Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

TL;DR

This paper introduces a Hamiltonian-guided method for compressing quantum many-body systems' evolution, reducing the effective dimension based on system properties and time-bandwidth considerations, with potential applications in machine learning and simulation.

Contribution

It demonstrates a systematic approach to quantum system compression using proper orthogonal decomposition guided by Hamiltonian properties, revealing an effective model dimension independent of specific simulators.

Findings

Compression dimension scales with time-bandwidth product

Autoencoders can enhance the compression

Numerical examples confirm theoretical predictions

Abstract

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 \emph{constant} Hamiltonian system of Hilbert space dimension $n$ whose frequencies range from $f_{\min}$ to $f_{\max}$, 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 $\delf=(f_{\max}-f_{\min})T$. 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 \emph{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. However, finding an efficient solution procedure \emph{is} dependent on the simulator implementation{\color{black}, 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 multi-layer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high dimensional Schr{\"o}dinger equations.