Quantum Constrained Hamiltonian Descent

The gist: The authors propose Quantum Constrained Hamiltonian Descent (QCHD), a continuous-time quantum framework for optimization under smooth equality constraints that avoids penalty terms by constructing an intrinsic quantum Hamiltonian with a Laplace–Beltrami operator on the constraint manifold.

Classical and Quantum Mechanics of Constrained Systems

The paper begins by reviewing the classical and quantum mechanics of constrained systems, which forms the basis of QCHD. The classical formulation involves a point particle moving in an N-dimensional Euclidean space R N subject to M independent holonomic constraints f A(x) = 0. To handle the singular nature of this system, Dirac's prescription is employed to construct the Dirac bracket, which defines a Poisson structure compatible with the second-class constraints The resulting Hamiltonian for this constrained system is given by H = 1/2m Mijpipj + V(x) on Md where Mij is the projector onto the tangent space of Md.

Quantum Constrained Hamiltonian Descent (QCHD)

QCHD treats parameters x as particle positions and the cost function V(x) as potential energy, simulating quantum dynamics to find minima. The core idea is to apply Dirac's prescription to construct an intrinsic quantum Hamiltonian with a Laplace–Beltrami operator in a kinetic term determined by the induced metric on the constraint manifold. In local coordinates, this Hamiltonian coincides with that of Quantum Riemannian Hamiltonian Descent (QRHD). The evolution preserves the constraints exactly without penalty terms.

Relation to QHD with a Penalty Term

The paper analyzes the relation between QCHD and QHD supplemented by a strong penalty term (QHD+Penalty). 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, showing that intrinsic quantization (QCHD) and quantized system with a large penalty contribution (QHD+Penalty) generally lead to different quantum dynamics.

Numerical Comparison of QCHD and QHD+Penalty

Numerical comparisons were conducted for four constrained optimization problems, including spherical constraints with quadratic and Ackley potentials, elliptic constraints with a quadratic potential, and two simultaneous constraints with a nonconvex potential. The results show that QCHD converges faster than QHD+Penalty while satisfying the constraints exactly in all examples. For instance, in the spherical constraint example (5.2), QCHD reached convergence time tε of 4.45, whereas QHD+Penalty took longer for various penalty strengths κ. In the elliptic constraint example (5.4), QCHD showed faster convergence than QHD+Penalty for all tested penalty strengths.

Geometric Formulation and Effective Hamiltonian

The geometric formulation in tubular coordinates reveals that the ambient Laplacian decomposes into tangential and normal parts, with the tangential part involving the normal-bundle covariant derivative given by D alpha = ∂ alpha - A a b∂ a. The effective Hamiltonian for QHD+Penalty in the strong penalty limit is decomposed as H pen(t) = H q(t) + H ξ(t) + R (4.11> where H q(t) describes the tangential motion and H ξ(t) describes the normal motion. The geometric potential Vgeo is determined by the extrinsic curvature of Md, and for two or more constraints, by a coupling to the normal-bundle connection.

Conclusion

QCHD is a coherent quantum approach that requires no penalty term, whereas QHD+Penalty retains information about the embedding through the geometric potential and the normal-bundle connection. The numerical simulations confirm that QCHD satisfies constraints exactly and reaches convergence thresholds faster than QHD+Penalty in all tested scenarios.

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Quantum Constrained Hamiltonian Descent

Yoshihiko Abe1,2,3∗ and Ryo Nagai3,4†. In this section we ask how QHD+Penalty is related to QCHD. For a strong penalty, the wave function of QHD+Penalty is squeezed into a thin neighborhood of Md, and its motion separates into an oscillation perpendicular to Md and a motion along Md. We derived the effective Hamiltonian for the motion along Md in the strong penalty limit (κ → ∞), assuming that the penalty is the squared-distance from Md, that the perpendicular oscillation stays within the thin neighborhood, and that a(t) varies slowly compared to this oscillation. The effective Hamiltonian differs from that of QCHD by the geometric potential Vgeo, which is determined by the extrinsic curvature of Md, and, for two or more constraints, by a coupling to the normal-bundle connection, which describes how the perpendicular directions rotate along Md. Both terms survive the limit κ → ∞. Thus QCHD, which quantizes the dynamics on Md, and QHD+Penalty, which keeps the quantum dynamics in R N near Md by a penalty, lead to different quantum dynamics in general, even though they describe the same classical constrained motion in this limit. When Vgeo is constant on the manifold and the connection term is absent, the difference (4.29) reduces to an overall phase when Vgeo is constant and the connection term is absent, as for a sphere. For the penalty κ P(f A) squared used in practice, the zero-point energy of the perpendicular oscillation adds a position-dependent potential whenever ∇f varies along Md. This potential grows as √κ and is not removed by normalizing the constraint functions by their gradients, so that the result of QHD+Penalty depends on the functions f A chosen to represent Md.

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Elliptic constraint with a quadratic potential

We take the ellipse M1: x Ra squared + y Rb squared = 1, R a = 0.6, R b = 0.4, (5.8) as the constraint manifold. The geometry is the same as in Sec. 5.2, so that again the large-κ effective theory coincides with QCHD up to a phase. Both Vgeo and ∇f depend on the position. By Sec. 4, the zero-point energy 1/2nabla f p 2κ/m of the normal oscillation is therefore position dependent and grows as √κ, so that the large-κ limit of QHD+Penalty does not approach QCHD.

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Two constraints in R cubed with a nonconvex potential

Finally, we consider an example with N = 3 and M = 2. This example has two constraints (M = 2) and a one-dimensional constraint manifold in a three-dimensional ambient space. Its geometry is again such that the large-κ effective theory coincides with QCHD up to a constant.

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Example Indicator QCHD QHD+Penalty Ratio

(1) sphere, quadratic tε (ε = 3 × 10-2) 4.5 8.5–11.5 1.9–2.

1. Quantum Constrained Hamiltonian Descent (QCHD) enables optimization under smooth equality constraints without requiring intermediate measurements or tuning penalty terms, allowing AI systems to solve constrained problems directly using intrinsic quantum dynamics on the constraint manifold Md.

2. The effective Hamiltonian for QHD+Penalty can be derived from the strong penalty limit to identify an extrinsic-curvature potential and a coupling to the normal-bundle connection, which can guide AI systems in understanding how constraint enforcement affects quantum dynamics compared to unconstrained simulations.

3. AI systems using QCHD can achieve faster optimization and improved concentration near global minima for initial states, as shown numerically for circular, elliptical, and two simultaneous constraints when compared against methods supplemented by a strong penalty term.

4. The framework allows AI systems to distinguish between intrinsic quantization (QCHD) and quantized system with a large penalty contribution (QHD+Penalty), revealing that intrinsic quantization (QCHD) and quantized system with a large penalty contribution (QHD+Penalty) generally lead to different quantum dynamics.