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Quantum SDP-Solvers: Better upper and lower bounds

Joran van Apeldoorn, András Gilyén, Sander Gribling, Ronald de Wolf

TL;DR

The paper advances quantum SDP-solving by presenting an improved solver that achieves substantially better dependence on width-related parameters than prior work, notably replacing Gibbs-state-based procedures with purified Gibbs sampling and leveraging a 2-sparse dual update. It introduces two broadly useful tools—a Fourier-series/Hamiltonian-simulation framework for smoothly applying functions of Hamiltonians and a generalized minimum-finding method—that have potential beyond SDP-solving. The authors also establish quantitative lower bounds showing linear-in-$mn$ scaling for worst-case quantum LP/SDP solvers when $m oughly n$, highlighting fundamental limits and the need for problem-specific oracle design to achieve speedups. Overall, the results sharpen the landscape of quantum SDP-solvers, provide practical algorithmic ingredients, and set the stage for targeted, structure-aware quantum optimization methods with potential real-world impact in regimes where $Rr/varepsilon$ is small relative to problem size.

Abstract

Brandão and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.

Quantum SDP-Solvers: Better upper and lower bounds

TL;DR

The paper advances quantum SDP-solving by presenting an improved solver that achieves substantially better dependence on width-related parameters than prior work, notably replacing Gibbs-state-based procedures with purified Gibbs sampling and leveraging a 2-sparse dual update. It introduces two broadly useful tools—a Fourier-series/Hamiltonian-simulation framework for smoothly applying functions of Hamiltonians and a generalized minimum-finding method—that have potential beyond SDP-solving. The authors also establish quantitative lower bounds showing linear-in- scaling for worst-case quantum LP/SDP solvers when , highlighting fundamental limits and the need for problem-specific oracle design to achieve speedups. Overall, the results sharpen the landscape of quantum SDP-solvers, provide practical algorithmic ingredients, and set the stage for targeted, structure-aware quantum optimization methods with potential real-world impact in regimes where is small relative to problem size.

Abstract

Brandão and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension of the problem and the number of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with when , which is the same as classical.

Paper Structure

This paper contains 44 sections, 51 theorems, 189 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Semidefinite programs
  3. Classical solvers for LPs and SDPs
  4. Quantum SDP-solvers: the Brandão-Svore algorithm
  5. Our results
  6. Improved quantum SDP-solver
  7. Tools that may be of more general interest
  8. Implementing smooth functions of a given Hamiltonian.
  9. A generalized minimum-finding algorithm.
  10. Lower bounds
  11. Subsequent work.
  12. Organization.
  13. An improved quantum SDP-solver
  14. Notation/Assumptions.
  15. Input oracles:
  16. ...and 29 more sections

Key Result

Theorem 1

Instantiating Meta-Algorithm alg:AKSDP using the trace calculation algorithm from Section sec:trCalc and the oracle from Section sec:oracle (with width-bound $w:=r+1$), and using this to do a binary search for $\hbox{\rm OPT}\in[-R,R]$ (using different guesses $\alpha$ for $\hbox{\rm OPT}$), gives a queries to the input matrices and the same order of other gates.

Figures (2)

  • Figure 1: The region $\mathcal{G}$ in light blue. The borders of two quadrants $\mathcal{G}_N$ have been drawn by thick dashed blue lines. The red dot at the beginning of the arrow is the point $(\alpha/r,c^{\prime}/r)$.
  • Figure 2: Illustration of $\mathcal{G}$ with the points $p_j,p_k$ and the angles $\angle \ell_j L_1,\angle L_1 L_2,\angle L_2\ell_k$ drawn in. Clearly the line $\overline{p_jp_k}$ only crosses $\mathcal{G}$ when the total angle is less than $\pi$.

Theorems & Definitions (104)

  • Theorem 1
  • Theorem 2: arora2016CombPrimDualSDP
  • Lemma 3: arora2016CombPrimDualSDP
  • Definition 4: Width of Oracle$_{\varepsilon}$
  • Theorem 5: arora2016CombPrimDualSDP
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Corollary 8
  • ...and 94 more