Classical algorithm solves sparse semidefinite programs faster than before
Solving Sparse SDPs in Sublinear Time: A Classical Algorithm Inspired by the Quantum OR Lemma
Data Structures and Algorithms
Summary
Solving certain mathematical problems called sparse semidefinite programs (SDPs) usually takes a long time, especially as the problem sizes grow. Previously, only quantum algorithms could solve these problems significantly faster, but the authors have developed a new classical method that matches much of that speedup. Their approach uses a clever way to estimate many related values all at once, inspired by quantum techniques but done with classical computation. This means some advanced problems can now be tackled faster on regular computers without needing quantum hardware.
What this means in practice
- •For optimization engineers: Solve large sparse semidefinite programs faster on classical hardware for systems modeling and control problems.
- •For machine learning engineers: Improve runtime of training models that rely on semidefinite programming without quantum resources.
Authors
Fernando G. S. L. Brandão, Alexander M. Dalzell, András Gilyén, Francisca Vasconcelos
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.