# BQP

In computational complexity theory, **bounded-error quantum polynomial time** (**BQP**) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances.[1] It is the quantum analogue of 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.

BQP algorithm (1 run) | ||
---|---|---|

Answer produced Correct answer |
Yes | No |

Yes | ≥ 2/3 | ≤ 1/3 |

No | ≤ 1/3 | ≥ 2/3 |

BQP algorithm (k runs) | ||

Answer producedCorrect answer |
Yes | No |

Yes | > 1 − 2^{−ck} |
< 2^{−ck} |

No | < 2^{−ck} |
> 1 − 2^{−ck} |

for some constant c > 0 |

## 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 , such that

- For all ,
*Q*takes_{n}*n*qubits as input and outputs 1 bit - For all
*x*in*L*, - For all
*x*not in*L*,

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]

## Quantum computation

The number of qubits in the computer is allowed to be a polynomial function of the instance size. For example, algorithms are known for factoring an *n*-bit integer using just over 2*n* qubits (Shor's algorithm).

Usually, computation on a quantum computer ends with a measurement. This leads to a collapse of quantum state to one of the basis states. It can be said that the quantum state is measured to be in the correct state with high probability.

Quantum computers have gained widespread interest because some problems of practical interest are known to be in **BQP**, but suspected to be outside **P**. Some prominent examples are:

- Integer factorization (see Shor's algorithm)[4]
- Discrete logarithm[4]
- Simulation of quantum systems (see universal quantum simulator)
- Approximating the Jones polynomial at certain roots of unity

## Relationship to other complexity classes

Unsolved problem in computer science:What is the relationship between (more unsolved problems in computer science)BQP and NP? |

This class is defined for a quantum computer and its natural corresponding class for an ordinary computer (or a Turing machine plus a source of randomness) 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 as a subroutine polynomially many polynomial time algorithms, the resulting algorithm is still polynomial time.

**BQP** contains **P** and **BPP** and is contained in **AWPP**,[5] **PP**[6] 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:

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[7] which showed that, relative to an oracle, BQP was not contained in PH.

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

## See also

- Hidden subgroup problem
- Polynomial hierarchy (PH).
- Quantum complexity theory, mainly P and NP.

## References

- Michael Nielsen and Isaac Chuang (2000).
*Quantum Computation and Quantum Information*. Cambridge: Cambridge University Press. ISBN 0-521-63503-9. - Bernstein, Ethan; Vazirani, Umesh (October 1997). "Quantum Complexity Theory".
*SIAM Journal on Computing*.**26**(5): 1411–1473. CiteSeerX 10.1.1.655.1186. doi:10.1137/S0097539796300921. - Barak, Sanjeev Arora, Boaz (2009).
*Computational Complexity: A Modern Approach / Sanjeev Arora and Boaz Barak*. Cambridge. p. 122. Retrieved 24 July 2018. - arXiv:quant-ph/9508027v2
*Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer*, Peter W. Shor - Fortnow, Lance; Rogers, John (1999). "Complexity limitations on Quantum computation" (PDF).
*J. Comput. Syst. Sci*.**59**(2): 240–252. doi:10.1006/jcss.1999.1651. ISSN 0022-0000. - L. Adleman, J. DeMarrais, and M.-D. Huang. Quantum computability. SIAM J. Comput., 26(5):1524–1540, 1997.
- George, Michael Goderbauer, Stefan. "ECCC - TR18-107".
*eccc.weizmann.ac.il*. Retrieved 2018-08-03. - Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time".
*Proceedings of the Royal Society A*.**461**(2063): 3473–3482. arXiv:quant-ph/0412187. Bibcode:2005RSPSA.461.3473A. doi:10.1098/rspa.2005.1546.. Preprint available at - Aaronson, Scott (2004-01-11). "Complexity Class of the Week: PP".
*Computational Complexity Weblog*. Retrieved 2008-05-02.