Energy Spectra of Compressed Quantum States

TL;DR

This paper investigates the energy spectra of compressed quantum states, revealing that their spectral support is large if most eigenstates are entangled, and explaining why certain compressed states do not exhibit exponential energy spectrum decay.

Contribution

It provides theoretical insights into the energy spectra of entanglement-compressed quantum states, connecting spectral support to entanglement and energy minimization.

Findings

Energy spectrum support is large for highly entangled eigenstates.

Energy spectrum decays inversely with squared eigenvalues under certain conditions.

Explains empirical observations about matrix product state spectra.

Abstract

Quantum algorithms for estimating the ground state energy of a quantum system often operate by preparing a classically accessible quantum state and then applying quantum phase estimation. Whether this approach yields quantum advantage hinges on the state's energy spectrum, that is, the sequence of the state's overlaps with the energy eigenstates of the system Hamiltonian. We show that the energy spectrum of any entanglement-compressed quantum state must have large support if most energy eigenstates are highly entangled, an assumption supported by the eigenstate thermalization hypothesis. Furthermore, we show that if the compressed quantum state minimizes expected energy, then its energy spectrum decays with the inverse-squared energy eigenvalues under a convex relaxation of the compression constraint. This explains the main empirical finding of Silvester, Carleo, and White (Physical Review Letters, 2025) that the energy spectra of matrix product states do not decay exponentially.