Equilibrium Learning of Quantum Circuits

Equilibrium learning trains a physical system using observables of its relaxed states. We extend this principle to gate-based quantum circuits by embedding a parametrized circuit in a Feynman-Kitaev (FK) Hamiltonian and imposing the compilation target either through an output clamping term or through a target-dependent cyclic condition on the clock register. The ground state energies and the overlaps of the corresponding ground states with the free history state define faithful cost functions for unitary and state compilation. We show that these costs are nonlinear functions of the compilation infidelity and therefore reshape the optimization landscape. We show that starting from the free history state, quantum phase estimation can prepare the relevant ground states and estimate the cost functions with polynomial resources. In two-qubit benchmarks, optimization with the equilibrium cost functions reaches a fixed fidelity threshold in fewer sweeps than standard compilation infidelity cost function for many instances. The results demonstrate the potential for encoding quantum learning problems as computation of equilibrium observables.