Approximate Span Programs

Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. For any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, but finding such an algorithm is generally challenging. We consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a problem $f$ can also be used to decide “property testing” versions of $f$, or more generally, approximate the span program witness size, a property of the input related to $f$. For example, using our techniques, the span program for OR, which can be used to design an optimal algorithm for the OR function, can also be used to design optimal algorithms for: threshold functions, in which we want to decide if the Hamming weight of a string is above a threshold or far below, given the promise that one of these is true; and approximate counting, in which we want to estimate the Hamming weight of the input. We achieve these results by relaxing the requirement that 1-inputs hit some target exactly in the span program, which could make design of span programs easier. We also give an exposition of span program structure, which increases the understanding of this important model. One implication is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in some cases. As applications, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between $s$ and $t$, $R_{s,t}(G)$, of $\tilde O(\frac{1}{\epsilon^{3/2}}n\sqrt{R_{s,t}(G)})$, and, when $\mu$ is a lower bound on $\lambda_2(G)$, by our phase gap lower bound, we can obtain $\tilde O(\frac{1}{\epsilon}n\sqrt{R_{s,t}(G)/\mu})$, both using $O(\log n)$ space.

💡 Research Summary

This paper significantly expands the scope and utility of span programs, a model of computation used to design quantum algorithms with optimal query complexity. The central innovation is the introduction of “approximate span programs,” which relax the traditional requirement that positive inputs must exactly hit a target vector within a certain subspace. This relaxation enables a single span program to address not only its original decision problem but also a broader class of “property testing” problems related to the function.

The authors develop a general framework showing that any span program deciding a function f can be repurposed to solve threshold versions of f (e.g., deciding if the Hamming weight of a string is above a threshold or far below) and, more generally, to estimate the “witness size” associated with the input. Specifically, they construct quantum algorithms that estimate the positive witness size w+(x) to relative accuracy ε using O( (1/ε^(3/2)) * sqrt(w+(x) * \widetilde{W}-) ) queries, where \widetilde{W}- is the approximate negative witness complexity. This framework is powerful when w+(x) corresponds to a quantity of interest, such as 1/|x| for the OR function or the effective resistance R_s,t(G) in a graph for st-connectivity.

A major technical contribution is a deep structural analysis of span programs. The authors examine the unitary operator U = (2Π_A - I)(2Π_B - I) derived from a span program and its associated “discriminant” matrix D. They establish a precise connection between the eigenvalues of U and the singular values of D. Crucially, they prove a lower bound on the phase gap Δ(U) (the smallest non-zero eigenvalue phase of U) in terms of the singular values of the span program operator A and its input-dependent restriction A(x): Δ(U) ≥ 2 * σ_min(D) and σ_min(D) ≥ σ_min(A(x)) / σ_max(A). This analysis provides an alternative pathway for witness size estimation via phase estimation, complementing previous methods based on effective spectral gap analysis.

The practical impact of these theoretical advances is demonstrated through a concrete application: estimating the effective resistance R_s,t(G) between two vertices in a graph within the adjacency query model. By interpreting the known span program for st-connectivity