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See also: [[Computational complexity]] and [[Complexity class]]

A [[complexity class]] is a collection of [[Computational problem|computational problems]] that can be solved by a computational model under certain resource constraints. For instance, the complexity class [[P (complexity)|P]] is defined as the set of problems solvable by a [[Turing machine]] in [[polynomial time]]. Similarly, quantum complexity classes may be defined using quantum models of computation, such as the [[Quantum computer|quantum circuit model]] or the equivalent [[quantum Turing machine]]. One of the main aims of quantum complexity theory is to find out how these classes relate to classical complexity classes such as [[P (complexity)|P]], [[NP (complexity)|NP]], [[BPP (complexity)|BPP]], and [[PSPACE]].

One of the reasons quantum complexity theory is studied are the implications of quantum computing for the modern [[Church–Turing thesis|Church-Turing thesis]]. In short the modern Church-Turing thesis states that any computational model can be simulated in polynomial time with a [[probabilistic Turing machine]].<ref name=":0">{{Cite journal|last=Vazirani|first=Umesh V.|date=2002|title=A survey of quantum complexity theory|url=http://dx.doi.org/10.1090/psapm/058/1922899|journal=Proceedings of Symposia in Applied Mathematics|pages=193–217|doi=10.1090/psapm/058/1922899|issn=2324-7088}}</ref><ref name=":3">{{Cite book|last=Nielsen, Michael A., 1974-|url=https://www.worldcat.org/oclc/665137861|title=Quantum computation and quantum information|date=2010|publisher=Cambridge University Press|others=Chuang, Isaac L., 1968-|isbn=978-1-107-00217-3|edition=10th anniversary ed|___location=Cambridge|oclc=665137861}}</ref> However, questions around the Church-Turing thesis arise in the context of quantum computing. It is unclear whether the Church-Turing thesis holds for the quantum computation model. There is much evidence that the thesis does not hold. and that itIt may not be possible for a probabilistic Turing machine to simulate quantum computation models in polynomial time.<ref name=":0" />

Both quantum computational complexity of functions and classical computational complexity of functions are often expressed with [[asymptotic notation]]. Some common forms of asymptotic notion of functions are <math>O(T(n))</math>, <math>\Omega(T(n))</math>, and <math>\Theta(T(n))</math>. <math>O(T(n))</math> expresses that something is bounded above by <math>cT(n)</math> where <math>c</math> is a constant such that <math>c>0</math> and <math>T(n)</math> is a function of <math>n</math>, <math>\Omega(T(n))</math> expresses that something is bounded below by <math>cT(n)</math> where <math>c</math> is a constant such that <math>c>0</math> and <math>T(n)</math> is a function of <math>n</math>, and <math>\Theta(T(n))</math> expresses both <math>O(T(n))</math> and <math>\Omega(T(n))</math>.<ref name=":1" /> These notations also their own names. <math>O(T(n))</math> is called [[Big O notation]], <math>\Omega(T(n))</math> is called Big Omega notation, and <math>\Theta(T(n))</math> is called Big Theta notation.

== Complexity Classes Overview ==

Some important complexity classes are P, BPP, BQP, PP, and P-Space. To define these we first define a promise problem. A promise problem is a decision problem which has an input assumed to be selected from the set of all possible input strings. A promise problem is a pair <math>A=(A_{yes},A_{no})</math>. <math>A_{yes}</math> is the set of yes instances, <math>A_{no}</math> is the set of no instances, and the intersection of these sets is such that <math>A_{yes}\cap A_{no}=\emptyset</math>. All of the previous complexity classes contain promise problems.<ref name=":2">{{Cite journal|last=Watrous|first=John|date=2008-04-21|title=Quantum Computational Complexity|url=http://arxiv.org/abs/0804.3401|journal=arXiv:0804.3401 [quant-ph]}}</ref>

{| class="wikitable"

While there is no known way to efficiently simulate a quantum computer with a classical computer, it is possible to efficiently simulate a classical computer with a quantum computer. This is evident from the fact that <math>BBP\subseteq BQP</math>.<ref>{{Cite journal|last=Watrous|first=John|date=2008-04-21|title=Quantum Computational Complexity|url=http://arxiv.org/abs/0804.3401|journal=arXiv:0804.3401 [quant-ph]}}</ref>

|+

!Complexity Class

!Criteria

|-

|P

|Promise problems for which a polynomial time deterministic Turing machine accepts all strings in <math>A_{yes}</math> and rejects all strings in <math>A_{no}</math><ref name=":2" />

|-

|BPP

|Promise problems for which a polynomial time Probabilistic Turing machine accepts every string in <math>A_{yes}</math> with a probability of at least <math>\frac{2}{3}</math>, and accepts every string in <math>A_{no}</math> with a probability of at most <math>\frac{1}{3}</math><ref name=":2" />

|-

|BQP

|Promise problems such that for functions <math>a,b:\mathbb {N}\rightarrow[0,1]</math>, there exists a polynomial time generated family of quantum circuits <math>Q={\{Q_n:n\in \mathbb{N}\}}</math>, where <math>Q_n</math> is a circuit which accepts <math>n</math> qubits and gives an output of one qubit. An element <math>x</math> of <math>A_{yes}</math> is accepted by <math>Q</math> with a probability greater than or equal to <math>a(\left \vert x \right \vert) </math>. An element <math>x</math> of <math>A_{no}</math> is accepted by <math>Q</math> with a probability less than or equal to <math>b(\left \vert x\right \vert)</math>. <ref name=":2" />

|-

|PP

|Promise problems for which a polynomial time Probabilistic Turing machine accepts every string in <math>A_{yes}</math> with a probability greater than <math>\frac{1}{2}</math>, and accepts every string in <math>A_{no}</math> with a probability of at most <math>\frac{1}{2}</math><ref name=":2" />

|-

|P-SPACE

|Promise problems for which a deterministic Turing machine, that runs in polynomial space, accepts every string in <math>A_{yes}</math> and rejects all strings in <math>A_{no}</math><ref name=":2" />

|}

== Simulating Quantum Circuits ==

Quantum gates always have a degree of imperfection in their implementation. We can not implement a quantum gate without some inaccuracy, so there is a need to approximately apply quantum gates.<ref name=":1" />

There is no known way to efficiently simulate a quantum computational model with a classical computer. This means that a classical computer cannot simulate a quantum computational model in polynomial time. However, a [[quantum circuit]] of <math>S(n)</math> qubits with <math>O(T(n))</math> [[quantum gates]] can be simulated by a classical circuit with <math>O(2^{S(n)}T(n)^3)</math> [[Logic gate|classical gates]].<ref name=":1">{{Citation|last=Cleve|first=Richard|title=An Introduction to Quantum Complexity Theory|date=2000|url=http://dx.doi.org/10.1142/9789810248185_0004|work=Quantum Computation and Quantum Information Theory|volume=|pages=103–127|publisher=WORLD SCIENTIFIC|isbn=978-981-02-4117-9|access-date=October 10, 2020}}</ref> This number of classical gates is obtained by determining how many bit operations are necessary to simulate the quantum circuit. FirstIn order to do this, first the amplitudes associated with the <math>S(n)</math> qubits must be accounted for. Each of the the states of the <math>S(n)</math> qubits can be described by a two-dimensional complex vector, or a state vector. These state vectors can also be described a [[linear combination]] of its [[Euclidean vector|component vectors]] with coefficients called amplitudes. These amplitudes are complex numbers which are normalized to one, meaning the sum of the squares of the absolute values of the amplitudes must be one.<ref name=":1" /> The entries of the state vector are these amplitudes. Which entry each of the amplitudes are correspond to the none-zero component of the component vector which they are the coefficients of in the linear combination description. As an equation this is described as <math>\alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}</math> or <math>\alpha \left \vert 1 \right \rangle + \beta \left \vert 0 \right \rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}</math> using [[Bra–ket notation|Dirac notation]]. The state of the entire <math>S(n)</math> qubit system can be described by a single state vector. This state vector describing the entire system is the tensor product of the state vectors describing the individual qubits in the system. The result of the tensor products of the <math>S(n)</math> qubits is a single state vector which has <math>2^{S(n)}</math> dimensions and entries that are the amplitudes associated with each basis state or component vector. Therefore, <math>2^{S(n)}</math>amplitudes must be accounted for with a <math>2^{S(n)}</math> dimensional complex vector which itis the state vector for the <math>S(n)</math> qubit system.<ref>{{Cite journal|last=Häner|first=Thomas|last2=Steiger|first2=Damian S.|date=2017-11-12|title=0.5 petabyte simulation of a 45-qubit quantum circuit|url=http://dx.doi.org/10.1145/3126908.3126947|journal=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis|___location=New York, NY, USA|publisher=ACM|doi=10.1145/3126908.3126947|isbn=978-1-4503-5114-0}}</ref> NextIn order to obtain an upper bound for the applicationnumber of thegates <math>T(n)</math>required to simulate a quantum gatescircuit onwe need a sufficient upper bound for the amount data used to specify the information about each of the <math>2^{S(n)}</math> amplitudes. mustTo bedo accounted for. Therefore,this <math>O(T(n))</math> bits of precision willare be required requiredsufficient for encoding each amplitude.<ref name=":1" /> So it takes <math>O(2^{S(n)}T(n))</math> classical bits to account for the state vector of the <math>S(n)</math> qubit system. Next the application of the <math>T(n)</math> quantum gates on <math>2^{S(n)}</math> amplitudes must be accounted for. The quantum gates can be represented as <math>2^{S(n)}\times2^{S(n)}</math> [[Sparse matrix|sparse matrices]].<ref name=":1" /> So to account for the each application of alleach of the <math>T(n)</math> quantum gates, the state vector must be multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix for everyeach of the <math>T(n)</math> quantum gategates. Every time the state vector is multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix, <math>O(2^{S(n)})</math> arithmetic operations must be preformed.<ref name=":1" /> Therefore, there are <math>O(2^{S(n)}T(n)^2)</math> bit operations for every quantum gate applied to the state vector. So <math>O(2^{S(n)}T(n)^2)</math> classical gate are needed to simulate <math>S(n)</math> qubit circuit with just one quantum gate. Therefore, <math>O(2^{S(n)}T(n)^3)</math> classical gates canare needed to simulate a quantum circuit of <math>S(n)</math> qubits with <math>O(T(n))</math> quantum gates.<ref name=":1" /> While there is no known way to efficiently simulate a quantum computer with a classical computer, it is possible to efficiently simulate a classical computer with a quantum computer. This is evident from the belief that <math>BBP\subseteq BQP</math>.<ref name=":2" />

=== Simulating Quantum Circuits ===

There is no known way to efficiently simulate a quantum computational model with a classical computer. This means that a classical computer cannot simulate a quantum computational model in polynomial time. However, a [[quantum circuit]] of <math>S(n)</math> qubits with <math>O(T(n))</math> [[quantum gates]] can be simulated by a classical circuit with <math>O(2^{S(n)}T(n)^3)</math> [[Logic gate|classical gates]].<ref name=":1">{{Citation|last=Cleve|first=Richard|title=An Introduction to Quantum Complexity Theory|date=2000|url=http://dx.doi.org/10.1142/9789810248185_0004|work=Quantum Computation and Quantum Information Theory|volume=|pages=103–127|publisher=WORLD SCIENTIFIC|isbn=978-981-02-4117-9|access-date=October 10, 2020}}</ref> This number of classical gates is obtained by determining how many bit operations are necessary to simulate the quantum circuit. First the amplitudes associated with the <math>S(n)</math> qubits must be accounted for. Therefore, <math>2^{S(n)}</math>amplitudes must be accounted for with a <math>2^{S(n)}</math> dimensional complex vector which it the state vector for the <math>S(n)</math> qubit system.<ref>{{Cite journal|last=Häner|first=Thomas|last2=Steiger|first2=Damian S.|date=2017-11-12|title=0.5 petabyte simulation of a 45-qubit quantum circuit|url=http://dx.doi.org/10.1145/3126908.3126947|journal=Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis|___location=New York, NY, USA|publisher=ACM|doi=10.1145/3126908.3126947|isbn=978-1-4503-5114-0}}</ref> Next the application of the <math>T(n)</math> quantum gates on <math>2^{S(n)}</math> amplitudes must be accounted for. Therefore, <math>O(T(n))</math> bits of precision will be required required for encoding each amplitude.<ref name=":1" /> So it takes <math>O(2^{S(n)}T(n))</math> classical bits to account for the state vector. The quantum gates can be represented as <math>2^{S(n)}\times2^{S(n)}</math> [[Sparse matrix|sparse matrices]].<ref name=":1" /> So to account for the each application of all of the quantum gates, the state vector must be multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix for every quantum gate. Every time the state vector is multiplied by a <math>2^{S(n)}\times2^{S(n)}</math> sparse matrix <math>O(2^{S(n)})</math> arithmetic operations must be preformed.<ref name=":1" /> Therefore, there are <math>O(2^{S(n)}T(n)^2)</math> bit operations for every quantum gate applied to the state vector. So <math>O(2^{S(n)}T(n)^2)</math> classical gate are needed to simulate <math>S(n)</math> qubit circuit with just one quantum gate. Therefore, <math>O(2^{S(n)}T(n)^3)</math> classical gates can simulate a quantum circuit of <math>S(n)</math> qubits with <math>O(T(n))</math> quantum gates.<ref name=":1" />

== PrimeQuery FactorizationComplexity ==

=== Deutsch-Jozsa Algorithm ===

The Deutsch-Jozsa algorithm is a quantum algorithm designed to solve a toy problem with a smaller query complexity than is possible with a classical algorithm. The toy problem asks whether a function <math>f:\{0,1\}^n\rightarrow\{0,1\}</math> is constant or balanced, those being the only two possibilities.<ref name=":3" /> The only way to evaluate the function <math>f</math> is to consult a [[black box]] or [[Oracle machine|oracle]]. A classical [[deterministic algorithm]] will have to check more than half of the possible inputs to be sure of whether or not the function is constant or balanced. With <math>2^n</math> possible inputs, the query complexity of the most efficient classical deterministic algorithm is <math>2^{n-1}+1</math>.<ref name=":3" /> The Deutsch-Jozsa algorithm takes advantage of quantum parallelism to check all of the elements of the ___domain at once and only needs to query the oracle once. Meaning the query complexity of the Deutsch-Jozsa algorithm is <math>1</math>.<ref name=":3" />

== References ==