Fully quantum training of parametric circuits with Quantum Hamiltonian Descent

Abstract

Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.

Quantum Hamiltonian Descent (QHD) optimizes continuous parameters by evolving a wavefunction under $H(t)=a(t)p^2/2+b(t)f(x)$. This paper studies its digital implementation and extends it to circuit-defined expectation-value costs without first estimating the cost landscape classically. The simulation integrates the kinetic term exactly using Fourier transforms and discretizes the interaction-picture potential. A state-dependent error analysis bounds the kinetic energy by $\langle T_0\rangle\leq (b/a)\operatorname{osc}(f)$ for nondecreasing $b/a$ and uniform initialization. For sufficiently smooth periodic costs, this gives a first-order simulation-error bound independent of the number $q$ of qubits per parameter. For Pauli-rotation circuits, the trigonometric structure yields $O(dR\Delta t^2)$ error at fixed parameter-reuse degree and observable norm. Reparametrizing time by $\tau=\int a(t)dt$ separates the ideal search curve $r(\tau)=b/a$ from a gauge used to minimize simulation cost. An online potential step coherently controls the circuit with the parameter register, applies observable evolution, uncomputes, and resets an auxiliary register. Its dephasing error scales as $\sum_j\varphi_j^2$, independently of $d$ and $q$. For exponential search curves, constant $b$ minimizes this bound, giving $\delta_{\rm on}\lesssim(r_f-r_0)^2/(k^2R)$. Five simulated circuit costs with one or two parameters achieve global-basin probabilities of $0.74$–$0.94$ using $2^{15}$ steps. No optimization speedup or equal-circuit-evaluation comparison with classical optimizers is established. Search complexity remains landscape-gap dependent; slower searches and higher precision increase online cost substantially, motivating local classical refinement. Larger-scale learning applications and tighter error analyses remain open.