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{{Short description|AlgorithmsAlgorithm to be run on quantum computers, typically relying on superposition and/or entanglement}}

{{Use American English|date=January 2019}}

{{Use dmy dates|date=December 2020}}

In [[quantum computing]], a '''quantum algorithm''' is an [[algorithm]] whichthat runs on a realistic model of [[quantum computation]], the most commonly used model being the [[quantum circuit]] model of computation.<ref>{{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}}</ref><ref>{{cite arXiv|eprint=0808.0369|class=quant-ph|first=M.|last=Mosca|author-link=Michele Mosca|title=Quantum Algorithms|date=2008}}</ref> 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,<ref>{{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}}</ref>{{rp|126}} the term quantum algorithm is usuallygenerally usedreserved for those algorithms whichthat seem inherently quantum, or use some essential feature of quantum computation such as [[quantum superposition]] or [[quantum entanglement]].

Problems whichthat are [[Undecidable problem|undecidable]] using classical computers remain undecidable using quantum computers.<ref name=nielchuan>{{cite book|title=Quantum Computation and Quantum Information|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|publisher=Cambridge University Press|year=2010|isbn=978-1-107-00217-3|edition=2nd|___location=Cambridge|author-link=Michael A. Nielsen|author-link2=Isaac Chuang|url=https://books.google.com/books?id=-s4DEy7o-a0C}}</ref>{{rp|127}} What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms because the quantum superposition and quantum entanglement that quantum algorithms exploit probablygenerally cannot be efficiently simulated on classical computers (see [[Quantum supremacy]]).

The best-known algorithms are [[Shor's algorithm]] for factoring and [[Grover's algorithm]] for searching an unstructured database or an unordered list. Shor's algorithmsalgorithm runs much (almost exponentially) faster than the best-most efficient known classical algorithm for factoring, the [[general number field sieve]].<ref>{{Cite web|url=https://quantum-computing.ibm.com/docs/iqx/guide/shors-algorithm|title = Shor's algorithm|access-date=21 October 2020|archive-date=12 January 2023|archive-url=https://web.archive.org/web/20230112064753/https://quantum-computing.ibm.com/docs/iqx/guide/shors-algorithm|url-status=dead}}</ref> Grover's algorithm runs quadratically faster than the best possible classical algorithm for the same task,<ref>{{citationcite web needed|url=https://quantum-computing.ibm.com/composer/docs/iqx/guide/grovers-algorithm |title=IBM quantum composer user guide: Grover's algorithm |access-date=February7 June 2022 |website=quantum-computing.ibm.com |archive-date=27 September 2022 |archive-url=https://web.archive.org/web/20220927192200/https://quantum-computing.ibm.com/composer/docs/iqx/guide/grovers-algorithm |url-status=dead 2018}}</ref> a [[linear search]].

==Overview==

Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a [[quantum circuit]] whichthat acts on some input [[qubit]]s and terminates with a [[measurement]]. A quantum circuit consists of simple [[quantum gate]]s, which act on at most a fixed numbereach of qubits.which Theacts numberon ofsome qubits has to be fixed because a changingfinite number of qubits implies non-unitary evolution. Quantum algorithms may also be stated in other models of quantum computation, such as the [[Hamiltonian oracle model]].<ref name=Hamiltonian_NAND_Tree>{{cite arXivjournal

| last1 = Farhi | first1 = E.Edward

| last2 = Goldstone |first2=J.Jeffrey

| last3 = Gutmann |first3=S.Sam

| date = 20072008

| title = A Quantum Algorithm for the Hamiltonian NAND Tree

| journal=Theory of Computing

| eprint = quant-ph/0702144

| volume=4

}}</ref>

| pages=169–190

| eprintarxiv = quant-ph/0702144

| doi=10.4086/toc.2008.v004a008 | doi-access=free}}</ref>

Quantum algorithms can be categorized by the main techniques usedinvolved byin the algorithm. Some commonly used techniques/ideas in quantum algorithms include [[phase kick-back]], [[quantum phase estimation algorithm|phase estimation]], the [[quantum Fourier transform]], [[quantum walk]]s, [[amplitude amplification]] and [[topological quantum field theory]]. Quantum algorithms may also be grouped by the type of problem solved, for instance; see, e.g., the survey on quantum algorithms for algebraic problems.<ref>{{cite journal|last1=Childs|first1=Andrew M.|author-link=Andrew Childs|last2=van Dam|first2=W.|year=2010|title=Quantum algorithms for algebraic problems|journal=[[Reviews of Modern Physics]]|volume=82|issue=1|pages=1–52|arxiv=0812.0380|bibcode=2010RvMP...82....1C|doi=10.1103/RevModPhys.82.1|s2cid=119261679}}</ref>

==Algorithms based on the quantum Fourier transform==

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[[File:Deutsch-Jozsa-algorithm-quantum-circuit.png|thumb|Deutsch-Jozsa algorithm]]

The Deutsch–Jozsa algorithm solves a [[black-box]] problem which probablythat requires exponentially many queries to the black box for any deterministic classical computer, but can be done with exactlya onesingle query by a quantum computer. If weHowever, allowwhen bothcomparing bounded-error quantumclassical and classicalquantum algorithms, then there is no speedup, since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. The algorithm determines whether a function ''f'' is either constant (0 on all inputs or 1 on all inputs) or balanced (returns 1 for half of the input ___domain and 0 for the other half).

===Bernstein–Vazirani algorithm===

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{{main|Simon's algorithm}}

Simon's algorithm solves a black-box problem exponentially faster than any classical algorithm, including bounded-error probabilistic algorithms. This algorithm, which achieves an exponential speedup over all classical algorithms that we consider efficient, was the motivation for [[Shor's factoring algorithm]] for factoring.

===Quantum phase estimation algorithm===

{{main|Quantum phase estimation algorithm}}

The [[quantum phase estimation algorithm]] is used to determine the eigenphase of an eigenvector of a unitary gate, given a quantum state proportional to the eigenvector and access to the gate. The algorithm is frequently used as a subroutine in other algorithms.

===Shor's algorithm===

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| doi=10.1137/s0097539795293172

| s2cid = 2337707

}}</ref> whereas the best known classical algorithms take super-polynomial time. TheseIt problemsis areunknown notwhether knownthese toproblems beare in [[P (complexity)|P]] or [[NP-complete]]. It is also one of the few quantum algorithms that solves a non&ndash;-black-box problem in polynomial time, where the best known classical algorithms run in super-polynomial time.

===Hidden subgroup problem===

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|publisher=[[Springer-Verlag]]

|isbn=3-540-60221-6

}}</ref> The more general hidden subgroup problem, where the group isn'tis not necessarily abelian, is a generalization of the previously mentioned problems, as well andas [[graph isomorphism]] and certain [[lattice problems]]. Efficient quantum algorithms are known for certain non-abelian groups. However, no efficient algorithms are known for the [[symmetric group]], which would give an efficient algorithm for graph isomorphism<ref>{{cite arXiv

|last1=Moore |first1=C.|author1-link=Cris Moore

|last2=Russell |first2=A.

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| eprint = cs/0304005

}}</ref>

===Boson sampling problem===

{{main|Boson sampling}}

The Boson Sampling Problem in an experimental configuration assumes<ref>{{cite journal|last1=Ralph|first1=T.C.|title=Figure 1: The boson-sampling problem|url=http://www.nature.com/nphoton/journal/v7/n7/fig_tab/nphoton.2013.175_F1.html|journal=Nature Photonics|date=July 2013|volume=7|issue=7|pages=514–515|publisher=Nature|doi=10.1038/nphoton.2013.175|access-date=12 September 2014}}</ref> an input of [[boson]]s (ex. photons of light) of moderate number getting randomly scattered into a large number of output modes constrained by a defined [[unitarity]]. The problem is then to produce a fair sample of the [[probability distribution]] of the output which is dependent on the input arrangement of bosons and the Unitarity.<ref>{{cite journal|last1=Lund|first1=A.P.|last2=Laing|first2=A.|last3=Rahimi-Keshari|first3=S.|last4=Rudolph|first4=T.|last5=O'Brien|first5=J.L.|last6=Ralph|first6=T.C.|title=Boson Sampling from Gaussian States|journal=Phys. Rev. Lett.|date=5 September 2014|volume=113|issue=10|doi=10.1103/PhysRevLett.113.100502|arxiv = 1305.4346 |bibcode = 2014PhRvL.113j0502L|pmid=25238340|page=100502|s2cid=27742471}}</ref> Solving this problem with a classical computer algorithm requires computing the [[Permanent (mathematics)|permanent]] of the unitary transform matrix, which may be either impossible or take a prohibitively long time. In 2014, it was proposed<ref>{{cite web|title=The quantum revolution is a step closer|url=http://phys.org/news/2014-09-quantum-revolution-closer.html|website=Phys.org|publisher=Omicron Technology Limited|access-date=12 September 2014}}</ref> that existing technology and standard probabilistic methods of generating single photon states could be used as input into a suitable quantum computable [[Linear optical quantum computing|linear optical network]] and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted<ref>{{cite journal|last1=Seshadreesan|first1=Kaushik P.|last2=Olson|first2=Jonathan P.|last3=Motes|first3=Keith R.|last4=Rohde|first4=Peter P.|last5=Dowling|first5=Jonathan P.|title=Boson sampling with displaced single-photon Fock states versus single-photon-added coherent states: The quantum-classical divide and computational-complexity transitions in linear optics|journal=Physical Review A|volume=91|issue=2|page=022334|doi=10.1103/PhysRevA.91.022334|arxiv = 1402.0531 |bibcode = 2015PhRvA..91b2334S |year=2015|s2cid=55455992}}</ref> the sampling problem had similar complexity for inputs other than [[Fock state]] photons and identified a transition in [[Quantum complexity theory|computational complexity]] from classically simulatable to just as hard as the Boson Sampling Problem, dependent on the size of coherent amplitude inputs.

===Estimating Gauss sums===

A [[Gauss sum]] is a type of [[exponential sum]]. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.<ref>

{{Cite arXiv

| last1=van Dam |firstfirst1=W.

| last2=Seroussi |first2=G.

| year =2002

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===Fourier fishing and Fourier checking===

We haveConsider an [[Oracle machine|oracle]] consisting of ''n'' random Boolean functions mapping ''n''-bit strings to a Boolean value., Wewith arethe requiredgoal toof findfinding n ''n''-bit strings ''z''₁,..., ''z_n'' such that for the Hadamard-Fourier transform, at least 3/4 of the strings satisfy

:<math>| \tilde{f}(z_i)| \geqslant 1</math>

and at least 1/4 satisfiessatisfy

:<math>| \tilde{f}(z_i) | \geqslant 2.</math>

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==Algorithms based on amplitude amplification==

[[Amplitude amplification]] is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered to beas a generalization of Grover's algorithm.{{citation needed|date=February 2024}}

===Grover's algorithm===

{{main|Grover's algorithm}}

Grover's algorithm searches an unstructured database (or an unordered list) with N entries, for a marked entry, using only <math>O(\sqrt{N})</math> queries instead of the <math>O({N})</math> queries required classically.<ref>

{{Cite arXiv|eprint=quant-ph/9605043|first=Lov K.|last=Grover|author-link=Lov Grover|title=A fast quantum mechanical algorithm for database search|date=1996}}</ref> Classically, <math>O({N})</math> queries are required even allowing bounded-error probabilistic algorithms.

Theorists have considered a hypothetical generalization of a standard quantum computer that could access the histories of the hidden variables in [[De Broglie–Bohm theory|Bohmian mechanics]]. (Such a computer is completely hypothetical and would ''not'' be a standard quantum computer, or even possible under the standard theory of quantum mechanics.) Such a hypothetical computer could implement a search of an N-item database in at most in <math>O(\sqrt[3]{N})</math> steps. This is slightly faster than the <math>O(\sqrt{N})</math> steps taken by [[Grover's algorithm]]. NeitherHowever, neither search method would allow either model of quantum computer to solve [[NP-completeness|NP-complete]] problems in polynomial time.<ref>{{Cite web|url=https://www.scottaaronson.com/papers/qchvpra.pdf|title=Quantum Computing and Hidden Variables|last=Aaronson|first=Scott}}</ref>

===Quantum counting===

[[Quantum counting]] solves a generalization of the search problem. It solves the problem of counting the number of marked entries in an unordered list, instead of just detecting ifwhether one exists. Specifically, it counts the number of marked entries in an <math>N</math>-element list, with an error of at most <math>\varepsilon</math> by making only <math>\Theta\left(\fracvarepsilon^{-1}{\varepsilon} \sqrt{\frac{N}{/k}}\right)</math> queries, where <math>k</math> is the number of marked elements in the list.<ref>

{{Cite book

|last1 = Brassard |first1 = G.

|last2=Hoyer |first2=P.

|last3=Tapp |first3=A.

|chapter = Quantum counting

|date = 1998

|title = QuantumAutomata, CountingLanguages and Programming

|arxiv = quant-ph/9805082

|doi=10.1007/BFb0055105

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|bibcode=2000quant.ph..5055B|isbn=9780821821404

|s2cid=54753

}}</ref> More precisely, the algorithm outputs an estimate <math>k'</math> for <math>k</math>, the number of marked entries, with the following accuracy: <math>|k-k'| \leq \varepsilon k</math>.

==Algorithms based on quantum walks==

{{main|Quantum walk}}

A quantum walk is the quantum analogue of a classical [[random walk]],. A classical random whichwalk can be described by a [[probability distribution]] over some states., Awhile a quantum walk can be described by a [[quantum superposition]] over states. Quantum walks are known to give exponential speedups for some black-box problems.<ref>

{{cite conference

|last1=Childs |first1=A. M.

Line 195:

|doi=10.1109/FOCS.2007.18

|isbn=978-0-7695-3010-9

}}</ref> They also provide polynomial speedups for many problems. A framework for the creation of quantum walk algorithms exists and is quite a versatile tool.<ref name="Search_via_quantum_walk" />

===Boson sampling problem===

{{main|Boson sampling}}

The Boson Sampling Problem in an experimental configuration assumes<ref>{{cite journal |last1=Ralph |first1=T.C. |date=July 2013 |title=Figure 1: The boson-sampling problem |url=http://www.nature.com/nphoton/journal/v7/n7/fig_tab/nphoton.2013.175_F1.html |journal=Nature Photonics |datepublisher=JulyNature 2013|volume=7 |issue=7 |pages=514–515|publisher=Nature |doi=10.1038/nphoton.2013.175 |s2cid=110342419 |access-date=12 September 2014|url-access=subscription }}</ref> an input of [[boson]]s (exe.g., photons of light) of moderate number gettingthat are randomly scattered into a large number of output modes, constrained by a defined [[unitarity]]. When individual photons are used, the problem is isomorphic to a multi-photon quantum walk.<ref>{{Cite journal |last1=Peruzzo |first1=Alberto |last2=Lobino |first2=Mirko |last3=Matthews |first3=Jonathan C. F. |last4=Matsuda |first4=Nobuyuki |last5=Politi |first5=Alberto |last6=Poulios |first6=Konstantinos |last7=Zhou |first7=Xiao-Qi |last8=Lahini |first8=Yoav |last9=Ismail |first9=Nur |last10=Wörhoff |first10=Kerstin |last11=Bromberg |first11=Yaron |last12=Silberberg |first12=Yaron |last13=Thompson |first13=Mark G. |last14=OBrien |first14=Jeremy L. |date=2010-09-17 |title=Quantum Walks of Correlated Photons |url=https://www.science.org/doi/10.1126/science.1193515 |journal=Science |language=en |volume=329 |issue=5998 |pages=1500–1503 |doi=10.1126/science.1193515 |pmid=20847264 |arxiv=1006.4764 |bibcode=2010Sci...329.1500P |hdl=10072/53193 |s2cid=13896075 |issn=0036-8075}}</ref> The problem is then to produce a fair sample of the [[probability distribution]] of the output whichthat is dependentdepends on the input arrangement of bosons and the Unitarityunitarity.<ref>{{cite journal |last1=Lund |first1=A.P. |last2=Laing |first2=A. |last3=Rahimi-Keshari |first3=S. |last4=Rudolph |first4=T. |last5=O'Brien |first5=J.L. |last6=Ralph |first6=T.C. |date=5 September 2014 |title=Boson Sampling from Gaussian States |journal=Phys. Rev. Lett.|date=5 September 2014|volume=113 |issue=10 |doipage=10.1103/PhysRevLett.113.100502 |arxiv = 1305.4346 |bibcode = 2014PhRvL.113j0502L |doi=10.1103/PhysRevLett.113.100502 |pmid=25238340|page=100502 |s2cid=27742471}}</ref> Solving this problem with a classical computer algorithm requires computing the [[Permanent (mathematics)|permanent]] of the unitary transform matrix, which may be either impossible or take a prohibitively long time or be outright impossible. In 2014, it was proposed<ref>{{cite web |title=The quantum revolution is a step closer |url=http://phys.org/news/2014-09-quantum-revolution-closer.html |access-date=12 September 2014 |website=Phys.org |publisher=Omicron Technology Limited|access-date=12 September 2014}}</ref> that existing technology and standard probabilistic methods of generating single -photon states could be used as an input into a suitable quantum computable [[Linear optical quantum computing|linear optical network]] and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted<ref>{{cite journal |last1=Seshadreesan |first1=Kaushik P. |last2=Olson |first2=Jonathan P. |last3=Motes |first3=Keith R. |last4=Rohde |first4=Peter P. |last5=Dowling |first5=Jonathan P. |year=2015 |title=Boson sampling with displaced single-photon Fock states versus single-photon-added coherent states: The quantum-classical divide and computational-complexity transitions in linear optics |journal=Physical Review A |volume=91 |issue=2 |page=022334|doi=10.1103/PhysRevA.91.022334 |arxiv = 1402.0531 |bibcode = 2015PhRvA..91b2334S |yeardoi=201510.1103/PhysRevA.91.022334 |s2cid=55455992}}</ref> the sampling problem had similar complexity for inputs other than [[Fock state|Fock-state]] photons and identified a transition in [[Quantum complexity theory|computational complexity]] from classically simulatablesimulable to just as hard as the Boson Sampling Problem, dependentdepending on the size of coherent amplitude inputs.

===Element distinctness problem===

{{main|Element distinctness problem}}

The element distinctness problem is the problem of determining whether all the elements of a list are distinct. Classically, Ω<math>\Omega(''N'')</math> queries are required for a list of size ''<math>N''.</math>; Howeverhowever, it can be solved in <math>\Theta(N^{2/3})</math> queries on a quantum computer. The optimal algorithm iswas put forth by [[Andris Ambainis]].,<ref>

{{cite journal

|last=Ambainis |first=A.

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|doi=10.1137/S0097539705447311

|s2cid=6581885

}}</ref> and [[Yaoyun Shi]] first proved a tight lower bound when the size of the range is sufficiently large.<ref>

{{cite conference

| last1=Shi | first1=Y.

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|doi=10.4086/toc.2005.v001a002

|doi-access=free

}}</ref> independently (and via different proofs) extended histhat work to obtain the lower bound for all functions.

===Triangle-finding problem===

{{main|Triangle finding problem}}

The triangle-finding problem is the problem of determining whether a given graph contains a triangle (a [[clique (graph theory)|clique]] of size 3). The best-known lower bound for quantum algorithms is Ω<math>\Omega(''N'')</math>, but the best algorithm known requires O(''N''^1.297) queries,<ref>{{cite arXiv| eprint=1105.4024| author1=Aleksandrs Belovs| title=Span Programs for Functions with Constant-Sized 1-certificates| class=quant-ph| year=2011}}</ref> an improvement over the previous best O(''N''^1.3) queries.<ref name=Search_via_quantum_walk>

{{cite conference

|last1=Magniez |first1=F.

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|work=Shtetl-Optimized

|access-date=2009-12-17

}}</ref> This type of formula requires Θ(''N''<supmath>\Theta(N^c)</supmath>) queries using randomness,<ref>

{{cite conference

|last1=Saks |first1=M.E.

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|doi=10.1109/SFCS.1986.44

|isbn=0-8186-0740-8

}}</ref> where <math>c = \log_2(1+\sqrt{33})/4 \approx 0.754</math>. With a quantum algorithm, however, it can be solved in Θ<math>\Theta(''N''<sup>0.5^{1/2})</supmath>) queries. No better quantum algorithm for this case was known until one was found for the unconventional Hamiltonian oracle model.<ref name=Hamiltonian_NAND_Tree/> The same result for the standard setting soon followed.<ref>

{{cite arXiv

|last=Ambainis |first=A.

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===Group commutativity===

The problem is to determine if a [[black box group|black-box group]], given by ''k'' generators, is [[Commutativity|commutative]]. A black -box group is a group with an oracle function, which must be used to perform the group operations (multiplication, inversion, and comparison with identity). WeThe areinterest in this context interestedlies in the query complexity, which is the number of oracle calls needed to solve the problem. The deterministic and randomized query complexities are <math>\Theta(k^2)</math> and <math>\Theta(k)</math>, respectively.<ref>

{{cite journal

|last=Pak |first=Igor |author1-link=Igor Pak

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|doi=10.1112/S1461157012000046

|doi-access=free

}}</ref> A quantum algorithm requires <math>\Omega(k^{2/3})</math> queries, butwhile the best -known classical algorithm uses <math>O(k^{2/3} \log k)</math> queries.<ref>

{{cite journal

|last1=Magniez |first1=F.

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===Quantum simulation===

{{main|Hamiltonian simulation}}

The idea that quantum computers might be more powerful than classical computers originated in Richard Feynman's observation that classical computers seem to require exponential time to simulate many-particle quantum systems, yet quantum many-body systems are able to "solve themselves."<ref>

{{Cite journal

| last1=Feynman | first1=R. P.

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| citeseerx=10.1.1.45.9310

| s2cid=124545445

}}</ref> Since then, the idea that quantum computers can simulate quantum physical processes exponentially faster than classical computers has been greatly fleshed out and elaborated. Efficient (that isi.e., polynomial-time) quantum algorithms have been developed for simulating both Bosonic and Fermionic systems,<ref>

{{Cite journal

| last1=Abrams |first1=D. S.

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| doi=10.1103/PhysRevLett.79.2586

|s2cid=18231521

}}</ref> andas inwell particular,as the simulation of chemical reactions beyond the capabilities of current classical supercomputers requiresusing only a few hundred qubits.<ref>

{{Cite journal

| last1=Kassal | first1=I.

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}}</ref>

===Solving a linear systemssystem of equations===

{{main|Quantum algorithm for linear systems of equations}}

In 2009, [[Aram Harrow]], Avinatan Hassidim, and [[Seth Lloyd]], formulated a quantum algorithm for solving [[System of linear equations|linear systems]]. The [[Quantum algorithm for linear systems of equations|algorithm]] estimates the result of a scalar measurement on the solution vector to a given linear system of equations.<ref name="Quantum algorithm for solving linear systems of equations by Harrow et al.">{{Cite journal|arxiv = 0811.3171|last1 = Harrow|first1 = Aram W|title = Quantum algorithm for solving linear systems of equations|journal = Physical Review Letters|volume = 103|issue = 15|pages = 150502|last2 = Hassidim|first2 = Avinatan|last3 = Lloyd|first3 = Seth|year = 2008|doi = 10.1103/PhysRevLett.103.150502|pmid = 19905613|bibcode = 2009PhRvL.103o0502H|s2cid = 5187993}}</ref>

Provided that the linear system is a [[sparse matrix|sparse]] and has a low [[condition number]] <math>\kappa</math>, and that the user is interested in the result of a scalar measurement on the solution vector, (instead of the values of the solution vector itself), then the algorithm has a runtime of <math>O(\log(N)\kappa^2)</math>, where <math>N</math> is the number of variables in the linear system. This offers an exponential speedup over the fastest classical algorithm, which runs in <math>O(N\kappa)</math> (or <math>O(N\sqrt{\kappa})</math> for positive semidefinite matrices).

==Hybrid quantum/classical algorithms==

Hybrid Quantum/Classical Algorithms combine quantum state preparation and measurement with classical optimization.<ref>{{cite journal|last1=Moll|first1=Nikolaj|last2=Barkoutsos|first2=Panagiotis|last3=Bishop|first3=Lev S.|last4=Chow|first4=Jerry M.|last5=Cross|first5=Andrew|last6=Egger|first6=Daniel J.|last7=Filipp|first7=Stefan|last8=Fuhrer|first8=Andreas|last9=Gambetta|first9=Jay M.|last10=Ganzhorn|first10=Marc|last11=Kandala|first11=Abhinav|last12=Mezzacapo|first12=Antonio|last13=Müller|first13=Peter|last14=Riess|first14=Walter|last15=Salis|first15=Gian|last16=Smolin|first16=John|last17=Tavernelli|first17=Ivano|last18=Temme|first18=Kristan|title=Quantum optimization using variational algorithms on near-term quantum devices|journal=Quantum Science and Technology|date=2018|volume=3|issue=3|pages= 030503|doi=10.1088/2058-9565/aab822|arxiv=1710.01022|bibcode=2018QS&T....3c0503M|s2cid=56376912}}</ref> These algorithms generally aim to determine the ground -state eigenvector and eigenvalue of a Hermitian Operatoroperator.

=== QAOA ===

The [[quantum approximate optimization algorithm]] istakes inspiration from quantum annealing, performing a toydiscretized modelapproximation of quantum annealing whichusing a quantum circuit. It can be used to solve problems in graph theory.<ref>{{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}}</ref> The algorithm makes use of classical optimization of quantum operations to maximize an "objective function."

=== Variational quantum eigensolver ===

The [[variational quantum eigensolver]] (VQE) algorithm applies classical optimization to minimize the energy expectation value of an [[Ansatz|ansatz state]] to find the ground state energy of a Hermitian operator, such as a molecule's Hamiltonian.<ref>{{Cite journal |last1=Peruzzo |first1=Alberto |last2=McClean |first2=Jarrod |last3=Shadbolt |first3=Peter |last4=Yung |first4=Man-Hong |last5=Zhou |first5=Xiao-Qi |last6=Love |first6=Peter J. |last7=Aspuru-Guzik |first7=Alán |last8=O’Brien |first8=Jeremy L. |date=2014-07-23 |title=A variational eigenvalue solver on a photonic quantum processor |journal=Nature Communications |language=En |volume=5 |issue=1 |pages=4213 |arxiv=1304.3061 |bibcode=2014NatCo...5.4213P |doi=10.1038/ncomms5213|pmid=25055053|pmc=4124861 |issn=2041-1723 |arxivpmc=1304.30614124861 |bibcodepmid=2014NatCo...5E4213P25055053}}</ref> ThisIt can also be extended to find excited energies of moleculesmolecular Hamiltonians.<ref>{{cite journal|last1=Higgott|first1=Oscar|last2=Wang|first2=Daochen|last3=Brierley|first3=Stephen|title=Variational Quantum Computation of Excited States|journal=Quantum|year=2019|volume=3|pages=156|doi=10.22331/q-2019-07-01-156|arxiv=1805.08138|bibcode=2019Quant...3..156H |s2cid=119185497}}</ref>

=== Contracted quantum eigensolver ===

The contracted quantum eigensolver (CQE) algorithm minimizes the residual of a contraction (or projection) of the Schrödinger equation onto the space of two (or more) electrons to find the ground- or excited-state energy and two-electron reduced density matrix of a molecule.<ref>{{Cite journal|last1=Smart|first1=Scott|last2=Mazziotti|first2=David|date=2021-02-18|title=Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices|journal=Phys. Rev. Lett.|language=En|volume=125|issue=7|pages=070504|doi=10.1103/PhysRevLett.126.070504|pmid=33666467|arxiv=2004.11416|bibcode=2021PhRvL.126g0504S|s2cid=216144443}}</ref> It is based on classical methods for solving energies and two-electron reduced density matrices directly from the anti-Hermitian contracted Schrödinger equation.<ref>{{Cite journal|last1=Mazziotti|first1=David|date=2006-10-06|title=Anti-Hermitian Contracted Schrödinger Equation: Direct Determination of the Two-Electron Reduced Density Matrices of Many-Electron Molecules|journal=Phys. Rev. Lett.|language=En|volume=97|issue=14|pages=143002|doi=10.1103/PhysRevLett.97.143002|pmid=17155245|bibcode=2006PhRvL..97n3002M}}</ref>

==See also==

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* The [https://quantumalgorithmzoo.org Quantum Algorithm Zoo]: A comprehensive list of quantum algorithms that provide a speedup over the fastest known classical algorithms.

* [https://www.cs.umd.edu/~amchilds/qa/ Andrew Childs' lecture notes on quantum algorithms]

*[https://bastion.center/the-quantum-search-algorithm/ The Quantum search algorithm - brute force] {{Webarchive|url=https://web.archive.org/web/20180901034619/https://bastion.center/the-quantum-search-algorithm/ |date=1 September 2018 }}.

===Surveys===

* {{Citecite bookarXiv | last1 = SmithDalzell | first1 =Alexander JM. | last2 = MoscaMcArdle | first2 = M.Sam | doi title=Quantum 10.1007/978-3-540-92910-9_43algorithms: |A chaptersurvey =of Algorithmsapplications forand Quantumend-to-end Computerscomplexities | title date= Handbook of Natural Computing2023 | pages class= 1451quant-ph | year eprint= 20122310.03011 | isbn = 978display-3-540-92909-3 | s2cid authors= 16565723 1}}

* {{Cite book | last1 = Smith | first1 = J. | last2 = Mosca | first2 = M. | doi = 10.1007/978-3-540-92910-9_43 | chapter = Algorithms for Quantum Computers | title = Handbook of Natural Computing | pages = 1451–1492 | year = 2012 | arxiv = 1001.0767 | isbn = 978-3-540-92909-3 | s2cid = 16565723 }}

* {{Cite journal | last1 = Childs | first1 = A. M. | last2 = Van Dam | first2 = W. | doi = 10.1103/RevModPhys.82.1 | title = Quantum algorithms for algebraic problems | journal = Reviews of Modern Physics | volume = 82 | pages = 1–52 | year = 2010 | issue = 1 |bibcode = 2010RvMP...82....1C | arxiv = 0812.0380 | s2cid = 119261679 }}

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{{DEFAULTSORT:Quantum Algorithm}}

[[Category:Quantum computing]]

[[Category:Quantum information science]]

[[Category:Theoretical computer science]]

[[Category:Quantum algorithms| ]]

[[Category:Emerging technologies]]