Quantum algorithms for spectral sums
This work addresses computational bottlenecks in matrix analysis for researchers in quantum computing and spectral graph theory, though it is incremental as it builds on prior quantum algorithms.
The authors tackled the problem of estimating spectral sums of matrices, such as von Neumann entropy and log-determinant, by proposing new quantum algorithms that achieve sub-linear runtime in matrix size and compete with classical randomized methods, offering polynomial improvements over existing quantum approaches.
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.