Solving Sparse SDPs in Sublinear Time: A Classical Algorithm Inspired by the Quantum OR Lemma

2026-09-30

AI Summary

This study addresses the inefficiency of classical solvers for sparse semidefinite programs (SDPs) and their reliance on quantum speedups by proposing the first sublinear-time classical solver for sparse SDPs without low-rank assumptions. Methodologically, the approach decouples computational costs by classically adapting the sampling-reuse mechanism of the quantum OR lemma, introduces a randomized online learning framework to optimize precision dependence, and combines randomized Lanczos filtering with sampling estimators to efficiently approximate Gibbs state expectations. The results demonstrate that the proposed algorithm achieves near-optimal complexity at constant precision, thereby establishing that no super-quadratic quantum advantage exists for solving general sparse SDPs.

Abstract

We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved $\widetilde{O}(\sqrt{n}+\sqrt{m})$ dependence on matrix dimension $n$ and constraint number $m$. Compared with the $\widetilde{O}(mn)$ runtime of existing classical methods, this suggests a quartic quantum speedup when $m \approx n$. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let $s$ denote the the input matrix sparsity and $γ:=Rr/\varepsilon$ capture dependence on the primal $(R)$ and dual $(r)$ radii as well as target accuracy $(\varepsilon)$. When $γ^2\leq\min\{m,n/s\}$, our solver runs in time $\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)$. For $γ=O(1)$, this is $\widetilde{O}\left((n+m)s\right)$ and sublinear in the $O(mns)$ input size. Similar to the quantum algorithms, this matches known lower bounds with respect to $m$ and $n$, up to logarithmic factors. This implies that, with respect to dimensions $m$ and $n$, there is no super-quadratic quantum advantage for generic sparse SDP solving.

Problem

• Sparse Semidefinite Programs
• Sublinear Time
• Bounded-Radius Regime
• Quantum Speedup

Innovation

• Sublinear-time classical algorithm
• Sparse semidefinite programs
• Quantum OR lemma
• Randomized Lanczos filtering
• Stochastic online learning