# Sharp Quantum vs. Classical Query Complexity Separations

@article{Beaudrap2000SharpQV, title={Sharp Quantum vs. Classical Query Complexity Separations}, author={J. Niel de Beaudrap and Richard Cleve and John Watrous}, journal={arXiv: Quantum Physics}, year={2000} }

We obtain the strongest separation between quantum and classical query complexity known to date -- specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations). The problem is simple to define and the quantum algorithm solving it is also simple when described in terms of certain quantum Fourier transforms (QFTs) that…

## 15 Citations

Characterization of Non-Deterministic Quantum Query and Quantum Communication Complexity

- Computer ScienceComputational Complexity Conference
- 2000

It is shown that the non-deterministic quantum query complexity of f is linearly related to the degree of a "non-Deterministic" polynomial for f, and that it can be exponentially smaller than its classical counterpart.

Optimal quantum query bounds for almost all Boolean functions

- Computer Science, MathematicsSTACS
- 2013

It is shown that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms, and that van Dam's oracle interrogation is essentially optimal for almost all functions.

Quantum algorithms for algebraic problems

- Computer Science
- 2010

This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

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.

Nondeterministic Quantum Query and Communication Complexities

- Computer ScienceSIAM J. Comput.
- 2003

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.

Uselessness for an Oracle model with internal randomness

- Computer Science, MathematicsQuantum Inf. Comput.
- 2014

In the oracle model with internal randomness where the goal is to gain any nonzero advantage over guessing, it is proved (roughly speaking) that k quantum queries are equivalent in power to 2k classical queries, thus extending results of Meyer and Pommersheim.

Optimal Parallel Quantum Query Algorithms

- Computer ScienceAlgorithmica
- 2016

It is proved that quantum and classical p-parallel query complexity are polynomially related for all total functions f when p is small compared to f’s block sensitivity.

Quantum algorithms for some hidden shift problems

- Computer ScienceSODA '03
- 2003

The hidden coset problem is defined, which generalizes the hidden shift problem and the hidden subgroup problem and provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

Quantum algorithm for multivariate polynomial interpolation

- Mathematics, Computer ScienceProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2018

This work presents and analyzes quantum algorithms for this multivariate polynomial interpolation problem over the fields Fq, R and C and finds a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2.

Exponential algorithmic speedup by a quantum walk

- Computer ScienceSTOC '03
- 2003

A black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer is constructed and it is proved that no classical algorithm can solve the problem in subexponential time.

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