Reflections for quantum query algorithms

TL;DR

This paper demonstrates that any boolean function can be evaluated optimally by a quantum query algorithm using a universal two-reflection structure, establishing the tightness of the general adversary bound and linking span programs to quantum algorithms.

Contribution

It proves the universality of a two-reflection quantum query structure and shows the general adversary bound is tight, connecting span programs with quantum query complexity.

Findings

Quantum algorithms can evaluate any boolean function with a universal two-reflection structure.

The general adversary bound is tight and can be expressed as an SDP for quantum query complexity.

Quantum query complexity of composed functions equals the product of individual complexities.

Abstract

We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f(g,...,g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It further shows that span programs are equivalent to quantum query algorithms.