Glossary of quantum computing

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

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{{glossary}}

{{term|Bacon–Shor code}}

{{defn|is a Subsystem error correcting code.{{Cite journal|last=Bacon|first=Dave|date=2006-01-30|title=Operator quantum error-correcting subsystems for self-correcting quantum memories|journal=Physical Review A|volume=73|issue=1|pages=012340|doi=10.1103/PhysRevA.73.012340|arxiv=quant-ph/0506023|bibcode=2006PhRvA..73a2340B|s2cid=118968017}} 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.

{{Cite book

|author = Aly Salah A., Klappenecker, Andreas

|title = 2008 IEEE International Symposium on Information Theory

|chapter = Subsystem code constructions

|arxiv = 0712.4321

|year = 2008

|pages = 369–373

|doi = 10.1109/ISIT.2008.4595010

|isbn = 978-1-4244-2256-2

|s2cid = 14063318

}} This simplicity led to the first demonstration of fault tolerant circuits on a quantum computer.

{{Cite journal

|title = Fault-tolerant control of an error-corrected qubit.

|author = Egan, L., Debroy, D.M., Noel, C.

|journal = Phys. Rev. Lett.

|arxiv=2009.11482

|issue = 7880

|pages = 281–286

|year = 2021

|volume = 598

|publisher = Nature

|doi = 10.1038/s41586-021-03928-y

|pmid = 34608286

|bibcode = 2021Natur.598..281E

|s2cid = 238357892

}} }}

{{term|BQP}}

{{defn|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.Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. {{isbn|0-521-63503-9}}. 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.}}

{{term|Classical shadow}}

{{defn|is a protocol for predicting functions of a quantum state using only a logarithmic number of measurements.{{Cite journal | arxiv=2002.08953|last1=Huang |first1=Hsin-Yuan |last2=Kueng |first2=Richard |last3=Preskill|first3=John |title =Predicting many properties of a quantum system from very few measurements|year=2020 |journal=Nat. Phys.|volume=16 | issue=10 |pages=1050–1057|doi=10.1038/s41567-020-0932-7|bibcode=2020NatPh..16.1050H |s2cid=211205098 }} Given an unknown state $\backslash rho$, a tomographically complete set of gates $U$ (e.g Clifford gates), a set of $M$ observables $\backslash \{O\_\{i\}\backslash \}$ and a quantum channel $M$ (defined by randomly sampling from $U$, applying it to $\backslash rho$ and measuring the resulting state); predict the expectation values $\backslash operatorname\{tr\}(O\_\{i\}\; \backslash rho)$.{{cite journal | arxiv=2011.11580|last1= Koh |first1=D. E. |last2=Grewal|first2= Sabee |title=Classical Shadows with Noise|journal= Quantum |year=2022|volume= 6 |page= 776 |doi= 10.22331/q-2022-08-16-776 |bibcode= 2022Quant...6..776K |s2cid= 227127118 }} A list of classical shadows $S$ is created using $\backslash rho$, $U$ and $M$ by running a Shadow generation algorithm. When predicting the properties of $\backslash rho$, a Median-of-means estimation algorithm is used to deal with the outliers in $S$.{{Cite journal | arxiv=2008.05234|last1=Struchalin |first1=G.I. |last2=Zagorovskii |first2=Ya. A. |last3=Kovlakov |first3=E.V. |last4=Straupe | first4=S.S. |last5=Kulik |first5= S.P.|title=Experimental Estimation of Quantum State Properties from Classical Shadows|year=2021 |journal=PRX Quantum|volume=2 | issue=1 |page=010307 |doi=10.1103/PRXQuantum.2.010307|s2cid=221103573 }} Classical shadow is useful for direct fidelity estimation, entanglement verification, estimating correlation functions, and predicting entanglement entropy.}}

{{term|Cloud-based quantum computing}}

{{defn|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.}}

{{term|Cross-entropy benchmarking}}

{{defn|(also referred to as XEB), is quantum benchmarking protocol which can be used to demonstrate quantum supremacy.{{Cite journal

|last1=Boixo |first1=S.

|display-authors=etal

|year = 2018

|title = Characterizing Quantum Supremacy in Near-Term Devices

|journal = Nature Physics

|volume = 14

|issue = 6

|pages = 595–600

|arxiv=1608.00263

|bibcode = 2018NatPh..14..595B

|doi = 10.1038/s41567-018-0124-x

|s2cid = 4167494

}} In XEB, a random quantum circuit is executed on a quantum computer multiple times in order to collect a set of $k$ samples in the form of bitstrings $\backslash \{x\_\{1\},\; \backslash dots,\; x\_\{k\}\backslash \}$. The bitstrings are then used to calculate the cross-entropy benchmark fidelity ($F\_\{\backslash rm\; XEB\}$) via a classical computer, given by

:$F\_\{\backslash rm\; XEB\}=\; 2^\{n\}\; \backslash langle\; P(x\_\{i\})\; \backslash rangle\_\{k\}\; -\; 1\; =\; \backslash frac\{2^\{n\}\}\{k\}\; \backslash left(\backslash sum\_\{i=1\}^\{k\}|\backslash langle\; 0^\{n\}|C|x\_\{i\}\backslash rangle|^\{2\}\backslash right)\; -\; 1$,

where $n$ is the number of qubits in the circuit and $P(x\_\{i\})$ is the probability of a bitstring $\{x\_\{i\}\}$ for an ideal quantum circuit $C$. If $F\_\{XEB\}\; =\; 1$, the samples were collected from a noiseless quantum computer. If $F\_\{\backslash rm\; XEB\}\; =\; 0$, then the samples could have been obtained via random guessing.

{{Cite arXiv

|last1=Aaronson |first1=S.

|title = Open Problems Related to Quantum Query Complexity

|eprint=2109.06917

|year = 2021

|class=quant-ph

}} 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.

{{Cite journal

|last1=Arute |first1=F.

|display-authors=etal

|year = 2019

|title = Quantum supremacy using a programmable superconducting processor

|journal = Nature

|volume = 574

|issue = 7779

|pages = 505–510

|arxiv = 1910.11333

|bibcode = 2019Natur.574..505A

|doi = 10.1038/s41586-019-1666-5

|pmid = 31645734

|s2cid = 204836822

}}}}

{{term|Eastin–Knill theorem}}

{{defn|is a no-go theorem that states: "No quantum error correcting code can have a continuous symmetry which acts transversely on physical qubits".{{Cite journal | arxiv = 0811.4262 |last1= Eastin | first1= Bryan |last2= Knill|first2= Emanuel |title = Restrictions on Transversal Encoded Quantum Gate Sets|journal = Physical Review Letters|volume = 102|issue = 11|pages = 110502|year = 2009|doi = 10.1103/PhysRevLett.102.110502|pmid = 19392181|bibcode = 2009PhRvL.102k0502E|s2cid= 44457708 }} 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 {{math|{H, S, CNOT, T }}} gates can't be implemented transversally. For example, the T gate can't be implemented transversely in the Steane code.{{Cite journal | arxiv = quant-ph/0403025 |last1=Campbell |first1=Earl T. |last2=Terhal | first2=Barbara M. |last3=Vuillot |first3= Christophe |title = Roads towards fault-tolerant universal quantum computation|year = 2016 |journal=Nature |volume=549 |issue=7671 |pages=172–179 |doi=10.1038/nature23460|pmid=28905902 |bibcode=2017Natur.549..172C |s2cid=4446310 }} 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{{Cite journal | arxiv = 1902.07725 |last1= Woods|first1=Mischa |last2= Alhambra | first2= Alvaro M.|title = Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames|year = 2020 |journal=Quantum |volume=4 |pages=245 |doi= 10.22331/q-2020-03-23-245|bibcode= 2020Quant...4..245W|s2cid= 119302752}} or many-body theory.{{Cite journal | arxiv = 1902.07714 |last1 = Faist |first1 = Philippe |last2 = Nezami | first2 = Sepehr | last3 = V. Albert | first3 = Victor |last4 = Salton | first4= Grant |last5 = Pastawski | first5= Fernando |last6 = Hayden | first6= Patrick |last7 = Preskill | first7= John |title = Continuous Symmetries and Approximate Quantum Error Correction|journal = Physical Review X |year = 2020|volume = 10 |issue = 4 |page = 041018 |doi = 10.1103/PhysRevX.10.041018 |bibcode = 2020PhRvX..10d1018F |s2cid = 119207861 }}}}

{{term|Five-qubit error correcting code}}

{{defn|is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error.{{cite arXiv | eprint=0904.2557|last1=Gottesman |first1=Daniel|title =An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation|year=2009|class=quant-ph }} In this code, 5 physical qubits are used to encode the logical qubit.

{{Cite journal

|title = Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code

|author = Knill, E. and Laflamme, R. and Martinez, R. and Negrevergne, C.

|journal = Phys. Rev. Lett.

|arxiv=quant-ph/0101034

|volume = 86

|issue = 25

|pages = 5811–5814

|year = 2001

|publisher = American Physical Society

|doi = 10.1103/PhysRevLett.86.5811

|pmid = 11415364

|bibcode = 2001PhRvL..86.5811K

|s2cid = 119440555

|url = https://link.aps.org/doi/10.1103/PhysRevLett.86.5811

}} With $X$ and $Z$ being Pauli matrices and $I$ the Identity matrix, this code's generators are $\backslash langle\; XZZXI,\; IXZZX,\; XIXZZ,ZXIXZ\; \backslash rangle$. Its logical operators are $\backslash bar\{X\}\; =\; XXXXX$ and $\backslash bar\{Z\}\; =\; ZZZZZ$.{{cite arXiv

|title=Stabilizer Codes and Quantum Error Correction

|author=D. Gottesman

|arxiv=quant-ph/9705052

|year=1997

}} 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.

{{Cite journal

|author = Roffe Joschka

|title = Quantum error correction: an introductory guide

|arxiv=1907.11157

|journal = Contemporary Physics

|volume = 60

|number = 3

|pages = 226–245

|year= 2019

|publisher=Taylor & Francis

|doi = 10.1080/00107514.2019.1667078

|bibcode = 2019ConPh..60..226R

|s2cid = 198893630

|url=https://doi.org/10.1080/00107514.2019.1667078

}}}}

{{term|Hadamard test (quantum computation)}}

{{defn| is a method used to create a random variable whose expected value is the expected real part $\backslash mathrm\{Re\}\backslash langle\backslash psi|\; U|\backslash psi\backslash rangle$, where $|\backslash psi\backslash rangle$ is a quantum state and $U$ is a unitary gate acting on the space of $|\backslash psi\backslash rangle$.

{{cite journal

| author = Dorit Aharonov Vaughan Jones, Zeph Landau

| title = A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

| journal = Algorithmica

| volume = 55

| issue = 3

| pages = 395–421

| year = 2009

| doi = 10.1007/s00453-008-9168-0

| arxiv = quant-ph/0511096

| s2cid = 7058660

}}

The Hadamard test produces a random variable whose image is in $\backslash \{\backslash pm\; 1\backslash \}$ and whose expected value is exactly $\backslash mathrm\{Re\}\backslash langle\backslash psi|\; U|\backslash psi\backslash rangle$. It is possible to modify the circuit to produce a random variable whose expected value is $\backslash mathrm\{Im\}\backslash langle\backslash psi|\; U|\backslash psi\backslash rangle$.}}

{{term|Magic state distillation}}

{{defn|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{{cite journal |last1=Campbell |first1=Earl T. |last2=Terhal |first2=Barbara M. |last3=Vuillot |first3=Christophe |title=Roads towards fault-tolerant universal quantum computation |journal=Nature |date=14 September 2017 |volume=549 |issue=7671 |pages=172–179 |doi=10.1038/nature23460|pmid=28905902 |arxiv=1612.07330 |bibcode=2017Natur.549..172C |s2cid=4446310 |url=http://eprints.whiterose.ac.uk/121343/1/1612.07330.pdf }} to be one of the leading proposals for achieving fault tolerant quantum computation. Magic state distillation has also been used to argue {{cite journal |last1=Howard |first1=Mark |last2=Wallman |first2=Joel |last3=Veitch |first3=Victor |last4=Emerson |first4=Joseph |title=Contextuality supplies the 'magic' for quantum computation |journal=Nature |date=11 June 2014 |volume=510 |issue=7505 |pages=351–355 |doi=10.1038/nature13460|pmid=24919152 |arxiv=1401.4174 |bibcode=2014Natur.510..351H |s2cid=4463585 }} that quantum contextuality may be the "magic ingredient" responsible for the power of quantum computers.{{cite journal |last1=Bartlett |first1=Stephen D. |title=Powered by magic |journal=Nature |date=11 June 2014 |volume=510 |issue=7505 |pages=345–347 |doi=10.1038/nature13504|pmid=24919151 |doi-access=free }} }}

{{term|Mølmer–Sørensen gate}}

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

{{term|Quantum algorithm}}

{{defn|is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation.{{cite book|title=Quantum Computation and Quantum Information|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|publisher=Cambridge University Press|year=2000|isbn=978-0-521-63503-5|author-link=Michael Nielsen|author-link2=Isaac Chuang|title-link=Quantum Computation and Quantum Information}}{{cite arXiv|eprint=0808.0369|class=quant-ph|first=M.|last=Mosca|author-link=Michele Mosca|title=Quantum Algorithms|date=2008}} 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,{{Cite book|title = Quantum Computer Science|url = https://books.google.com/books?id=-wkJIuw0YRsC&q=quantum%2520computer%2520equivalent%2520classical%2520computer&pg=PA23|publisher = Morgan & Claypool Publishers|date = 2009-01-01|isbn = 9781598297324|first1 = Marco|last1 = Lanzagorta|first2 = Jeffrey K.|last2 = Uhlmann}}{{rp|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.}}

{{term|Quantum computing}}

{{defn|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.{{cite book | last=Hidary | first=Jack | title=Quantum computing : an applied approach | publisher=Springer | publication-place=Cham | date=2019 | isbn=978-3-030-23922-0 | oclc=1117464128 | page=3}}{{sfn|Nielsen|Chuang|2010|p=1}} 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.

}}

{{term|Quantum volume}}

{{defn|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.}}

{{term|Quantum error correction}}

{{defn|(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.}}

{{term|Quantum image processing}}

{{defn|(QIMP), is using quantum computing or quantum information processing to create and work with quantum images.{{cite thesis |last= Venegas-Andraca |first= Salvador E.|date= 2005 |title= Discrete Quantum Walks and Quantum Image Processing|type= DPhil thesis|publisher= The University of Oxford|url= https://ora.ox.ac.uk/objects/uuid:2baab08b-ee68-4ce5-8e68-8201f086a1ba}}{{cite journal |title=Towards realising secure and efficient image and video processing applications on quantum computers |journal=Entropy |volume=15 |issue=8 |pages=2874–2974 |year=2013 |last1=Iliyasu |first1=A.M.|bibcode=2013Entrp..15.2874I |doi=10.3390/e15082874 |doi-access=free }}

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.{{cite journal |title=Quantum image processing: A review of advances in its security technologies |journal=International Journal of Quantum Information |volume=15 |issue=3 |pages=1730001–44 |year=2017 |last1=Yan |first1=F.|last2=Iliyasu |first2=A.M.|last3=Le |first3=P.Q.|doi=10.1142/S0219749917300017 |bibcode=2017IJQI...1530001Y |doi-access=free }}}}

{{term|Quantum programming}}

{{defn|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.{{Cite book| author=Jarosław Adam Miszczak |title= High-level Structures in Quantum Computing | isbn=9781608458516|year= 2012 |publisher= Morgan & Claypool Publishers }} 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.{{Cite web|url=https://github.com/qosf/awesome-quantum-software|title=Comprehensive list of quantum open-source projects|website=Github|access-date=2022-01-27}}}}

{{term|Quantum simulator}}

{{defn| 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.{{cite journal

|doi=10.1140/epjqt10

|title=What is a quantum simulator?

|year=2014

|last1=Johnson

|first1=Tomi H.

|last2=Clark

|first2=Stephen R.

|last3=Jaksch

|first3=Dieter

|journal=EPJ Quantum Technology

|volume=1

|number=10|page=10

|arxiv=1405.2831

|bibcode=2014EPJQT...1...10J

|s2cid=120250321

}}

{{NIST-PD

|article=NIST Physicists Benchmark Quantum Simulator with Hundreds of Qubits

|Date= May 2, 2012

|url=https://www.nist.gov/public_affairs/tech-beat/tb20120502.cfm/

|author=Michael E. Newman

|accessdate=2013-02-22}}

{{cite journal

|doi=10.1038/nature10981

|url=http://tf.boulder.nist.gov/general/pdf/2614.pdf

|arxiv=1204.5789

|bibcode = 2012Natur.484..489B

|title=Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins

|year=2012

|last1=Britton

|first1=Joseph W.

|last2=Sawyer

|first2=Brian C.

|last3=Keith

|first3=Adam C.

|last4=Wang

|first4=C.-C. Joseph

|last5=Freericks

|first5=James K.

|last6=Uys

|first6=Hermann

|last7=Biercuk

|first7=Michael J.

|last8=Bollinger

|first8=John J.

|journal=Nature

|volume=484

|issue=7395

|pages=489–92

|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. Quantum simulators may be contrasted with generally programmable "digital" quantum computers, which would be capable of solving a wider class of quantum problems.}}

{{term|Quantum state discrimination}}

{{defn|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 $(E\_i)\_i$ on a given unknown state $\backslash rho$, under the promise that the state received is an element of a collection of states $\backslash \{\backslash sigma\_i\backslash \}\_i$, with $\backslash sigma\_i$ occurring with probability $p\_i$, that is, $\backslash rho=\backslash sum\_i\; p\_i\; \backslash sigma\_i$. The task is then to find the probability of the POVM $(E\_i)\_i$ correctly guessing which state was received. Since the probability of the POVM returning the $i$-th outcome when the given state was $\backslash sigma\_j$ has the form $\backslash text\{Prob\}(i|j)\; =\; \backslash operatorname\{tr\}(E\_i\; \backslash sigma\_j)$, it follows that the probability of successfully determining the correct state is $P\_\{\backslash rm\; success\}\; =\; \backslash sum\_\{i\}\; p\_\{i\}\backslash operatorname\{tr\}(\backslash sigma\_\{i\}\; E\_\{i\})$.{{Cite journal | arxiv=1707.02571|last1=Bae |first1=Joonwoo |last2=Kwek |first2=Leong-Chuan |title =Quantum state discrimination and its applications|year =2015 |journal= Journal of Physics A: Mathematical and Theoretical

|volume=48 |issue=8 |page=083001 |doi=10.1088/1751-8113/48/8/083001|bibcode=2015JPhA...48h3001B |s2cid=119199057 }}

}}

{{term|Quantum supremacy}}

{{defn|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).{{cite arXiv |last=Preskill |first=John |date=2012-03-26 |title=Quantum computing and the entanglement frontier |eprint=1203.5813 |class=quant-ph}}{{Cite journal|last=Preskill|first=John|date=2018-08-06|title=Quantum Computing in the NISQ era and beyond|journal=Quantum|volume=2|pages=79|doi=10.22331/q-2018-08-06-79|arxiv=1801.00862 |bibcode=2018Quant...2...79P |doi-access=free}}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. 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.{{Cite journal|last1=Harrow|first1=Aram W.|last2=Montanaro|first2=Ashley|date=September 2017|title=Quantum computational supremacy|journal=Nature|volume=549|issue=7671|pages=203–209|doi=10.1038/nature23458|pmid=28905912|issn=1476-4687|arxiv=1809.07442|bibcode=2017Natur.549..203H|s2cid=2514901}}{{Cite journal |last1=Papageorgiou |first1=Anargyros |last2=Traub |first2=Joseph F. |date=2013-08-12 |title=Measures of quantum computing speedup |journal=Physical Review A |volume=88 |issue=2 |pages=022316 |doi=10.1103/PhysRevA.88.022316 |issn=1050-2947 |arxiv=1307.7488|bibcode=2013PhRvA..88b2316P|s2cid=41867048 }} The term was coined by John Preskill in 2012,"John Preskill Explains 'Quantum Supremacy'". Quanta Magazine. 2 October 2019. Retrieved 2020-04-21. but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980){{cite book

| author=Manin, Yu. I.

| title=Vychislimoe i nevychislimoe

| trans-title=Computable and Noncomputable

| year=1980

| publisher=Sov.Radio

| url=http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5Bdjv%5D.zip

| pages=13–15

| language=ru

| access-date=2013-03-04

| url-status=dead

| archive-url=https://web.archive.org/web/20130510173823/http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).%5Bdjv%5D.zip

| archive-date=2013-05-10

}}

and Richard Feynman's (1981) proposals of quantum computing.{{Cite journal |last=Feynman |first=Richard P. |date=1982-06-01 |title=Simulating Physics with Computers |journal=International Journal of Theoretical Physics |volume=21 |pages=467–488 |issue=6–7 |doi=10.1007/BF02650179 |issn=0020-7748 |bibcode=1982IJTP...21..467F|citeseerx=10.1.1.45.9310 |s2cid=124545445 }} Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov,{{Cite book |last1=Aaronson |first1=Scott |last2=Arkhipov |first2=Alex |title=Proceedings of the forty-third annual ACM symposium on Theory of computing |chapter=The computational complexity of linear optics |date=2011 |series=STOC '11 |location=New York, NY, USA |publisher=ACM |pages=333–342 |doi=10.1145/1993636.1993682 |isbn=9781450306911 |arxiv=1011.3245|s2cid=681637 }} D-Wave's specialized frustrated cluster loop problems,{{cite arXiv |last1=King |first1=James |last2=Yarkoni |first2=Sheir |last3=Raymond |first3=Jack |last4=Ozfidan |first4=Isil |last5=King |first5=Andrew D. |last6=Nevisi |first6=Mayssam Mohammadi |last7=Hilton |first7=Jeremy P. |last8=McGeoch |first8=Catherine C. |date=2017-01-17 |title=Quantum Annealing amid Local Ruggedness and Global Frustration |eprint=1701.04579 |class=quant-ph}} and sampling the output of random quantum circuits.{{cite arXiv |last1=Aaronson |first1=Scott |last2=Chen |first2=Lijie |date=2016-12-18 |title=Complexity-Theoretic Foundations of Quantum Supremacy Experiments |eprint=1612.05903 |class=quant-ph}}{{Cite journal|last1=Bouland|first1=Adam|last2=Fefferman|first2=Bill|last3=Nirkhe|first3=Chinmay|last4=Vazirani|first4=Umesh|date=2018-10-29|title=On the complexity and verification of quantum random circuit sampling|url=http://dx.doi.org/10.1038/s41567-018-0318-2|journal=Nature Physics|volume=15|issue=2|pages=159–163|doi=10.1038/s41567-018-0318-2|arxiv=1803.04402|s2cid=125264133|issn=1745-2473}}}}

{{term|Quantum Turing machine}}

{{defn|(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.{{cite conference|author=Andrew Yao|author-link=Andrew Yao|title=Quantum circuit complexity|conference=34th Annual Symposium on Foundations of Computer Science|pages=352–361|year=1993}}{{cite journal|arxiv=1808.01701|author1=Abel Molina|author2=John Watrous|author-link2=John Watrous (computer scientist)|title=Revisiting the simulation of quantum Turing machines by quantum circuits|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|date=2018|volume=475|issue=2226|doi=10.1098/rspa.2018.0767|pmid=31293355|pmc=6598068}}{{rp|2}}}}

{{term|Qubit}}

{{defn|A qubit ({{IPAc-en|ˈ|k|juː|b|ɪ|t}}) 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.}}

{{term|Quil (instruction set architecture)}}

{{defn| 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.{{cite arXiv|last1=Smith|first1=Robert S.|last2=Curtis|first2=Michael J.|last3=Zeng|first3=William J.|date=2016-08-10|title=A Practical Quantum Instruction Set Architecture|eprint=1608.03355|class=quant-ph}} Many quantum algorithms (including quantum teleportation, quantum error correction, simulation,{{Cite journal|last1=McClean|first1=Jarrod R.|last2=Romero|first2=Jonathan|last3=Babbush|first3=Ryan|last4=Aspuru-Guzik|first4=Alán|date=2016-02-04|title=The theory of variational hybrid quantum-classical algorithms|arxiv=1509.04279|journal=New Journal of Physics|volume=18|issue=2|pages=023023|doi=10.1088/1367-2630/18/2/023023|issn=1367-2630|bibcode=2016NJPh...18b3023M|s2cid=92988541}}{{cite arXiv|last=Rubin|first=Nicholas C.|date=2016-10-21|title=A Hybrid Classical/Quantum Approach for Large-Scale Studies of Quantum Systems with Density Matrix Embedding Theory|eprint=1610.06910|class=quant-ph}} and optimization algorithms{{cite arXiv|last1=Farhi|first1=Edward|last2=Goldstone|first2=Jeffrey|last3=Gutmann|first3=Sam|date=2014-11-14|title=A Quantum Approximate Optimization Algorithm|eprint=1411.4028|class=quant-ph}}) require a shared memory architecture. Quil is being developed for the superconducting quantum processors developed by Rigetti Computing through the Forest quantum programming API.{{Cite web|url=https://spectrum.ieee.org/rigetti-launches-fullstack-quantum-computing-service-and-quantum-ic-fab|title=Rigetti Launches Full-Stack Quantum Computing Service and Quantum IC Fab|website=IEEE Spectrum: Technology, Engineering, and Science News|date=26 June 2017|language=en|access-date=2017-07-06}}{{Cite web|url=https://quantumcomputingreport.com/news/rigetti-quietly-releases-beta-of-forest-platform-for-quantum-programming-in-the-cloud/|title=Rigetti Quietly Releases Beta of Forest Platform for Quantum Programming in the Cloud {{!}} Quantum Computing Report|website=quantumcomputingreport.com|date=8 March 2017|language=en-US|access-date=2017-07-06}} 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.{{Cite web|url=https://ornl-qci.github.io/xacc/accelerators/rigetti|title=XACC Rigetti Accelerator|website=ornl-qci.github.io|access-date=2017-07-06|archive-date=2017-12-01|archive-url=https://web.archive.org/web/20171201032840/https://ornl-qci.github.io/xacc/accelerators/rigetti|url-status=dead}}{{Citation|last=Doiron|first=Nick|title=jsquil: Quantum computer instructions for JavaScript developers|date=2017-03-07|url=https://github.com/mapmeld/jsquil|access-date=2017-07-06}}}}

{{term|Qutrit}}

{{defn|(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.{{Cite journal|last1=Nisbet-Jones|first1=Peter B. R.|last2=Dilley|first2=Jerome|last3=Holleczek|first3=Annemarie|last4=Barter|first4=Oliver|last5=Kuhn|first5=Axel|date=2013|title=Photonic qubits, qutrits and ququads accurately prepared and delivered on demand|url=http://stacks.iop.org/1367-2630/15/i=5/a=053007|journal=New Journal of Physics|language=en|volume=15|issue=5|pages=053007|doi=10.1088/1367-2630/15/5/053007|issn=1367-2630|arxiv=1203.5614|bibcode=2013NJPh...15e3007N|s2cid=110606655 }}

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.{{cite web |url=https://spectrum.ieee.org/qudits-the-real-future-of-quantum-computing |title=Qudits: The Real Future of Quantum Computing? |date=28 June 2017 |publisher=IEEE Spectrum |access-date=2021-05-24}}}}

{{term|Solovay–Kitaev theorem}}

{{defn|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.{{Cite journal|last=Kitaev|first=A Yu|date=1997-12-31|title=Quantum computations: algorithms and error correction|url=http://dx.doi.org/10.1070/rm1997v052n06abeh002155|journal=Russian Mathematical Surveys|volume=52|issue=6|pages=1191–1249|doi=10.1070/rm1997v052n06abeh002155|bibcode=1997RuMaS..52.1191K |s2cid=250816585 |issn=0036-0279|url-access=subscription}} Solovay also gave a talk on his result at MSRI in 2000 but it was interrupted by a fire alarm.{{Cite AV media|last=Solovay|first=Robert|date=2000-02-08|title=Lie Groups and Quantum Circuits|url=https://www.msri.org/workshops/189/schedules/12826|publisher=MSRI}} Christopher M. Dawson and Michael Nielsen call the theorem one of the most important fundamental results in the field of quantum computation.{{Cite journal|last1=Dawson|first1=Christopher M.|last2=Nielsen|first2=Michael|date=2006-01-01|title=The Solovay-Kitaev algorithm|url=https://dl.acm.org/doi/abs/10.5555/2011679.2011685|journal=Quantum Information & Computation|volume=6|pages=81–95|doi=10.26421/QIC6.1-6|language=EN|arxiv=quant-ph/0505030}}}}

{{glossaryend}}