Energy Spectra of Compressed Quantum States.

Area of Science:

• Quantum Computing
• Quantum Many-Body Physics
• Quantum Information Theory

Background:

• Quantum algorithms for ground state energy estimation often use quantum phase estimation on classically accessible states.
• The success of these algorithms, termed quantum advantage, depends on the energy spectrum of the quantum state.
• The energy spectrum describes the overlap of a quantum state with the energy eigenstates of a Hamiltonian.

Purpose of the Study:

• To investigate the properties of the energy spectrum for entanglement-compressed quantum states.
• To explain the empirical observation that matrix product states do not show exponential energy spectrum decay.
• To connect the energy spectrum characteristics to the entanglement properties of quantum states and eigenstates.

Main Methods:

• Theoretical analysis of the energy spectrum for entanglement-compressed quantum states.
• Application of the eigenstate thermalization hypothesis to justify assumptions about highly entangled energy eigenstates.
• Convex relaxation of the compression constraint to derive spectral decay properties.

Main Results:

• The energy spectrum of any entanglement-compressed quantum state must have large support if energy eigenstates are highly entangled.
• If a compressed quantum state minimizes expected energy, its energy spectrum decays with the inverse-squared energy eigenvalues.
• This theoretical finding explains the non-exponential decay observed in the energy spectra of matrix product states.

Conclusions:

• The entanglement properties of quantum states fundamentally influence their energy spectral characteristics.
• The inverse-squared decay of energy spectra for energy-minimizing compressed states provides a theoretical basis for empirical observations.
• Understanding these spectral properties is crucial for assessing the potential for quantum advantage in ground state energy estimation.