Hybrid Oscillator–Qubit Framework for Efficient Simulation of Time Periodic Hamiltonian

Hybrid oscillator–qubit quantum processors are beneficial for problems involving both bosons and qubits. Here, we show that they also provide advantages for simulating time-periodic Hamiltonians within the Floquet–Hilbert space formalism, which replaces explicit time dependence with an auxiliary space. This formalism enables optimal simulation of time-periodic Hamiltonians and offers advantages for VQE algorithms, but becomes costly on fully qubit-based processors. We present three representations of the d-dimensional auxiliary matrices—using one, two, or d-oscillators. In the d-oscillator representation, an auxiliary matrix can be expressed through two sets of commuting beam-splitter generators. For VQE and Floquet-ADAPT-VQE algorithms, we show that the measurement cost of the auxiliary part of the extended Floquet Hamiltonian scales as O(1), in contrast to O(log(d)) for qubit-only processors. Our results open new avenues for hybrid oscillator–qubit framework in the quantum simulation of driven many-body systems.