8957: 8954: 8952: 8949: 8948: 8946: 8944: 8940: 8934: 8931: 8929: 8926: 8924: 8921: 8919: 8916: 8914: 8911: 8909: 8906: 8904: 8901: 8899: 8896: 8894: 8893:Quantum chaos 8891: 8889: 8886: 8884: 8881: 8880: 8878: 8876: 8872: 8866: 8863: 8861: 8860:Stern–Gerlach 8858: 8856: 8853: 8851: 8848: 8846: 8843: 8841: 8838: 8836: 8833: 8831: 8828: 8826: 8823: 8821: 8818: 8816: 8813: 8812: 8810: 8806: 8800: 8797: 8795: 8794:Transactional 8792: 8790: 8787: 8785: 8784:Quantum logic 8782: 8780: 8777: 8775: 8772: 8766: 8763: 8762: 8761: 8758: 8757: 8756: 8753: 8751: 8748: 8746: 8743: 8741: 8738: 8736: 8733: 8731: 8728: 8727: 8725: 8723: 8719: 8713: 8710: 8708: 8705: 8703: 8700: 8698: 8695: 8693: 8690: 8689: 8687: 8683: 8677: 8674: 8672: 8669: 8667: 8664: 8662: 8659: 8657: 8654: 8652: 8649: 8647: 8644: 8643: 8641: 8637: 8629: 8626: 8624: 8621: 8620: 8619: 8618:Wave function 8616: 8614: 8611: 8609: 8606: 8604: 8601: 8599: 8596: 8594: 8593:Superposition 8591: 8589: 8588:Quantum state 8586: 8584: 8581: 8579: 8576: 8574: 8571: 8569: 8566: 8564: 8561: 8559: 8556: 8552: 8549: 8547: 8544: 8542: 8541:Excited state 8539: 8537: 8534: 8533: 8532: 8529: 8527: 8524: 8522: 8519: 8517: 8514: 8512: 8509: 8508: 8506: 8502: 8496: 8493: 8491: 8488: 8486: 8483: 8479: 8476: 8475: 8474: 8471: 8469: 8466: 8465: 8463: 8459: 8455: 8448: 8443: 8441: 8436: 8434: 8429: 8428: 8425: 8413: 8405: 8403: 8395: 8394: 8391: 8385: 8382: 8380: 8377: 8375: 8372: 8370: 8367: 8365: 8361: 8358: 8356: 8352: 8348: 8345: 8344: 8342: 8340: 8334: 8324: 8321: 8319: 8316: 8314: 8311: 8309: 8306: 8305: 8303: 8301: 8297: 8291: 8288: 8286: 8283: 8281: 8280:Spin qubit QC 8278: 8276: 8273: 8272: 8270: 8267: 8263: 8257: 8254: 8252: 8249: 8248: 8246: 8244: 8240: 8234: 8231: 8229: 8226: 8224: 8221: 8219: 8216: 8215: 8213: 8211: 8207: 8204: 8198: 8192: 8189: 8185: 8184: 8180: 8178: 8175: 8173: 8170: 8168: 8165: 8163: 8160: 8158: 8155: 8153: 8150: 8148: 8145: 8144: 8142: 8141: 8139: 8137: 8131: 8125: 8122: 8120: 8117: 8113: 8110: 8109: 8108: 8105: 8101: 8098: 8097: 8096: 8093: 8089: 8088:cluster state 8086: 8085: 8084: 8081: 8079: 8076: 8074: 8071: 8070: 8068: 8066: 8060: 8052: 8048: 8044: 8042: 8038: 8034: 8033: 8032: 8029: 8025: 8022: 8021: 8020: 8017: 8015: 8012: 8010: 8007: 8006: 8004: 7998: 7992: 7989: 7987: 7984: 7982: 7979: 7977: 7974: 7972: 7969: 7968: 7966: 7964: 7958: 7952: 7949: 7947: 7944: 7942: 7939: 7937: 7934: 7932: 7929: 7927: 7924: 7922: 7919: 7917: 7914: 7912: 7909: 7907: 7904: 7902: 7899: 7897: 7896:Deutsch–Jozsa 7894: 7892: 7889: 7887: 7884: 7882: 7879: 7878: 7876: 7874: 7870: 7860: 7857: 7853: 7850: 7848: 7845: 7843: 7840: 7839: 7838: 7835: 7833: 7832:Quantum money 7830: 7828: 7825: 7823: 7820: 7819: 7817: 7815: 7811: 7805: 7802: 7798: 7795: 7794: 7793: 7790: 7786: 7783: 7782: 7781: 7778: 7776: 7773: 7771: 7768: 7766: 7763: 7759: 7756: 7754: 7751: 7750: 7749: 7746: 7745: 7743: 7741:communication 7737: 7731: 7728: 7726: 7723: 7721: 7718: 7716: 7713: 7711: 7708: 7706: 7703: 7701: 7698: 7696: 7693: 7691: 7688: 7686: 7683: 7681: 7678: 7676: 7673: 7671: 7668: 7666: 7663: 7661: 7658: 7656: 7653: 7652: 7650: 7646: 7638: 7635: 7634: 7633: 7630: 7626: 7623: 7622: 7621: 7618: 7616: 7613: 7611: 7608: 7606: 7603: 7599: 7596: 7595: 7594: 7591: 7589: 7586: 7584: 7581: 7580: 7578: 7574: 7570: 7563: 7558: 7556: 7551: 7549: 7544: 7543: 7540: 7532: 7528: 7524: 7523:Quantum Light 7520: 7516: 7515: 7505: 7501: 7496: 7491: 7487: 7483: 7478: 7474: 7470: 7461: 7457: 7452: 7448: 7443: 7439: 7435: 7431: 7430: 7412: 7408: 7401: 7393: 7389: 7385: 7381: 7377: 7373: 7369: 7365: 7360: 7355: 7352:(2): 022428. 7351: 7347: 7340: 7338: 7323: 7319: 7312: 7298: 7294: 7287: 7273: 7269: 7262: 7248: 7247:Physics World 7244: 7237: 7229: 7225: 7220: 7215: 7211: 7207: 7202: 7197: 7193: 7189: 7185: 7181: 7177: 7170: 7162: 7158: 7154: 7150: 7146: 7142: 7138: 7134: 7129: 7124: 7120: 7116: 7113:operations". 7108: 7100: 7096: 7091: 7086: 7082: 7078: 7074: 7070: 7066: 7059: 7051: 7047: 7043: 7039: 7035: 7031: 7027: 7023: 7019: 7015: 7011: 7007: 6999: 6985: 6981: 6974: 6966: 6962: 6958: 6954: 6950: 6946: 6942: 6938: 6934: 6930: 6925: 6920: 6916: 6912: 6904: 6889: 6885: 6879: 6871: 6867: 6863: 6859: 6855: 6851: 6847: 6843: 6838: 6833: 6829: 6825: 6821: 6814: 6806: 6802: 6798: 6794: 6790: 6786: 6781: 6776: 6772: 6768: 6760: 6752: 6748: 6744: 6740: 6736: 6732: 6728: 6724: 6719: 6714: 6710: 6706: 6699: 6691: 6687: 6683: 6679: 6675: 6671: 6667: 6663: 6659: 6655: 6648: 6640: 6636: 6632: 6628: 6624: 6620: 6616: 6612: 6608: 6604: 6596: 6588: 6584: 6580: 6576: 6572: 6568: 6563: 6558: 6555:(5): 052332. 6554: 6550: 6543: 6535: 6531: 6527: 6523: 6519: 6515: 6510: 6505: 6501: 6497: 6490: 6482: 6478: 6474: 6470: 6466: 6462: 6458: 6454: 6449: 6444: 6440: 6436: 6429: 6421: 6417: 6413: 6409: 6405: 6401: 6396: 6391: 6387: 6383: 6376: 6367: 6362: 6355: 6347: 6343: 6339: 6335: 6331: 6327: 6322: 6317: 6314:(3): 032324. 6313: 6309: 6302: 6294: 6290: 6286: 6282: 6275: 6267: 6263: 6259: 6255: 6251: 6247: 6242: 6237: 6234:(1): 012340. 6233: 6229: 6222: 6214: 6210: 6206: 6202: 6197: 6192: 6188: 6184: 6177: 6169: 6165: 6161: 6157: 6153: 6149: 6145: 6141: 6137: 6133: 6129: 6125: 6117: 6115: 6106: 6102: 6098: 6094: 6090: 6086: 6082: 6078: 6074: 6070: 6065: 6060: 6056: 6052: 6044: 6042: 6033: 6029: 6025: 6021: 6017: 6013: 6008: 6003: 6000:(3): 032346. 5999: 5995: 5987: 5979: 5975: 5971: 5967: 5963: 5959: 5954: 5949: 5946:(3): 031006. 5945: 5941: 5934: 5932: 5930: 5921: 5917: 5913: 5909: 5905: 5901: 5896: 5891: 5888:(1): 012310. 5887: 5883: 5876: 5868: 5864: 5860: 5856: 5852: 5848: 5844: 5840: 5835: 5830: 5827:(4): 045014. 5826: 5822: 5814: 5812: 5803: 5799: 5795: 5791: 5787: 5783: 5779: 5775: 5771: 5767: 5762: 5757: 5753: 5749: 5742: 5740: 5731: 5727: 5723: 5719: 5715: 5711: 5706: 5701: 5697: 5693: 5686: 5684: 5675: 5671: 5667: 5663: 5659: 5655: 5651: 5647: 5643: 5639: 5634: 5629: 5626:(7): 076001. 5625: 5621: 5617: 5610: 5602: 5598: 5594: 5588: 5580: 5576: 5572: 5568: 5564: 5560: 5556: 5552: 5545: 5537: 5533: 5529: 5525: 5521: 5517: 5513: 5509: 5505: 5499: 5497: 5489: 5484: 5479: 5475: 5471: 5466: 5461: 5457: 5453: 5449: 5442: 5438: 5429: 5426: 5424: 5421: 5420: 5414: 5405: 5401: 5397: 5395: 5390: 5388: 5383: 5378: 5376: 5372: 5368: 5364: 5359: 5356: 5353: 5349: 5344: 5342: 5338: 5328: 5326: 5321: 5319: 5318:quantum gates 5315: 5311: 5307: 5303: 5295: 5291: 5288: 5284: 5280: 5276: 5272: 5268: 5265: 5262: 5258: 5254: 5250: 5249:Alexei Kitaev 5247: 5244: 5240: 5236: 5232: 5228: 5225: 5221: 5217: 5213: 5209: 5205: 5201: 5197: 5194: 5193:Andrew Steane 5190: 5186: 5182: 5178: 5174: 5170: 5166: 5163: 5159: 5155: 5152: 5149: 5145: 5144:Andrew Steane 5142: 5139: 5136: 5135: 5134: 5126: 5124: 5119: 5114: 5112: 5060: 5036: 5015: 5004: 5000: 4989: 4985: 4982: 4975: 4972: 4966: 4950: 4865: 4827: 4823: 4816:General codes 4813: 4797: 4787: 4757: 4749: 4739: 4713: 4703: 4690: 4667: 4659: 4649: 4642: 4639: 4633: 4630: 4604: 4594: 4576: 4573: 4550: 4547: 4537: 4520: 4514: 4511: 4508: 4500: 4494: 4491: 4483: 4481: 4471: 4466: 4462: 4449: 4443: 4440: 4432: 4426: 4418: 4416: 4406: 4401: 4397: 4384: 4378: 4375: 4372: 4364: 4358: 4355: 4347: 4345: 4335: 4330: 4326: 4313: 4307: 4304: 4296: 4290: 4282: 4280: 4270: 4265: 4261: 4243: 4241: 4217: 4212: 4208: 4199: 4191: 4186: 4182: 4145: 4140: 4136: 4127: 4119: 4114: 4110: 4093: 4084: 4073: 4070: 4069:photon number 4054: 4048: 4020: 3995: 3986: 3976: 3960: 3954: 3946: 3932: 3922: 3916: 3908: 3900: 3894: 3883: 3869: 3855: 3845: 3837:Binomial code 3834: 3831: 3824:Bosonic codes 3821: 3807: 3804: 3801: 3798: 3778: 3775: 3772: 3752: 3749: 3746: 3738: 3734: 3729: 3712: 3707: 3701: 3698: 3693: 3686: 3681: 3675: 3670: 3668: 3663: 3656: 3651: 3645: 3640: 3633: 3630: 3625: 3619: 3614: 3612: 3607: 3600: 3595: 3589: 3584: 3577: 3572: 3566: 3561: 3559: 3554: 3543:are given by 3542: 3524: 3520: 3497: 3493: 3470: 3466: 3443: 3439: 3418: 3413: 3409: 3405: 3402: 3397: 3393: 3389: 3386: 3381: 3377: 3373: 3370: 3365: 3361: 3357: 3354: 3334: 3311: 3297: 3293: 3290: 3271: 3263: 3257: 3246: 3237: 3229: 3223: 3212: 3203: 3195: 3189: 3173: 3168: 3164: 3159: 3145: 3114: 3106: 3100: 3089: 3080: 3072: 3066: 3055: 3046: 3038: 3032: 3016: 3011: 3007: 3002: 2988: 2957: 2953: 2942: 2938: 2934: 2926: 2922: 2911: 2907: 2903: 2896: 2893: 2864: 2854: 2850: 2846: 2840: 2830: 2826: 2822: 2816: 2802: 2786: 2774: 2770: 2766: 2757: 2737: 2713: 2682: 2672: 2659: 2652: 2647: 2642: 2631: 2627: 2623: 2616: 2611: 2606: 2595: 2591: 2587: 2580: 2577: 2548: 2538: 2534: 2530: 2524: 2514: 2510: 2506: 2500: 2486: 2473: 2468: 2462: 2452: 2444: 2438: 2427: 2421: 2394: 2388: 2378: 2370: 2364: 2353: 2347: 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8046: 8036: 7730:Purification 7660:Eastin–Knill 7522: 7485: 7481: 7472: 7455: 7446: 7437: 7434:Daniel Lidar 7414:. 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9269:algorithms 9109:Extensions 8943:Technology 8789:Relational 8740:Copenhagen 8651:Heisenberg 8598:Tunnelling 8461:Background 8379:libquantum 8313:Flux qubit 8218:Cavity QED 8167:Bacon–Shor 8157:stabilizer 7685:No-cloning 7416:2022-11-26 7359:2109.03533 7327:2024-07-01 7302:2024-07-01 7277:2023-04-01 7252:2023-04-01 7128:2111.12654 6989:2021-10-05 6924:2006.03071 6894:2021-08-30 6837:2001.07998 6366:2312.04522 6007:1708.05010 5953:1602.00008 5465:2010.08699 5434:References 5428:Soft error 5343:) qubits. 5257:toric code 5216:Peter Shor 5212:Eric Rains 5189:Peter Shor 5138:Peter Shor 119:projective 95:Peter Shor 60:redundancy 54:Classical 38:and other 9416:simulator 9304:computing 9274:amplifier 8815:Bell test 8685:Equations 8511:Born rule 8285:NV center 7720:Threshold 7700:No-hiding 7665:Gleason's 7392:237442177 7384:2469-9926 7210:1476-4687 7161:244527180 7099:245442996 7050:238357892 7034:0028-0836 6965:219401398 6949:1476-4687 6870:210861230 6820:Zeng, Bei 6780:1006.3941 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8000:Quantum 7901:Grover's 7675:Holevo's 7648:Theorems 7598:timeline 7588:NISQ era 7527:Archived 7322:Phys.org 7228:36813892 7153:35614250 7042:34608286 6984:phys.org 6957:33442044 6888:phys.org 6862:36654309 6805:55090423 6743:22297844 6690:32268350 6682:21617070 6631:15577904 6587:11679660 6534:11662810 6473:15744292 6160:27437573 6097:25043007 6032:51691343 5978:29518512 5920:18995200 5802:19929020 5794:24093235 5666:23787909 5599:(2000). 5417:See also 5341:transmon 5314:provided 4885:, where 4076:Cat code 2975:, where 2897:′ 2581:′ 2171:fidelity 2056:′ 2032:′ 2015:′ 1936:′ 1922:′ 1728:′ 1714:′ 1643:′ 1612:′ 1593:fidelity 772:′ 738:′ 633:′ 471:encoding 131:identity 76:decoding 72:encoding 9411:sensing 9391:network 9376:machine 9346:imaging 9294:circuit 9289:channel 9261:Quantum 9157:Related 9136:History 8875:Science 8707:Rydberg 8473:History 8337:Quantum 8275:Kane QC 8134:Quantum 8062:Quantum 7991:PostBQP 7961:Quantum 7946:Simon's 7739:Quantum 7576:General 7531:YouTube 7500:Bibcode 7364:Bibcode 7219:9946823 7188:Bibcode 7133:Bibcode 7077:Bibcode 7075:. 184. 7069:Physics 7014:Bibcode 6929:Bibcode 6842:Bibcode 6785:Bibcode 6751:2610639 6723:Bibcode 6662:Bibcode 6654:Science 6611:Bibcode 6567:Bibcode 6514:Bibcode 6481:4420858 6453:Bibcode 6400:Bibcode 6326:Bibcode 6246:Bibcode 6213:1215697 6132:Bibcode 6069:Bibcode 6012:Bibcode 5958:Bibcode 5900:Bibcode 5867:7179816 5839:Bibcode 5766:Bibcode 5710:Bibcode 5638:Bibcode 5579:9896493 5559:Bibcode 5536:9912632 5516:Bibcode 5470:Bibcode 5085:. 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