Quantum algorithms for spectral sums
Alessandro Luongo, Changpeng Shao
TL;DR
This work presents quantum algorithms for estimating spectral sums of positive semidefinite matrices, including the log-determinant, Schatten $p$-norms, von Neumann entropy, and the trace of the inverse, by leveraging block-encodings and quantum singular value transformation. The methods yield sub-linear dependence on matrix size and polylogarithmic factors in key parameters like the condition number $\kappa$ and error $\epsilon$, with concrete applications to graph problems such as counting triangles, estimating effective resistance, and counting spanning trees. The authors develop multiple algorithmic branches (log-determinant, Schatten norms, entropy, inverse trace) and provide detailed runtime analyses, error bounds, and comparisons to classical approaches, along with discussions of practical few-depth and one-clean-qubit regimes. Overall, the paper demonstrates that quantum spectral-sum estimation can outperform several classical randomized techniques and offers a versatile toolkit for spectral graph theory tasks and related domains.
Abstract
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. The spectral sum of an PSD matrix $A$, for a function $f$, is defined as $ \text{Tr}[f(A)] = \sum_j f(λ_j)$, where $λ_j$ are the eigenvalues of $A$. Typical examples of spectral sums are the von Neumann entropy, the trace of $A^{-1}$, the log-determinant, and the Schatten $p$-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees within a graph.