Quantum system compression: A Hamiltonian guided walk through Hilbert space

We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert-space dimension with the number of particles. For a constant Hamiltonian system of Hilbert-space dimension n with frequencies ranging from fmin to fmax, we show via a proper orthogonal decomposition that for a run time T the dominant dynamics are compressed in the neighborhood of a subspace whose dimension is the smallest integer larger than the time-bandwidth product Δ=(fmax-fmin)T. We also show how the distribution of initial states can further compress the system dimension. Under the stated conditions, the time-bandwidth estimate reveals the existence of an effective compressed model whose dimension is derived solely from system properties and not dependent on the particular implementation of a variational simulator, such as a machine learning system, or quantum device, or possibly even specially adapting traditional methods of solving the time-dependent Schrödinger equation. However, finding an efficient solution procedure is dependent on the simulator implementation, which is not discussed in this paper. In addition, we show that the compression rendered by the proper orthogonal decomposition encoding method can be further strengthened via a multilayer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. The essential time-bandwidth product is also simply estimated for a wide class of physical systems, where typically localized high-frequency motion occurs at or around each of the many particles, and with low-frequency dynamics associated with globally distributed characteristic degrees of freedom. This estimate for the bandwidth has a generic character indicating the wide significance of expected quantum system dynamics compression. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high-dimensional Schrödinger equations.