[Paper Review] Quantum Hamiltonian Descent

Jiaqi Leng, Ethan Hickman+2 more|arXiv (Cornell University)|Mar 2, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces Quantum Hamiltonian Descent (QHD), a novel quantum optimization algorithm derived from the path integral formulation of classical gradient descent dynamics. By leveraging quantum tunneling through non-classical trajectories, QHD outperforms classical solvers and the quantum adiabatic algorithm on non-convex quadratic programs up to 75 dimensions when implemented on D-Wave's quantum processor, demonstrating a significant quantum advantage in solution quality and time-to-solution.

ABSTRACT

Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to quantum speedups in optimization relies on the quantum acceleration of intermediate steps of classical algorithms, while keeping the overall algorithmic trajectory and solution quality unchanged. We propose Quantum Hamiltonian Descent (QHD), which is derived from the path integral of dynamical systems referring to the continuous-time limit of classical gradient descent algorithms, as a truly quantum counterpart of classical gradient methods where the contribution from classically-prohibited trajectories can significantly boost QHD's performance for non-convex optimization. Moreover, QHD is described as a Hamiltonian evolution efficiently simulatable on both digital and analog quantum computers. By embedding the dynamics of QHD into the evolution of the so-called Quantum Ising Machine (including D-Wave and others), we empirically observe that the D-Wave-implemented QHD outperforms a selection of state-of-the-art gradient-based classical solvers and the standard quantum adiabatic algorithm, based on the time-to-solution metric, on non-convex constrained quadratic programming instances up to 75 dimensions. Finally, we propose a "three-phase picture" to explain the behavior of QHD, especially its difference from the quantum adiabatic algorithm.

Motivation & Objective

• To develop a truly quantum counterpart of classical gradient descent that leverages quantum tunneling to escape spurious local minima.
• To identify a quantum algorithm that improves solution quality beyond classical methods by modifying the algorithmic trajectory, not just accelerating subroutines.
• To design a quantum optimization framework efficiently simulatable on both digital and analog quantum computers.
• To empirically validate QHD’s performance against state-of-the-art classical and quantum solvers on constrained non-convex quadratic programming problems.
• To provide a theoretical and operational framework explaining QHD’s behavior through a 'three-phase picture' distinct from the quantum adiabatic algorithm.

Proposed method

• QHD is derived via path integral quantization of the Bregman-Lagrangian dynamical system, representing the continuous-time limit of classical gradient descent.
• The algorithm is formulated as a Hamiltonian evolution governed by the Schrödinger equation, enabling efficient simulation on both digital and analog quantum computers.
• The dynamics are embedded into the Quantum Ising Machine (e.g., D-Wave), allowing empirical evaluation on real hardware.
• A product-formula method is used for digital implementation, with spatial discretization controlled by qubit precision (q=3,16,32).
• Quantum subroutines such as floating-point adders, multipliers, and approximate QFT are used, with T-counts estimated for fault-tolerant cost analysis.
• The algorithm uses iterative evolution steps involving phase estimation and controlled operations on the Hamiltonian, with a time evolution parameter T≈681 in D-Wave experiments.

Experimental results

Research questions

• RQ1 Can a quantum algorithm be constructed that fundamentally alters the trajectory of gradient descent by including classically prohibited paths?
• RQ2 Does the inclusion of non-classical trajectories in quantum optimization lead to improved solution quality and faster convergence on non-convex problems?
• RQ3 How does QHD compare to classical gradient-based solvers and the standard quantum adiabatic algorithm in terms of time-to-solution on constrained quadratic programs?
• RQ4 What is the resource cost of implementing QHD on near-term digital quantum computers, and is it feasible for real-world problems?
• RQ5 What explains the performance difference between QHD and the quantum adiabatic algorithm, and can a three-phase model explain this behavior?

Key findings

• QHD outperforms a selection of state-of-the-art classical gradient-based solvers and the standard quantum adiabatic algorithm on non-convex constrained quadratic programming instances up to 75 dimensions, based on the time-to-solution metric.
• The D-Wave-implemented QHD achieves better solution quality and faster convergence than classical solvers, particularly on problems with high non-convexity and sparsity.
• The digital implementation of QHD requires over 5×10⁸ T-gates even at low precision (3-qubit resolution), exceeding the capabilities of near-term digital quantum hardware.
• The T-count scales significantly with problem dimension and sparsity, with estimates reaching 2.67×10¹⁰ for 75-dimensional problems at 32-qubit precision.
• The proposed 'three-phase picture' explains QHD’s behavior as distinct from the adiabatic algorithm, highlighting its ability to exploit quantum tunneling and non-classical path contributions.
• QHD’s performance advantage stems from its ability to access classically forbidden trajectories, enabling escape from local minima that trap classical gradient descent.