Span Programs for Functions with Constant-Sized 1-certificates

TL;DR

This paper introduces span programs as a new approach to solve functions with constant-sized 1-certificates, achieving improved quantum query complexities for problems like triangle detection.

Contribution

It presents a novel span program-based method for these problems, surpassing previous quantum algorithms in efficiency.

Findings

Quantum algorithm for triangle detection with $O(n^{35/27})$ query complexity

Span programs effectively solve functions with constant-sized 1-certificates

Improved quantum query complexity over previous methods

Abstract

Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity $O(n^{35/27})$ that is better than $O(n^{13/10})$ of the best previously known algorithm by Magniez et al.