Quantum Constrained Hamiltonian Descent
Abstract
We propose Quantum Constrained Hamiltonian Descent (QCHD), a continuous-time quantum framework for optimization under smooth equality constraints. The unconstrained counterpart, Quantum Hamiltonian Descent (QHD), minimizes a cost function by simulating the Schrödinger dynamics of a particle in the potential given by the cost function. Applying Dirac's formalism to the classical dynamics with constraints, we construct an intrinsic quantum Hamiltonian with a Laplace--Beltrami operator on the constraint manifold in a kinetic term. Its evolution preserves the constraints exactly without penalty terms and is expected to solve quantum-dynamically the constrained optimization problem. We analyze its relation to Quantum Hamiltonian Descent supplemented by a strong penalty term. For a squared-distance penalty, the effective tangential dynamics acquires an extrinsic-curvature potential and, in nontrivial normal-rotation sectors, a coupling to the normal-bundle connection. These embedding-dependent terms remain in the strong-penalty limit, generally yielding quantum dynamics different from QCHD. Numerical comparisons for circular, elliptical, and two simultaneous constraints show faster optimization and improved concentration near global minima for the initial states and finite penalty strengths considered.
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BibTeX
@misc{abe2026quantum,
title = {{Quantum Constrained Hamiltonian Descent}},
author = {Yoshihiko Abe and Ryo Nagai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31170},
eprint = {2610.31170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31170},
}
RIS
TY - UNPB
TI - Quantum Constrained Hamiltonian Descent
AU - Yoshihiko Abe
AU - Ryo Nagai
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31170
UR - https://arxiv.org/abs/2610.31170
AB - We propose Quantum Constrained Hamiltonian Descent (QCHD), a continuous-time quantum framework for optimization under smooth equality constraints. The unconstrained counterpart, Quantum Hamiltonian Descent (QHD), minimizes a cost function by simulating the Schrödinger dynamics of a particle in the potential given by the cost function. Applying Dirac's formalism to the classical dynamics with constraints, we construct an intrinsic quantum Hamiltonian with a Laplace--Beltrami operator on the constraint manifold in a kinetic term. Its evolution preserves the constraints exactly without penalty terms and is expected to solve quantum-dynamically the constrained optimization problem. We analyze its relation to Quantum Hamiltonian Descent supplemented by a strong penalty term. For a squared-distance penalty, the effective tangential dynamics acquires an extrinsic-curvature potential and, in nontrivial normal-rotation sectors, a coupling to the normal-bundle connection. These embedding-dependent terms remain in the strong-penalty limit, generally yielding quantum dynamics different from QCHD. Numerical comparisons for circular, elliptical, and two simultaneous constraints show faster optimization and improved concentration near global minima for the initial states and finite penalty strengths considered.
ER -