Glossary of quantum computing

This glossary of quantum computing is a list of definitions of terms and concepts used in quantum computing, its sub-disciplines, and related fields.

Bacon–Shor code

is a Subsystem error correcting code. ^[1] In a Subsystem code, information is encoded in a subsystem of a Hilbert space. Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space. ^[2] This simplicity led to the first demonstration of fault tolerant circuits on a quantum computer. ^[3]

BQP

In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. ^[4] It is the quantum analogue to the complexity class BPP . A decision problem is a member of BQP if there exists a quantum algorithm (an algorithm that runs on a quantum computer) that solves the decision problem with high probability and is guaranteed to run in polynomial time. A run of the algorithm will correctly solve the decision problem with a probability of at least 2/3.

Classical shadow

is a protocol for predicting functions of a quantum state using only a logarithmic number of measurements. ^[5] Given an unknown state ${\displaystyle \rho }$, a tomographically complete set of gates ${\displaystyle U}$ (e.g Clifford gates), a set of ${\displaystyle M}$ observables ${\displaystyle \{O_{i}\}}$ and a quantum channel ${\displaystyle M}$ (defined by randomly sampling from ${\displaystyle U}$, applying it to ${\displaystyle \rho }$ and measuring the resulting state); predict the expectation values ${\displaystyle \operatorname {tr} (O_{i}\rho )}$. ^[6] A list of classical shadows ${\displaystyle S}$ is created using ${\displaystyle \rho }$, ${\displaystyle U}$ and ${\displaystyle M}$ by running a Shadow generation algorithm. When predicting the properties of ${\displaystyle \rho }$, a Median-of-means estimation algorithm is used to deal with the outliers in ${\displaystyle S}$. ^[7] Classical shadow is useful for direct fidelity estimation, entanglement verification, estimating correlation functions, and predicting entanglement entropy. ^[5]

Cloud-based quantum computing

is the invocation of quantum emulators, simulators or processors through the cloud. Increasingly, cloud services are being looked on as the method for providing access to quantum processing. Quantum computers achieve their massive computing power by initiating quantum physics into processing power and when users are allowed access to these quantum-powered computers through the internet it is known as quantum computing within the cloud.

Cross-entropy benchmarking

(also referred to as XEB), is quantum benchmarking protocol which can be used to demonstrate quantum supremacy. ^[8] In XEB, a random quantum circuit is executed on a quantum computer multiple times in order to collect a set of ${\displaystyle k}$ samples in the form of bitstrings ${\displaystyle \{x_{1},\dots ,x_{k}\}}$. The bitstrings are then used to calculate the cross-entropy benchmark fidelity (${\displaystyle F_{\rm {XEB}}}$) via a classical computer, given by

${\displaystyle F_{\rm {XEB}}=2^{n}\langle P(x_{i})\rangle _{k}-1={\frac {2^{n}}{k}}\left(\sum _{i=1}^{k}|\langle 0^{n}|C|x_{i}\rangle |^{2}\right)-1}$,

where ${\displaystyle n}$ is the number of qubits in the circuit and ${\displaystyle P(x_{i})}$ is the probability of a bitstring ${\displaystyle {x_{i}}}$ for an ideal quantum circuit ${\displaystyle C}$. If ${\displaystyle F_{XEB}=1}$, the samples were collected from a noiseless quantum computer. If ${\displaystyle F_{\rm {XEB}}=0}$, then the samples could have been obtained via random guessing. ^[9] This means that if a quantum computer did generate those samples, then the quantum computer is too noisy and thus has no chance of performing beyond-classical computations. Since it takes an exponential amount of resources to classically simulate a quantum circuit, there comes a point when the biggest supercomputer that runs the best classical algorithm for simulating quantum circuits can't compute the XEB. Crossing this point is known as achieving quantum supremacy; and after entering the quantum supremacy regime, XEB can only be estimated. ^[10]

Eastin–Knill theorem

is a no-go theorem that states: "No quantum error correcting code can have a continuous symmetry which acts transversely on physical qubits". ^[11] In other words, no quantum error correcting code can transversely implement a universal gate set. Since quantum computers are inherently noisy, quantum error correcting codes are used to correct errors that affect information due to decoherence. Decoding error corrected data in order to perform gates on the qubits makes it prone to errors. Fault tolerant quantum computation avoids this by performing gates on encoded data. Transversal gates, which perform a gate between two "logical" qubits each of which is encoded in N "physical qubits" by pairing up the physical qubits of each encoded qubit ("code block"), and performing independent gates on each pair, can be used to perform fault tolerant but not universal quantum computation because they guarantee that errors don't spread uncontrollably through the computation. This is because transversal gates ensure that each qubit in a code block is acted on by at most a single physical gate and each code block is corrected independently when an error occurs. Due to the Eastin–Knill theorem, a universal set like { H, S , CNOT, T } gates can't be implemented transversally. For example, the T gate can't be implemented transversely in the Steane code. ^[12] This calls for ways of circumventing Eastin–Knill in order to perform fault tolerant quantum computation. In addition to investigating fault tolerant quantum computation, the Eastin–Knill theorem is also useful for studying quantum gravity via the AdS/CFT correspondence and in condensed matter physics via quantum reference frame ^[13] or many-body theory. ^[14]

Five-qubit error correcting code

is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error. ^[15] In this code, 5 physical qubits are used to encode the logical qubit. ^[16] With ${\displaystyle X}$ and ${\displaystyle Z}$ being Pauli matrices and ${\displaystyle I}$ the Identity matrix, this code's generators are ${\displaystyle \langle XZZXI,IXZZX,XIXZZ,ZXIXZ\rangle }$. Its logical operators are ${\displaystyle {\bar {X}}=XXXXX}$ and ${\displaystyle {\bar {Z}}=ZZZZZ}$. ^[17] Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors. ^[18]

Hadamard test (quantum computation)

is a method used to create a random variable whose expected value is the expected real part ${\displaystyle \mathrm {Re} \langle \psi |U|\psi \rangle }$, where ${\displaystyle |\psi \rangle }$ is a quantum state and ${\displaystyle U}$ is a unitary gate acting on the space of ${\displaystyle |\psi \rangle }$. ^[19] The Hadamard test produces a random variable whose image is in ${\displaystyle \{\pm 1\}}$ and whose expected value is exactly ${\displaystyle \mathrm {Re} \langle \psi |U|\psi \rangle }$. It is possible to modify the circuit to produce a random variable whose expected value is ${\displaystyle \mathrm {Im} \langle \psi |U|\psi \rangle }$. ^[19]

Magic state distillation

is a process that takes in multiple noisy quantum states and outputs a smaller number of more reliable quantum states. It is considered by many experts ^[20] to be one of the leading proposals for achieving fault tolerant quantum computation. Magic state distillation has also been used to argue ^[21] that quantum contextuality may be the "magic ingredient" responsible for the power of quantum computers. ^[22]

Mølmer–Sørensen gate

(or MS gate), is a two qubit gate used in trapped ion quantum computing. It was proposed by Klaus Mølmer and Anders Sørensen. ^[23] Their proposal also extends to gates on more than two qubits.

Quantum algorithm

is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. ^{[24] [25]} A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, ^{[26] : 126} the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement.

Quantum computing

is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. ^{[27] [28]} Though current quantum computers are too small to outperform usual (classical) computers for practical applications, larger realizations are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science.

Quantum volume

is a metric that measures the capabilities and error rates of a quantum computer. It expresses the maximum size of square quantum circuits that can be implemented successfully by the computer. The form of the circuits is independent from the quantum computer architecture, but compiler can transform and optimize it to take advantage of the computer's features. Thus, quantum volumes for different architectures can be compared.

Quantum error correction

(QEC), is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault-tolerant quantum computation that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements.

Quantum image processing

(QIMP), is using quantum computing or quantum information processing to create and work with quantum images. ^{[29] [30]} Due to some of the properties inherent to quantum computation, notably entanglement and parallelism, it is hoped that QIMP technologies will offer capabilities and performances that surpass their traditional equivalents, in terms of computing speed, security, and minimum storage requirements. ^{[30] [31]}

Quantum programming

is the process of assembling sequences of instructions, called quantum programs, that are capable of running on a quantum computer. Quantum programming languages help express quantum algorithms using high-level constructs. ^[32] The field is deeply rooted in the open-source philosophy and as a result most of the quantum software discussed in this article is freely available as open-source software. ^[33]

Quantum simulator

Quantum simulators permit the study of quantum system in a programmable fashion. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. ^{[34] [35] [36]} Quantum simulators may be contrasted with generally programmable "digital" quantum computers, which would be capable of solving a wider class of quantum problems.

Quantum state discrimination

In quantum information science, quantum state discrimination refers to the task of inferring the quantum state that produced the observed measurement probabilities. More precisely, in its standard formulation, the problem involves performing some POVM ${\displaystyle (E_{i})_{i}}$ on a given unknown state ${\displaystyle \rho }$, under the promise that the state received is an element of a collection of states ${\displaystyle \{\sigma _{i}\}_{i}}$, with ${\displaystyle \sigma _{i}}$ occurring with probability ${\displaystyle p_{i}}$, that is, ${\displaystyle \rho =\sum _{i}p_{i}\sigma _{i}}$. The task is then to find the probability of the POVM ${\displaystyle (E_{i})_{i}}$ correctly guessing which state was received. Since the probability of the POVM returning the ${\displaystyle i}$-th outcome when the given state was ${\displaystyle \sigma _{j}}$ has the form ${\displaystyle {\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})}$, it follows that the probability of successfully determining the correct state is ${\displaystyle P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})}$. ^[37]

Quantum supremacy

or quantum advantage, is the goal of demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time (irrespective of the usefulness of the problem). ^{[38] [39] [40]} Conceptually, quantum supremacy involves both the engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum computer and has a superpolynomial speedup over the best known or possible classical algorithm for that task. ^{[41] [42]} The term was coined by John Preskill in 2012, ^{[43] [44]} but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980) ^[45] and Richard Feynman's (1981) proposals of quantum computing. ^[46] Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov, ^[47] D-Wave's specialized frustrated cluster loop problems, ^[48] and sampling the output of random quantum circuits. ^{[49] [50]}

Quantum Turing machine

(QTM), or universal quantum computer, is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algorithm can be expressed formally as a particular quantum Turing machine. However, the computationally equivalent quantum circuit is a more common model. ^{[51] [52] : 2}

Qubit

A qubit ( /ˈkjuːbɪt/ ) or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.

Quil (instruction set architecture)

is a quantum instruction set architecture that first introduced a shared quantum/classical memory model. It was introduced by Robert Smith, Michael Curtis, and William Zeng in A Practical Quantum Instruction Set Architecture. ^[43] Many quantum algorithms (including quantum teleportation, quantum error correction, simulation, ^{[53] [54]} and optimization algorithms ^[55] ) require a shared memory architecture. Quil is being developed for the superconducting quantum processors developed by Rigetti Computing through the Forest quantum programming API. ^{[56] [57]} A Python library called pyQuil was introduced to develop Quil programs with higher level constructs. A Quil backend is also supported by other quantum programming environments. ^{[58] [59]}

Qutrit

(or quantum trit), is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states. ^[60] The qutrit is analogous to the classical radix-3 trit, just as the qubit, a quantum system described by a superposition of two orthogonal states, is analogous to the classical radix-2 bit. There is ongoing work to develop quantum computers using qutrits and qubits with multiple states. ^[61]

Solovay–Kitaev theorem

In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2) then that set is guaranteed to fill SU(2) quickly, which means any desired gate can be approximated by a fairly short sequence of gates from the generating set. Robert M. Solovay initially announced the result on an email list in 1995, and Alexei Kitaev independently gave an outline of its proof in 1997. ^[62] Solovay also gave a talk on his result at MSRI in 2000 but it was interrupted by a fire alarm. ^[63] Christopher M. Dawson and Michael Nielsen call the theorem one of the most important fundamental results in the field of quantum computation. ^[64]

References

1. Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A. 73 (1): 012340. arXiv: quant-ph/0506023 . Bibcode:2006PhRvA..73a2340B. doi:10.1103/PhysRevA.73.012340. S2CID   118968017.
2. Aly Salah A., Klappenecker, Andreas (2008). "Subsystem code constructions". 2008 IEEE International Symposium on Information Theory. pp. 369–373. arXiv: 0712.4321 . doi:10.1109/ISIT.2008.4595010. ISBN   978-1-4244-2256-2. S2CID   14063318.
3. Egan, L., Debroy, D.M., Noel, C. (2021). "Fault-tolerant control of an error-corrected qubit". Phys. Rev. Lett. 598 (7880). Nature: 281–286. arXiv: 2009.11482 . Bibcode:2021Natur.598..281E. doi:10.1038/s41586-021-03928-y. PMID   34608286. S2CID   238357892.
4. Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN   0-521-63503-9.
5. Huang, Hsin-Yuan; Kueng, Richard; Preskill, John (2020). "Predicting many properties of a quantum system from very few measurements". Nat. Phys. 16 (10): 1050–1057. arXiv: 2002.08953 . Bibcode:2020NatPh..16.1050H. doi:10.1038/s41567-020-0932-7. S2CID   211205098.
6. Koh, D. E.; Grewal, Sabee (2022). "Classical Shadows with Noise". Quantum. 6: 776. arXiv: 2011.11580 . Bibcode:2022Quant...6..776K. doi:10.22331/q-2022-08-16-776. S2CID   227127118.
7. Struchalin, G.I.; Zagorovskii, Ya. A.; Kovlakov, E.V.; Straupe, S.S.; Kulik, S.P. (2021). "Experimental Estimation of Quantum State Properties from Classical Shadows". PRX Quantum. 2 (1): 010307. arXiv: 2008.05234 . doi:10.1103/PRXQuantum.2.010307. S2CID   221103573.
8. Boixo, S.; et al. (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics. 14 (6): 595–600. arXiv: 1608.00263 . Bibcode:2018NatPh..14..595B. doi:10.1038/s41567-018-0124-x. S2CID   4167494.
9. Aaronson, S. (2021). "Open Problems Related to Quantum Query Complexity". arXiv: 2109.06917 [quant-ph].
10. Arute, F.; et al. (2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv: 1910.11333 . Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. PMID   31645734. S2CID   204836822.
11. Eastin, Bryan; Knill, Emanuel (2009). "Restrictions on Transversal Encoded Quantum Gate Sets". Physical Review Letters. 102 (11): 110502. arXiv: 0811.4262 . Bibcode:2009PhRvL.102k0502E. doi:10.1103/PhysRevLett.102.110502. PMID   19392181. S2CID   44457708.
12. Campbell, Earl T.; Terhal, Barbara M.; Vuillot, Christophe (2016). "Roads towards fault-tolerant universal quantum computation". Nature . 549 (7671): 172–179. arXiv: quant-ph/0403025 . Bibcode:2017Natur.549..172C. doi:10.1038/nature23460. PMID   28905902. S2CID   4446310.
13. Woods, Mischa; Alhambra, Alvaro M. (2020). "Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames". Quantum. 4: 245. arXiv: 1902.07725 . Bibcode:2020Quant...4..245W. doi:10.22331/q-2020-03-23-245. S2CID   119302752.
14. Faist, Philippe; Nezami, Sepehr; V. Albert, Victor; Salton, Grant; Pastawski, Fernando; Hayden, Patrick; Preskill, John (2020). "Continuous Symmetries and Approximate Quantum Error Correction". Physical Review X. 10 (4): 041018. arXiv: 1902.07714 . Bibcode:2020PhRvX..10d1018F. doi:10.1103/PhysRevX.10.041018. S2CID   119207861.
15. Gottesman, Daniel (2009). "An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation". arXiv: 0904.2557 [quant-ph].
16. Knill, E. and Laflamme, R. and Martinez, R. and Negrevergne, C. (2001). "Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code". Phys. Rev. Lett. 86 (25). American Physical Society: 5811–5814. arXiv: quant-ph/0101034 . Bibcode:2001PhRvL..86.5811K. doi:10.1103/PhysRevLett.86.5811. PMID   11415364. S2CID   119440555.
17. D. Gottesman (1997). "Stabilizer Codes and Quantum Error Correction". arXiv: quant-ph/9705052 .
18. Roffe Joschka (2019). "Quantum error correction: an introductory guide". Contemporary Physics. 60 (3). Taylor & Francis: 226–245. arXiv: 1907.11157 . Bibcode:2019ConPh..60..226R. doi:10.1080/00107514.2019.1667078. S2CID   198893630.
19. Dorit Aharonov Vaughan Jones, Zeph Landau (2009). "A Polynomial Quantum Algorithm for Approximating the Jones Polynomial". Algorithmica. 55 (3): 395–421. arXiv: quant-ph/0511096 . doi:10.1007/s00453-008-9168-0. S2CID   7058660.
20. Campbell, Earl T.; Terhal, Barbara M.; Vuillot, Christophe (14 September 2017). "Roads towards fault-tolerant universal quantum computation" (PDF). Nature. 549 (7671): 172–179. arXiv: 1612.07330 . Bibcode:2017Natur.549..172C. doi:10.1038/nature23460. PMID   28905902. S2CID   4446310.
21. Howard, Mark; Wallman, Joel; Veitch, Victor; Emerson, Joseph (11 June 2014). "Contextuality supplies the 'magic' for quantum computation". Nature. 510 (7505): 351–355. arXiv: 1401.4174 . Bibcode:2014Natur.510..351H. doi:10.1038/nature13460. PMID   24919152. S2CID   4463585.
22. Bartlett, Stephen D. (11 June 2014). "Powered by magic". Nature. 510 (7505): 345–347. doi: 10.1038/nature13504 . PMID   24919151.
23. Sørensen, Anders; Mølmer, Klaus (March 1, 1999). "Multi-particle entanglement of hot trapped ions". Physical Review Letters. 82 (9): 1835–1838. arXiv:quant-ph/9810040. Bibcode:1999PhRvL..82.1835M. doi:10.1103/PhysRevLett.82.1835. S2CID 49333990.
24. Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. ISBN   978-0-521-63503-5.
25. Mosca, M. (2008). "Quantum Algorithms". arXiv: 0808.0369 [quant-ph].
26. Lanzagorta, Marco; Uhlmann, Jeffrey K. (2009-01-01). Quantum Computer Science. Morgan & Claypool Publishers. ISBN   9781598297324.
27. Hidary, Jack (2019). Quantum computing : an applied approach. Cham: Springer. p. 3. ISBN   978-3-030-23922-0. OCLC   1117464128.
28. Nielsen & Chuang 2010, p. 1.
29. Venegas-Andraca, Salvador E. (2005). Discrete Quantum Walks and Quantum Image Processing (DPhil thesis). The University of Oxford.
30. Iliyasu, A.M. (2013). "Towards realising secure and efficient image and video processing applications on quantum computers". Entropy. 15 (8): 2874–2974. Bibcode:2013Entrp..15.2874I. doi: 10.3390/e15082874 .
31. Yan, F.; Iliyasu, A.M.; Le, P.Q. (2017). "Quantum image processing: A review of advances in its security technologies". International Journal of Quantum Information. 15 (3): 1730001–44. Bibcode:2017IJQI...1530001Y. doi: 10.1142/S0219749917300017 .
32. Jarosław Adam Miszczak (2012). High-level Structures in Quantum Computing. Morgan & Claypool Publishers. ISBN   9781608458516.
33. "Comprehensive list of quantum open-source projects". Github. Retrieved 2022-01-27.
34. Johnson, Tomi H.; Clark, Stephen R.; Jaksch, Dieter (2014). "What is a quantum simulator?". EPJ Quantum Technology. 1 (10). arXiv: 1405.2831 . doi:10.1140/epjqt10. S2CID   120250321.
35.  This article incorporates public domain material from Michael E. Newman. NIST Physicists Benchmark Quantum Simulator with Hundreds of Qubits. National Institute of Standards and Technology . Retrieved 2013-02-22.
36. Britton, Joseph W.; Sawyer, Brian C.; Keith, Adam C.; Wang, C.-C. Joseph; Freericks, James K.; Uys, Hermann; Biercuk, Michael J.; Bollinger, John J. (2012). "Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins" (PDF). Nature. 484 (7395): 489–92. arXiv: 1204.5789 . Bibcode:2012Natur.484..489B. doi:10.1038/nature10981. PMID   22538611. S2CID   4370334. Note: This manuscript is a contribution of the US National Institute of Standards and Technology and is not subject to US copyright.
37. Bae, Joonwoo; Kwek, Leong-Chuan (2015). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv: 1707.02571 . Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. S2CID   119199057.
38. Preskill, John (2012-03-26). "Quantum computing and the entanglement frontier". arXiv: 1203.5813 [quant-ph].
39. Preskill, John (2018-08-06). "Quantum Computing in the NISQ era and beyond". Quantum. 2: 79. arXiv: 1801.00862 . Bibcode:2018Quant...2...79P. doi: 10.22331/q-2018-08-06-79 .
40. Zhong, Han-Sen; Wang, Hui; Deng, Yu-Hao; Chen, Ming-Cheng; Peng, Li-Chao; Luo, Yi-Han; Qin, Jian; Wu, Dian; Ding, Xing; Hu, Yi; Hu, Peng (2020-12-03). "Quantum computational advantage using photons". Science. 370 (6523): 1460–1463. arXiv:2012.01625. Bibcode:2020Sci...370.1460Z. doi:10.1126/science.abe8770. ISSN 0036-8075. PMID 33273064. S2CID 227254333.
41. Harrow, Aram W.; Montanaro, Ashley (September 2017). "Quantum computational supremacy". Nature. 549 (7671): 203–209. arXiv: 1809.07442 . Bibcode:2017Natur.549..203H. doi:10.1038/nature23458. ISSN   1476-4687. PMID   28905912. S2CID   2514901.
42. Papageorgiou, Anargyros; Traub, Joseph F. (2013-08-12). "Measures of quantum computing speedup". Physical Review A. 88 (2): 022316. arXiv: 1307.7488 . Bibcode:2013PhRvA..88b2316P. doi:10.1103/PhysRevA.88.022316. ISSN   1050-2947. S2CID   41867048.
43. Smith, Robert S.; Curtis, Michael J.; Zeng, William J. (2016-08-10). "A Practical Quantum Instruction Set Architecture". arXiv: 1608.03355 [quant-ph].
44. "John Preskill Explains 'Quantum Supremacy'". Quanta Magazine. 2 October 2019. Retrieved 2020-04-21.
45. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 2013-05-10. Retrieved 2013-03-04.
46. Feynman, Richard P. (1982-06-01). "Simulating Physics with Computers". International Journal of Theoretical Physics. 21 (6–7): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX   10.1.1.45.9310 . doi:10.1007/BF02650179. ISSN   0020-7748. S2CID   124545445.
47. Aaronson, Scott; Arkhipov, Alex (2011). "The computational complexity of linear optics". Proceedings of the forty-third annual ACM symposium on Theory of computing. STOC '11. New York, NY, USA: ACM. pp. 333–342. arXiv: 1011.3245 . doi:10.1145/1993636.1993682. ISBN   9781450306911. S2CID   681637.
48. King, James; Yarkoni, Sheir; Raymond, Jack; Ozfidan, Isil; King, Andrew D.; Nevisi, Mayssam Mohammadi; Hilton, Jeremy P.; McGeoch, Catherine C. (2017-01-17). "Quantum Annealing amid Local Ruggedness and Global Frustration". arXiv: 1701.04579 [quant-ph].
49. Aaronson, Scott; Chen, Lijie (2016-12-18). "Complexity-Theoretic Foundations of Quantum Supremacy Experiments". arXiv: 1612.05903 [quant-ph].
50. Bouland, Adam; Fefferman, Bill; Nirkhe, Chinmay; Vazirani, Umesh (2018-10-29). "On the complexity and verification of quantum random circuit sampling". Nature Physics. 15 (2): 159–163. arXiv: 1803.04402 . doi:10.1038/s41567-018-0318-2. ISSN   1745-2473. S2CID   125264133.
51. Andrew Yao (1993). Quantum circuit complexity. 34th Annual Symposium on Foundations of Computer Science. pp. 352–361.
52. Abel Molina; John Watrous (2018). "Revisiting the simulation of quantum Turing machines by quantum circuits". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 475 (2226). arXiv: 1808.01701 . doi:10.1098/rspa.2018.0767. PMC   6598068 . PMID   31293355.
53. McClean, Jarrod R.; Romero, Jonathan; Babbush, Ryan; Aspuru-Guzik, Alán (2016-02-04). "The theory of variational hybrid quantum-classical algorithms". New Journal of Physics. 18 (2): 023023. arXiv: 1509.04279 . Bibcode:2016NJPh...18b3023M. doi:10.1088/1367-2630/18/2/023023. ISSN   1367-2630. S2CID   92988541.
54. Rubin, Nicholas C. (2016-10-21). "A Hybrid Classical/Quantum Approach for Large-Scale Studies of Quantum Systems with Density Matrix Embedding Theory". arXiv: 1610.06910 [quant-ph].
55. Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam (2014-11-14). "A Quantum Approximate Optimization Algorithm". arXiv: 1411.4028 [quant-ph].
56. "Rigetti Launches Full-Stack Quantum Computing Service and Quantum IC Fab". IEEE Spectrum: Technology, Engineering, and Science News. 26 June 2017. Retrieved 2017-07-06.
57. "Rigetti Quietly Releases Beta of Forest Platform for Quantum Programming in the Cloud | Quantum Computing Report". quantumcomputingreport.com. 8 March 2017. Retrieved 2017-07-06.
58. "XACC Rigetti Accelerator". ornl-qci.github.io. Archived from the original on 2017-12-01. Retrieved 2017-07-06.
59. Doiron, Nick (2017-03-07), jsquil: Quantum computer instructions for JavaScript developers , retrieved 2017-07-06
60. Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013). "Photonic qubits, qutrits and ququads accurately prepared and delivered on demand". New Journal of Physics. 15 (5): 053007. arXiv: 1203.5614 . Bibcode:2013NJPh...15e3007N. doi:10.1088/1367-2630/15/5/053007. ISSN   1367-2630. S2CID   110606655.
61. "Qudits: The Real Future of Quantum Computing?". IEEE Spectrum. 28 June 2017. Retrieved 2021-05-24.
62. Kitaev, A Yu (1997-12-31). "Quantum computations: algorithms and error correction". Russian Mathematical Surveys. 52 (6): 1191–1249. Bibcode:1997RuMaS..52.1191K. doi:10.1070/rm1997v052n06abeh002155. ISSN   0036-0279. S2CID   250816585.
63. Solovay, Robert (2000-02-08). Lie Groups and Quantum Circuits. MSRI.
64. Dawson, Christopher M.; Nielsen, Michael (2006-01-01). "The Solovay-Kitaev algorithm". Quantum Information & Computation. 6: 81–95. arXiv: quant-ph/0505030 . doi:10.26421/QIC6.1-6.

Further reading

Textbooks

• Aaronson, Scott (2013). Quantum Computing Since Democritus. Cambridge University Press. doi:10.1017/CBO9780511979309. ISBN   978-0-521-19956-8. OCLC   829706638.
• Akama, Seiki (2014). Elements of Quantum Computing: History, Theories and Engineering Applications. Springer. doi:10.1007/978-3-319-08284-4. ISBN   978-3-319-08284-4. OCLC   884786739.
• Benenti, Giuliano; Casati, Giulio; Rossini, Davide; Strini, Giuliano (2019). Principles of Quantum Computation and Information: A Comprehensive Textbook (2nd ed.). doi:10.1142/10909. ISBN   978-981-3237-23-0. OCLC   1084428655. S2CID   62280636.
• Bernhardt, Chris (2019). Quantum Computing for Everyone. MIT Press. ISBN   978-0-262-35091-4. OCLC   1082867954.
• Hidary, Jack D. (2021). Quantum Computing: An Applied Approach (2nd ed.). doi:10.1007/978-3-030-83274-2. ISBN   978-3-03-083274-2. OCLC   1272953643. S2CID   238223274.
• Hiroshi, Imai; Masahito, Hayashi, eds. (2006). Quantum Computation and Information: From Theory to Experiment. Topics in Applied Physics. Vol. 102. doi:10.1007/3-540-33133-6. ISBN   978-3-540-33133-9.
• Hughes, Ciaran; Isaacson, Joshua; Perry, Anastasia; Sun, Ranbel F.; Turner, Jessica (2021). Quantum Computing for the Quantum Curious (PDF). doi:10.1007/978-3-030-61601-4. ISBN   978-3-03-061601-4. OCLC   1244536372. S2CID   242566636.
• Jaeger, Gregg (2007). Quantum Information: An Overview. doi:10.1007/978-0-387-36944-0. ISBN   978-0-387-36944-0. OCLC   186509710.
• Johnston, Eric R.; Harrigan, Nic; Gimeno-Segovia, Mercedes (2019). Programming Quantum Computers: Essential Algorithms and Code Samples. O'Reilly Media, Incorporated. ISBN   978-1-4920-3968-6. OCLC   1111634190.
• Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007). An Introduction to Quantum Computing. OUP Oxford. ISBN   978-0-19-857000-4. OCLC   85896383.
• Kitaev, Alexei Yu.; Shen, Alexander H.; Vyalyi, Mikhail N. (2002). Classical and Quantum Computation. American Mathematical Soc. ISBN   978-0-8218-3229-5. OCLC   907358694.
• Mermin, N. David (2007). Quantum Computer Science: An Introduction. doi:10.1017/CBO9780511813870. ISBN   978-0-511-34258-5. OCLC   422727925.
• National Academies of Sciences, Engineering, and Medicine (2019). Grumbling, Emily; Horowitz, Mark (eds.). Quantum Computing : Progress and Prospects. Washington, DC. doi:10.17226/25196. ISBN   978-0-309-47970-7. OCLC   1091904777. S2CID   125635007.
• Nielsen, Michael; Chuang, Isaac (2010). Quantum Computation and Quantum Information (10th anniversary ed.). doi:10.1017/CBO9780511976667. ISBN   978-0-511-99277-3. OCLC   700706156. S2CID   59717455.
• Stolze, Joachim; Suter, Dieter (2004). Quantum Computing: A Short Course from Theory to Experiment. doi:10.1002/9783527617760. ISBN   978-3-527-61776-0. OCLC   212140089.
• Wichert, Andreas (2020). Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning (2nd ed.). doi:10.1142/11938. ISBN   978-981-12-2431-7. OCLC   1178715016. S2CID   225498497.
• Wong, Thomas (2022). Introduction to Classical and Quantum Computing (PDF). Rooted Grove. ISBN   979-8-9855931-0-5. OCLC   1308951401. Archived from the original (PDF) on 2022-01-29. Retrieved 2022-08-29.
• Zeng, Bei; Chen, Xie; Zhou, Duan-Lu; Wen, Xiao-Gang (2019). Quantum Information Meets Quantum Matter. arXiv: 1508.02595 . doi:10.1007/978-1-4939-9084-9. ISBN   978-1-4939-9084-9. OCLC   1091358969. S2CID   118528258.

Academic papers

• Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; Hamilton, Alexander R.; Ferry, David K.; Gea-Banacloche, Julio; Bezrukov, Sergey M.; Kish, Laszlo B. (2003). "Dreams versus Reality: Plenary Debate Session on Quantum Computing". Quantum Information Processing. 2 (6): 449–472. arXiv: quant-ph/0310130 . doi:10.1023/B:QINP.0000042203.24782.9a. hdl:2027.42/45526. S2CID   34885835.
• Berthiaume, Andre (1997). "Quantum Computation".
• DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. arXiv: quant-ph/0002077 . Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. S2CID   15439711.
• DiVincenzo, David P. (1995). "Quantum Computation". Science. 270 (5234): 255–261. Bibcode:1995Sci...270..255D. CiteSeerX   10.1.1.242.2165 . doi:10.1126/science.270.5234.255. S2CID   220110562. Table 1 lists switching and dephasing times for various systems.
• Feynman, Richard (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6–7): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX   10.1.1.45.9310 . doi:10.1007/BF02650179. S2CID   124545445.
• Jeutner, Valentin (2021). "The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers". Morals & Machines. 1 (1): 52–59. doi: 10.5771/2747-5174-2021-1-52 . S2CID   236664155.
• Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond".
• Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronics Engineers Computer Society Press.