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On Speedups for Convex Optimization via Quantum Dynamics

Shouvanik Chakrabarti, Dylan Herman, Jacob Watkins, Enrico Fontana, Brandon Augustino, Junhyung Lyle Kim, Marco Pistoia

TL;DR

The paper analyzes quantum speedups for unconstrained convex optimization by simulating Quantum Hamiltonian Descent (QHD) dynamics. It provides rigorous, dimension-aware resource bounds for real-space Schrödinger dynamics with black-box potentials via a pseudo-spectral real-space method and integrates these into a comprehensive optimization framework. The main result is a near-optimal upper bound of $ ilde{O}(d^{3/2}(GR/ ext{ε})^{2})$ queries for nonsmooth convex optimization via QHD, paired with a matching lower bound under a no-fast-forwarding assumption, implying no speedup in the noiseless zeroth-order setting. In noisy and stochastic settings, the work demonstrates super-quadratic quantum advantages over classical algorithms and constructs a quantum stochastic-optimization algorithm with superior high-dimensional performance. Collectively, these results establish the first rigorous quantum speedups for convex optimization achieved through a dynamical quantum algorithm, while clarifying the fundamental limits imposed by discretization and the potential benefits in noisy and stochastic regimes.

Abstract

We explore the potential for quantum speedups in convex optimization using discrete simulations of the Quantum Hamiltonian Descent (QHD) framework, as proposed by Leng et al., and establish the first rigorous query complexity bounds. We develop enhanced analyses for quantum simulation of Schrödinger operators with black-box potential via the pseudo-spectral method, providing explicit resource estimates independent of wavefunction assumptions. These bounds are applied to assess the complexity of optimization through QHD. Our findings pertain to unconstrained convex optimization in $d$ dimensions. In continuous time, we demonstrate that QHD, with suitable parameters, can achieve arbitrarily fast convergence rates. The optimization speed limit arises solely from the discretization of the dynamics, mirroring a property of the classical dynamics underlying QHD. Considering this cost, we show that a $G$-Lipschitz convex function can be optimized to an error of $ε$ with $\widetilde{\mathcal{O}}(d^{1.5}G^2 R^2/ε^2)$ queries. Moreover, under reasonable assumptions on the complexity of Hamiltonian simulation, $\widetildeΩ(d/ε^2)$ queries are necessary. Thus, QHD does not offer a speedup over classical zeroth order methods with exact oracles. However, we demonstrate that the QHD algorithm tolerates $\widetilde{\mathcal{O}}(ε^3/d^{1.5}G^2 R^2)$ noise in function evaluation. We show that QHD offers a super-quadratic query advantage over all known classical algorithms tolerating this level of evaluation noise in the high-dimension regime. Additionally, we design a quantum algorithm for stochastic convex optimization that provides a super-quadratic speedup over all known classical algorithms in the high-dimension regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization achieved through a dynamical algorithm.

On Speedups for Convex Optimization via Quantum Dynamics

TL;DR

The paper analyzes quantum speedups for unconstrained convex optimization by simulating Quantum Hamiltonian Descent (QHD) dynamics. It provides rigorous, dimension-aware resource bounds for real-space Schrödinger dynamics with black-box potentials via a pseudo-spectral real-space method and integrates these into a comprehensive optimization framework. The main result is a near-optimal upper bound of queries for nonsmooth convex optimization via QHD, paired with a matching lower bound under a no-fast-forwarding assumption, implying no speedup in the noiseless zeroth-order setting. In noisy and stochastic settings, the work demonstrates super-quadratic quantum advantages over classical algorithms and constructs a quantum stochastic-optimization algorithm with superior high-dimensional performance. Collectively, these results establish the first rigorous quantum speedups for convex optimization achieved through a dynamical quantum algorithm, while clarifying the fundamental limits imposed by discretization and the potential benefits in noisy and stochastic regimes.

Abstract

We explore the potential for quantum speedups in convex optimization using discrete simulations of the Quantum Hamiltonian Descent (QHD) framework, as proposed by Leng et al., and establish the first rigorous query complexity bounds. We develop enhanced analyses for quantum simulation of Schrödinger operators with black-box potential via the pseudo-spectral method, providing explicit resource estimates independent of wavefunction assumptions. These bounds are applied to assess the complexity of optimization through QHD. Our findings pertain to unconstrained convex optimization in dimensions. In continuous time, we demonstrate that QHD, with suitable parameters, can achieve arbitrarily fast convergence rates. The optimization speed limit arises solely from the discretization of the dynamics, mirroring a property of the classical dynamics underlying QHD. Considering this cost, we show that a -Lipschitz convex function can be optimized to an error of with queries. Moreover, under reasonable assumptions on the complexity of Hamiltonian simulation, queries are necessary. Thus, QHD does not offer a speedup over classical zeroth order methods with exact oracles. However, we demonstrate that the QHD algorithm tolerates noise in function evaluation. We show that QHD offers a super-quadratic query advantage over all known classical algorithms tolerating this level of evaluation noise in the high-dimension regime. Additionally, we design a quantum algorithm for stochastic convex optimization that provides a super-quadratic speedup over all known classical algorithms in the high-dimension regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization achieved through a dynamical algorithm.

Paper Structure

This paper contains 37 sections, 58 theorems, 329 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. Real space Hamiltonian simulation
  4. Convex Optimization
  5. Schedule Invariance and Lower Bounds on Query Complexity
  6. Prospects for Quantum Speedup for Convex Optimization Algorithms
  7. Comparison to Classical Algorithms for Noiseless Convex Optimization
  8. Comparison to Classical Algorithms for Noisy Convex Optimization
  9. Comparison to Quantum Algorithms for Noisy Convex Optimization
  10. Comparison to Algorithms for Stochastic Convex Optimization
  11. Related Work
  12. Background
  13. Preliminaries and Notation
  14. Hilbert Spaces
  15. Time Dependent Hamiltonian Simulation
  16. ...and 22 more sections

Key Result

Theorem 2.1

Consider the Schrödinger equation subject to initial data $\Phi(x,0) = \Phi_0(x)$, where $a, b$ are sufficiently smooth functions of time and $f$ is a $G$-Lipschitz function with bound $\Lambda \geq \left\| f \right\|_\infty$, accessed through a $\epsilon_f$-noisy binary quantum oracle $O_f$. If $\epsilon_f = \widetilde{{\cal O}}(\e

Theorems & Definitions (110)

  • Theorem 2.1: Informal version of Thm \ref{['thm:master_simulation_thm']}
  • Theorem 2.2
  • Definition 2.1: Exponential Schedules with Convergence Rate $e^{-ct}$
  • Definition 2.2: Polynomial Schedules with Convergence Rate $t^{-k}$
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1: $\epsilon_f$-accurate binary oracle
  • Definition 3.2: $\epsilon_p$-accurate phase oracle
  • Definition 4.1: $R$-restriction
  • Theorem 4.1
  • ...and 100 more