Koopman Operator Theory for Quantum Simulation of Classical Dynamics

We present a framework for simulating a measure-preserving, ergodic dynamical system by a finite-dimensional quantum system amenable to implementation on a quantum computer. The framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, H, of functions on classical state space. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on H such that quantum mechanical expectation values are consistent with pointwise function evaluation. Meanwhile, quantum states and observables on H evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. The state of the quantum system is projected onto a finite-rank density operator on a 2ⁿ-dimensional tensor product Hilbert space associated with n qubits. The finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an n-channel quantum circuit with O(n) gates. Furthermore, the circuit features a quantum Fourier transform stage with O(n²) gates, which makes predictions of observables possible by measurement in the standard computational basis. We illustrate our approach with quantum circuit simulations for low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One.