# Complexity limitations on quantum computation

@article{Fortnow1998ComplexityLO, title={Complexity limitations on quantum computation}, author={Lance Fortnow and John D. Rogers}, journal={Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247)}, year={1998}, pages={202-209} }

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. BQP is low for PP, i.e., PP/sup BQP/=PP. There exists a relativized world where P=BQP and the polynomial-time hierarchy is infinite. There exists a relativized world where BQP does not have complete sets. There exists a relativized world where P=BQP but P/spl ne/UP/spl cap/coUP and one…

## 174 Citations

Revisiting a limit on efficient quantum computation

- Computer ScienceACM-SE 44
- 2006

This paper offers an exposition of a theorem originally due to Adleman, Demarrais and Huang that shows that the quantum complexity class BQP is contained in the classical counting class PP (Probabilistic Polynomial time).

Quantum computing, postselection, and probabilistic polynomial-time

- Computer ScienceProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2005

It is shown that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently, or probabilistic polynomial-time, and implies, as an easy corollary, a celebrated theorem of Beigel, Reingold and Spielman that PP is closed under intersection.

Approximate Counting and Quantum Computation

- Computer Science, MathematicsCombinatorics, Probability and Computing
- 2005

A form of additive approximation which can be used to simulate a function in BQP is introduced and it is shown that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations.

Oracle Separations for Quantum Statistical Zero-Knowledge

- Mathematics, Computer ScienceArXiv
- 2018

These proofs rely on a bound on output state discrimination for relativized quantum circuits based on the quantum adversary method of Ambainis, following a technique similar to one used by Ben-David and Kothari to prove limitations on a query complexity variant of quantum statistical zero-knowledge.

Complexity of Quantum Computers

- Computer Science
- 2006

It is shown that quantum computers can e fficiently solve a few problems known to be in the complexity class BQP, which cannot be ge n ralized to all algorithms, and quantum computers only give a quadratic improvement for NP-complet e problems.

Quantum and classical complexity classes: Separations, collapses, and closure properties

- Computer ScienceInf. Comput.
- 2003

Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy

- PhysicsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 1999

It is shown that the complexity class NQP (a quantum analogue of NP) of Adleman, Demarrais and Huang, is equal to the counting class coC=P.

Quantum Complexity Classes

- Computer ScienceArXiv
- 2004

Inspired by BQP, new complexity classes are defined that incorporate the current important quantum algorithms and the importance of the unitarity constraint given by quantum mechanics is revealed.

The Acrobatics of BQP

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2021

It is shown that, in the black-box setting, the behavior of quantum polynomial-time ( BQP ) can be remarkably decoupled from that of classical complexity classes like NP .

Nondeterministic Quantum Query and Quantum Communication Complexities

- Computer ScienceArXiv
- 2000

The nondeterministic quantum algorithms for Boolean functions f have positive acceptance probability on input x iff f(x)=1, which implies that the quantum communication complexities of the equality and disjointness functions are n+1 if the authors do not allow any error probability.

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