# Complexity limitations on quantum computation

@article{Fortnow1999ComplexityLO, title={Complexity limitations on quantum computation}, author={Lance Fortnow and John D. Rogers}, journal={J. Comput. Syst. Sci.}, year={1999}, volume={59}, pages={240-252} }

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., PPBQP=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?UP?coUP and one-way functions…

## Topics from this paper

## 182 Citations

Quantum Computation Relative to Oracles

- Computer Science, MathematicsUMC
- 2000

New relativized worlds in which (i) co-RP ⊈ NQE, (ii) BQP and UP = EXP, (iii) P = EQP and RP =EXP, and (iv) EQP⊈ ∑ 2 P ⋃ ∏ 2 P are presented.

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 Science, MathematicsProceedings 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.

Oracle Separations for Quantum Statistical Zero-Knowledge

- Computer Science, PhysicsArXiv
- 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.

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.

Complexity of Quantum Computers

- 2006

A quantum computer uses quantum mechanical phenomena to per form calculations. This method of computation may enable us to solve certain problems faste r than any classical computer. In this paper,…

Limitations of quantum advice and one-way communication

- Computer Science, MathematicsProceedings. 19th IEEE Annual Conference on Computational Complexity, 2004.
- 2004

It is shown in three settings that quantum messages have only limited advantages over classical ones, and the polynomial method is used to give the first correct proof of a direct product theorem for quantum search.

Limitations of quantum advice and one-way communication

- Mathematics
- 2004

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three…

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

- Computer Science, 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 and classical complexity classes: Separations, collapses, and closure properties

- Mathematics, Computer ScienceInf. Comput.
- 2005

It is proved that WPP is closed under polynomial-time truth-table reductions, while an oracle relative to which WPP isn't closed is constructed, implying that proving the equality of the similar appearing classes LWPP and WPP would require nonrelativizable proof techniques.

## References

SHOWING 1-10 OF 32 REFERENCES

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

- Computer Science, 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 Computability

- Computer Science, MathematicsSIAM J. Comput.
- 1997

It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in P and PSPACE.

Quantum lower bounds by polynomials

- Mathematics, Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

Quantum Complexity Theory

- Computer ScienceSIAM J. Comput.
- 1997

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

Determining Acceptance Possibility for a Quantum Computation is Hard for PH

- Mathematics
- 1998

Abstract It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to…

Relationships between quantum and classical space-bounded complexity classes

- Mathematics, Computer ScienceProceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247)
- 1998

It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power, which implies that any space s QTM can be simulated deterministically in space O (s/sup 2/), and further that any (unbounded-error) QTM running in log-space can be simulation in NC/Sup 2/.

Strengths and Weaknesses of Quantum Computing

- Mathematics, PhysicsSIAM J. Comput.
- 1997

It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

- Computer Science, MathematicsSIAM Rev.
- 1999

Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems.

A fast quantum mechanical algorithm for database search

- Computer Science, PhysicsSTOC '96
- 1996

In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .

RELATIVIZABLE AND NONRELATIVIZABLE THEOREMS IN THE POLYNOMIAL THEORY OF ALGORITHMS

- Mathematics
- 1994

Starting with the paper of Baker, Gill, and Solovay [BGS 75] in complexity theory, many results have been proved that separate certain relativized complexity classes or show that they have no…