Span Program for Non-binary Functions

TL;DR

This paper extends span programs and learning graphs to non-binary functions, providing a framework that characterizes quantum query complexity for functions with larger input/output alphabets, advancing quantum algorithm design.

Contribution

It introduces a non-binary span program framework and generalizes learning graphs, enabling analysis of quantum query complexity for functions with non-binary inputs and outputs.

Findings

Non-binary span programs characterize quantum query complexity up to a constant factor.

Generalization of learning graphs provides upper bounds for non-binary functions.

Framework unifies binary and non-binary quantum query complexity analysis.

Abstract

Span programs characterize the quantum query complexity of binary functions $f:\{0,\ldots,\ell\}^n \to \{0,1\}$ up to a constant factor. In this paper we generalize the notion of span programs for functions with non-binary input/output alphabets $f: [\ell]^n \to [m]$. We show that non-binary span program characterizes the quantum query complexity of any such function up to a constant factor. We argue that this non-binary span program is indeed the generalization of its binary counterpart. We also generalize the notion of span programs for a special class of relations. Learning graphs provide another tool for designing quantum query algorithms for binary functions. In this paper, we also generalize this tool for non-binary functions, and as an application of our non-binary span program show that any non-binary learning graph gives an upper bound on the quantum query complexity.