Quantum compiler for 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 (quantum states) on a reproducing kernel Hilbert space $\mathcal{H}$. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on $\mathcal{H}$ such that quantum mechanical expectation values are consistent with pointwise function evaluation. Meanwhile, quantum states and observables on $\mathcal{H}$ evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. To achieve an exponential quantum advantage, the state of the quantum system is projected onto a density operator on a $2^n$-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 an $O(n)$ number of gates and no interchannel communication. Furthermore, the circuit features a quantum Fourier transform stage with $O(n^2)$ gates, which makes predictions of observables possible by measurement in the standard computational basis. In this talk, we describe this ``quantum compiler'' framework, and illustrate it with applications to low-dimensional dynamical systems.