  We present a framework for simulating measure-preserving, ergodic dynamical
systems on a quantum computer. Our approach is based on a quantum feature map
representing classical states by density operators (quantum states) on a
reproducing kernel Hilbert space (RKHS) $\mathcal H $ of functions on classical
state space. 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. Quantum
states and observables on $\mathcal H$ evolve under the action of a unitary
group of Koopman operators in a compatible manner with classical dynamical
evolution. To achieve quantum parallelism, the state of the quantum system is
projected onto a finite-rank density operator on a $2^N$-dimensional tensor
product Hilbert space associated with $N$ qubits. By employing discrete
Fourier-Walsh transforms of spectral functions, the Hamiltonian of the
finite-dimensional quantum system is decomposed into a sum of mutually
commuting operators of pure tensor product form, implementable as an
$N$-channel quantum circuit. In the special case of quasiperiodic dynamics, no
interchannel communication is necessary. Circuits of higher complexity are
required to simulate chaotic dynamics. The output of the quantum compiler is a
stochastic simulator of the evolution of classical observables, realized
through projective quantum measurements. Numerical examples, including a
quantum circuit implementation using the Qiskit software development kit,
demonstrate the consistency of the quantum compiler for simple dynamical
systems.
