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Quantum Optimization via Gradient-Based Hamiltonian Descent

Jiaqi Leng, Bin Shi

TL;DR

This work addresses unconstrained continuous optimization by augmenting Quantum Hamiltonian Descent (QHD) with gradient information to form gradient-based QHD. It develops a continuous-time convergence theory using Lyapunov functionals that yields $O(t^{-2})$ rates for both function-value convergence and gradient-norm convergence under appropriate parameter choices, and formulates a discrete-time quantum algorithm with gate complexity linear in problem dimension $d$. The authors show that the gradient-enhanced dynamics recover QHD as a special case and provide a practical quantum-complexity analysis via operator splitting and QSVT-based simulation. Numerical experiments in 2D demonstrate that gradient-based QHD consistently outperforms QHD and classical baselines on convex and non-convex landscapes, highlighting the potential of gradient information to boost quantum optimization. The work lays groundwork for scalable quantum optimization while outlining avenues for high-dimensional extensions and gradient-estimation enhancements.

Abstract

With rapid advancements in machine learning, first-order algorithms have emerged as the backbone of modern optimization techniques, owing to their computational efficiency and low memory requirements. Recently, the connection between accelerated gradient methods and damped heavy-ball motion, particularly within the framework of Hamiltonian dynamics, has inspired the development of innovative quantum algorithms for continuous optimization. One such algorithm, Quantum Hamiltonian Descent (QHD), leverages quantum tunneling to escape saddle points and local minima, facilitating the discovery of global solutions in complex optimization landscapes. However, QHD faces several challenges, including slower convergence rates compared to classical gradient methods and limited robustness in highly non-convex problems due to the non-local nature of quantum states. Furthermore, the original QHD formulation primarily relies on function value information, which limits its effectiveness. Inspired by insights from high-resolution differential equations that have elucidated the acceleration mechanisms in classical methods, we propose an enhancement to QHD by incorporating gradient information, leading to what we call gradient-based QHD. Gradient-based QHD achieves faster convergence and significantly increases the likelihood of identifying global solutions. Numerical simulations on challenging problem instances demonstrate that gradient-based QHD outperforms existing quantum and classical methods by at least an order of magnitude.

Quantum Optimization via Gradient-Based Hamiltonian Descent

TL;DR

This work addresses unconstrained continuous optimization by augmenting Quantum Hamiltonian Descent (QHD) with gradient information to form gradient-based QHD. It develops a continuous-time convergence theory using Lyapunov functionals that yields rates for both function-value convergence and gradient-norm convergence under appropriate parameter choices, and formulates a discrete-time quantum algorithm with gate complexity linear in problem dimension . The authors show that the gradient-enhanced dynamics recover QHD as a special case and provide a practical quantum-complexity analysis via operator splitting and QSVT-based simulation. Numerical experiments in 2D demonstrate that gradient-based QHD consistently outperforms QHD and classical baselines on convex and non-convex landscapes, highlighting the potential of gradient information to boost quantum optimization. The work lays groundwork for scalable quantum optimization while outlining avenues for high-dimensional extensions and gradient-estimation enhancements.

Abstract

With rapid advancements in machine learning, first-order algorithms have emerged as the backbone of modern optimization techniques, owing to their computational efficiency and low memory requirements. Recently, the connection between accelerated gradient methods and damped heavy-ball motion, particularly within the framework of Hamiltonian dynamics, has inspired the development of innovative quantum algorithms for continuous optimization. One such algorithm, Quantum Hamiltonian Descent (QHD), leverages quantum tunneling to escape saddle points and local minima, facilitating the discovery of global solutions in complex optimization landscapes. However, QHD faces several challenges, including slower convergence rates compared to classical gradient methods and limited robustness in highly non-convex problems due to the non-local nature of quantum states. Furthermore, the original QHD formulation primarily relies on function value information, which limits its effectiveness. Inspired by insights from high-resolution differential equations that have elucidated the acceleration mechanisms in classical methods, we propose an enhancement to QHD by incorporating gradient information, leading to what we call gradient-based QHD. Gradient-based QHD achieves faster convergence and significantly increases the likelihood of identifying global solutions. Numerical simulations on challenging problem instances demonstrate that gradient-based QHD outperforms existing quantum and classical methods by at least an order of magnitude.

Paper Structure

This paper contains 41 sections, 9 theorems, 80 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Warm-up: gradient-based QHD v.s. QHD
  3. Overview of contributions
  4. Organization.
  5. Related work
  6. NAG-related algorithms and ODEs.
  7. Quantum algorithms for unconstrained optimization.
  8. Gradient-based Hamiltonian dynamics
  9. Classical Hamiltonian flows with gradient
  10. Connection with high-resolution ODEs.
  11. Canonical quantization
  12. Connection with the original QHD.
  13. Convergence analysis
  14. Case 1: convergence to global minimum
  15. Case 2: convergence to first-order stationary point
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\beta = 0$ and $\gamma \ge \max(3\alpha, 0)$ for any $\alpha \in \mathbb{R}$. For any $1/\alpha \ge T_0 > 0$, we denote $\Psi(t,x)$ as the solution to the PDE eqn:grad-qhd for $t \ge T_0$. Let $X_t$ be a random variable distributed according to the probability density $|\Psi(t,x)|^2$. Then, for where $\mathscr{K}_0 = T^{-4}_0\langle \Psi(T_0)|(-\Delta)|\Psi(T_0)\rangle$ and In other words, $

Figures (6)

  • Figure 1: Numerical comparison of successful probability across iterations for both QHD and gradient-based QHD applied to the Styblinski-Tang function. $\pmb{P}_k$ denotes the success probability at iteration $k$.
  • Figure 2: Numerical performance comparison of various algorithms on the Styblinski-Tang function.
  • Figure 3: Numerical comparison of various optimization algorithms on the convex objective function \ref{['eqn:convex-obj']}, including function values and success probability.
  • Figure 4: Numerical comparison of various optimization algorithms on the Michalewicz function, including function values and success probability.
  • Figure 5: Numerical comparison of various optimization algorithms on the Cube-Wave function, including function values and success probability.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • Lemma 3: Commutation relations
  • Theorem 4
  • Remark 2
  • Lemma 5
  • Remark 3
  • Remark 4
  • Theorem 6
  • ...and 10 more