On exact quantum query complexity

TL;DR

This paper investigates the exact quantum query complexity of boolean functions, revealing relationships with classical complexity, limitations of certain quantum algorithms, and characterizations for symmetric and nonadaptive cases, supported by numerical analysis.

Contribution

It introduces families of boolean functions with quantum complexity ratios between 1/2 and 2/3 of classical, and characterizes nonadaptive and symmetric cases using coding theory.

Findings

Quantum algorithms cannot always be simplified to parity computations.

Exact quantum query complexity can be a constant fraction of classical complexity.

Numerical results for small functions provide bounds on success probabilities.

Abstract

We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs characterising quantum query complexity for small problem sizes. We include numerical results giving the optimal success probabilities achievable by quantum algorithms computing all boolean functions on up to 4 bits, and all symmetric boolean functions on up to 6 bits.