The Quantum Circuit Trains Its Parameters as Quantum Matter

Parametric quantum circuits are usually optimized by a classical loop that queries costs or gradients one point at a time. Quantum Hamiltonian Descent instead represents parameters as the position of a quantum particle moving through the cost landscape. This work places the circuit being trained on a second register and lets the parameter register control it coherently, creating a fully quantum online step. For smooth periodic extensions, the authors prove a simulation-error bound independent of grid resolution and at most linear in the number of circuit parameters. They also reduce the two schedule functions to one curve governing adiabaticity and use the remaining freedom to lower gate count. Benchmark simulations increased the probability of finding global minima at equal cost; hardware-scale advantage remains unshown.