Span-program-based quantum algorithm for evaluating formulas

TL;DR

This paper introduces a quantum algorithm for evaluating complex formulas using span programs and spectral graph methods, achieving optimal performance for certain balanced formulas and extending previous quantum evaluation techniques.

Contribution

It establishes a novel connection between span programs and weighted bipartite graphs, enabling optimal quantum evaluation of formulas over extended gate sets.

Findings

Optimal quantum evaluation for balanced ternary majority formulas.

Generalizes previous quantum algorithms for AND-OR formulas.

Provides a new spectral approach to formula evaluation.

Abstract

We give a quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority). The algorithm is optimal on read-once formulas for which each gate's inputs are balanced in a certain sense. The main new tool is a correspondence between a classical linear-algebraic model of computation, "span programs," and weighted bipartite graphs. A span program's evaluation corresponds to an eigenvalue-zero eigenvector of the associated graph. A quantum computer can therefore evaluate the span program by applying spectral estimation to the graph. For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of randomized alpha-beta pruning is known to be suboptimal. In contrast, our algorithm generalizes the optimal quantum AND-OR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula.