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.^[1] 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.

Definition

BQP can be viewed as the languages associated with certain bounded-error uniform families of quantum circuits.^[1] A language L is in BQP if and only if there exists a polynomial-time uniform family of quantum circuits ${\displaystyle \{Q_{n}\colon n\in \mathbb {N} \}}$, such that

• For all ${\displaystyle n\in \mathbb {N} }$, Q_n takes n qubits as input and outputs 1 bit
• For all x in L, ${\displaystyle \mathrm {Pr} (Q_{|x|}(x)=1)\geq {\tfrac {2}{3}}}$
• For all x not in L, ${\displaystyle \mathrm {Pr} (Q_{|x|}(x)=0)\geq {\tfrac {2}{3}}}$

Alternatively, one can define BQP in terms of quantum Turing machines. A language L is in BQP if and only if there exists a polynomial quantum Turing machine that accepts L with an error probability of at most 1/3 for all instances.^[2]

Similarly to other "bounded error" probabilistic classes, the choice of 1/3 in the definition is arbitrary. We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound. The complexity class is unchanged by allowing error as high as 1/2 − n^−c on the one hand, or requiring error as small as 2^−nc on the other hand, where c is any positive constant, and n is the length of input.^[3]

Relationship to other complexity classes

BQP is defined for quantum computers; the corresponding complexity class for classical computers (or more formally for probabilistic Turing machines) is BPP. Just like P and BPP, BQP is low for itself, which means BQP^BQP = BQP.^[2] Informally, this is true because polynomial time algorithms are closed under composition. If a polynomial time algorithm calls polynomial time algorithms as subroutines, the resulting algorithm is still polynomial time.

BQP contains P and BPP and is contained in AWPP,^[4] PP^[5] and PSPACE.^[2] In fact, BQP is low for PP, meaning that a PP machine achieves no benefit from being able to solve BQP problems instantly, an indication of the possible difference in power between these similar classes. The known relationships with classic complexity classes are:

${\displaystyle {\mathsf {P\subseteq BPP\subseteq BQP\subseteq AWPP\subseteq PP\subseteq PSPACE\subseteq EXP}}}$

As the problem of P ≟ PSPACE has not yet been solved, the proof of inequality between BQP and classes mentioned above is supposed to be difficult.^[2] The relation between BQP and NP is not known. In May 2018, computer scientists Ran Raz of Princeton University and Avishay Tal of Stanford University published a paper^[6] which showed that, relative to an oracle, BQP was not contained in PH. It can be proven that there exists an oracle A such that BQP^A ${\displaystyle \nsubseteq }$ PH^A.^[7] In an extremely informal sense, this can be thought of as giving PH and BQP an identical, but additional, capability and verifying that BQP with the oracle (BQP^A) can do things PH^A cannot. While an oracle separation has been proven, the fact that BQP is not contained in PH has not been proven. An oracle separation does not prove whether or not complexity classes are the same. The oracle separation gives intuition that BQP may not be contained in PH.

It has been suspected for many years that Fourier Sampling is a problem that exists within BQP, but not within the polynomial hierarchy. Recent conjectures have provided evidence that a similar problem, Fourier Checking, also exists in the class BQP without being contained in the polynomial hierarchy. This conjecture is especially notable because it suggests that problems existing in BQP could be classified as harder than NP-Complete problems. Paired with the fact that many practical BQP problems are suspected to exist outside of P (it is suspected and not verified because there is no proof that P ≠ NP), this illustrates the potential power of quantum computing in relation to classical computing.^[7]

Adding postselection to BQP results in the complexity class PostBQP which is equal to PP.^[8][9]

A complete problem for Promise-BQP

Promise-BQP is the class of promise problems that can be solved by a uniform family of quantum circuits (i.e., within BQP).^[10] Completeness proofs focus on this version of BQP. Similar to the notion of NP-completeness and other complete problems, we can define a complete problem as a problem that is in Promise-BQP and that every other problem in Promise-BQP reduces to it in polynomial time.

APPROX-QCIRCUIT-PROB

The APPROX-QCIRCUIT-PROB problem is complete for efficient quantum computation, and the version presented below is complete for the Promise-BQP complexity class (and not for the total BQP complexity class, for which no complete problems are known). APPROX-QCIRCUIT-PROB's completeness makes it useful for proofs showing the relationships between other complexity classes and BQP.

Given a description of a quantum circuit ${\displaystyle C}$ acting on ${\displaystyle n}$ qubits with ${\displaystyle m}$ gates, where ${\displaystyle m}$ is a polynomial in ${\displaystyle n}$ and each gate acts on one or two qubits, and two numbers ${\displaystyle \alpha ,\beta \in ,\alpha >\beta }$, distinguish between the following two cases:

• measuring the first qubit of the state ${\displaystyle C|0\rangle ^{\otimes n}}$ yields ${\displaystyle |1\rangle }$ with probability ${\displaystyle \geq \alpha }$
• measuring the first qubit of the state ${\displaystyle C|0\rangle ^{\otimes n}}$ yields ${\displaystyle |1\rangle }$ with probability ${\displaystyle \leq \beta }$

Here, there is a promise on the inputs as the problem does not specify the behavior if an instance is not covered by these two cases.

Claim. Any BQP problem reduces to APPROX-QCIRCUIT-PROB.

Proof. Suppose we have an algorithm ${\displaystyle A}$ that solves APPROX-QCIRCUIT-PROB, i.e., given a quantum circuit ${\displaystyle C}$ acting on ${\displaystyle n}$ qubits, and two numbers ${\displaystyle \alpha ,\beta \in ,\alpha >\beta }$, ${\displaystyle A}$ distinguishes between the above two cases. We can solve any problem in BQP with this oracle, by setting ${\displaystyle \alpha =2/3,\beta =1/3}$.

For any ${\displaystyle L\in \mathrm {BQP} }$, there exists family of quantum circuits ${\displaystyle \{Q_{n}\colon n\in \mathbb {N} \}}$ such that for all ${\displaystyle n\in \mathbb {N} }$, a state ${\displaystyle |x\rangle }$ of ${\displaystyle n}$ qubits, if ${\displaystyle x\in L,Pr(Q_{n}(|x\rangle )=1)\geq 2/3}$; else if ${\displaystyle x\notin L,Pr(Q_{n}(|x\rangle )=0)\geq 2/3}$. Fix an input ${\displaystyle |x\rangle }$ of ${\displaystyle n}$ qubits, and the corresponding quantum circuit ${\displaystyle Q_{n}}$. We can first construct a circuit ${\displaystyle C_{x}}$ such that ${\displaystyle C_{x}|0\rangle ^{\otimes n}=|x\rangle }$. This can be done easily by hardwiring ${\displaystyle |x\rangle }$ and apply a sequence of CNOT gates to flip the qubits. Then we can combine two circuits to get ${\displaystyle C^{\prime }=Q_n{C}_x}$Zdroj:https://en.wikipedia.org?pojem=BQP
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