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Quantum speedups for linear programming via interior point methods

Simon Apers, Sander Gribling

TL;DR

This work demonstrates a quantum interior-point method for linear programs with many constraints and few variables, achieving a sublinear quantum query complexity in tall-LP regimes by combining spectral approximation of Hessians with quantum gradient and matrix-vector techniques. The approach leverages three barrier families—logarithmic, volumetric, and Lewis-weight barriers—and introduces two independent quantum subroutines: (i) spectral approximation of $B^TB$ via leverage-score sampling and Grover search, and (ii) gradient estimation via quantum multivariate mean estimation with preconditioning to remove condition-number dependence. The main results include concrete gate/row-query complexities for each barrier, lower bounds matching the achieved scaling in key regimes, and a benchmark comparison to cutting-plane methods, highlighting both the potential and limitations of quantum approaches to tall LPs. The findings open avenues for faster quantum optimization in data-rich settings and motivate further work on improving the $d$-dependence, dynamic data-structure strategies, and broader application to LPs and regression-like problems.

Abstract

We describe a quantum algorithm based on an interior point method for solving a linear program with $n$ inequality constraints on $d$ variables. The algorithm explicitly returns a feasible solution that is $\varepsilon$-close to optimal, and runs in time $\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$ which is sublinear for tall linear programs (i.e., $n \gg d$). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [FOCS '14]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the \emph{spectral approximation} of $A^T A$ for a tall matrix $A \in \mathbb R^{n \times d}$. The algorithm uses leverage score sampling in combination with Grover search, and returns a $δ$-approximation by making $O(\sqrt{nd}/δ)$ row queries to $A$. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [FOCS '20]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi and Jerbi [STOC '22]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by pre-conditioning our random variable using our quantum algorithm for spectral approximation.

Quantum speedups for linear programming via interior point methods

TL;DR

This work demonstrates a quantum interior-point method for linear programs with many constraints and few variables, achieving a sublinear quantum query complexity in tall-LP regimes by combining spectral approximation of Hessians with quantum gradient and matrix-vector techniques. The approach leverages three barrier families—logarithmic, volumetric, and Lewis-weight barriers—and introduces two independent quantum subroutines: (i) spectral approximation of via leverage-score sampling and Grover search, and (ii) gradient estimation via quantum multivariate mean estimation with preconditioning to remove condition-number dependence. The main results include concrete gate/row-query complexities for each barrier, lower bounds matching the achieved scaling in key regimes, and a benchmark comparison to cutting-plane methods, highlighting both the potential and limitations of quantum approaches to tall LPs. The findings open avenues for faster quantum optimization in data-rich settings and motivate further work on improving the -dependence, dynamic data-structure strategies, and broader application to LPs and regression-like problems.

Abstract

We describe a quantum algorithm based on an interior point method for solving a linear program with inequality constraints on variables. The algorithm explicitly returns a feasible solution that is -close to optimal, and runs in time which is sublinear for tall linear programs (i.e., ). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [FOCS '14]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the \emph{spectral approximation} of for a tall matrix . The algorithm uses leverage score sampling in combination with Grover search, and returns a -approximation by making row queries to . This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [FOCS '20]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi and Jerbi [STOC '22]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by pre-conditioning our random variable using our quantum algorithm for spectral approximation.
Paper Structure (44 sections, 38 theorems, 91 equations, 3 tables, 7 algorithms)

This paper contains 44 sections, 38 theorems, 91 equations, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Interior point methods for linear programs
  3. Quantum algorithms
  4. Quantum spectral approximation
  5. Lewis weights
  6. Quantum gradient and matrix-vector approximation
  7. Main result and discussion
  8. Tall LPs.
  9. Lower bound.
  10. Benchmark.
  11. Open questions.
  12. Related work
  13. Quantum interior point methods.
  14. Quantum gradient and Hessian approximation.
  15. Quantum linear algebra.
  16. ...and 29 more sections

Key Result

Theorem 1

Consider query access to a matrix $B \in \mathbb{R}^{n \times d}$ with row sparsity $r$. For any $0 < \varepsilon \leq 1$ and $p \geq 2$, there is a quantum algorithm that provides query access to $\varepsilon$-multiplicative approximations of the Lewis weights $w^{(p)}_i(B)$. The algorithm has a pr

Theorems & Definitions (66)

  • Theorem : Quantum Lewis weights, informal version of \ref{['thm:quantum-Lewis']}
  • Theorem 1.0: Approximate matrix-vector product
  • Theorem 1.1: Quantum IPM for LP
  • Theorem 3.1: Quantum spectral approximation
  • Theorem 3.1: Quantum leverage score approximation
  • Remark 3.2: Graph sparsification
  • Lemma 3.3: Consequence of the matrix Chernoff bound Tropp11:matrixconcentration, cohen2015uniform
  • Theorem 3.4: cohen2015uniform
  • Lemma 3.5: Repeated halving
  • proof
  • ...and 56 more