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Energy Spectra of Compressed Quantum States

Daochen Wang

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.

Energy Spectra of Compressed Quantum States

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.

Paper Structure

This paper contains 8 sections, 4 theorems, 54 equations, 2 figures, 1 table.

Table of Contents

  1. Energy spectra via entanglement.
  2. Case study: 2D AFHM.
  3. Energy spectra and quantum advantage.
  4. Conclusion.
  5. Acknowledgments.
  6. Data availability.
  7. Volume law for entanglement min-entropy of typical eigenstates
  8. Two-level Hamiltonians

Key Result

Proposition 1

Let $k,n\in \mathbb{N}$. Let $m, M_1,\dots, M_k>0$. Suppose $\ket{\psi}$ and $\ket{\psi_1},\dots,\ket{\psi_k}$ are $n$-qubit states with Then, In particular,

Figures (2)

  • Figure 1: The entanglement min-entropy ($S_{\min}$) and entropy ($S_1$) of eigenstates of the 2D AFHM on $n=16$ spins. Each data point represents the mean of all $S_{\alpha}$ values corresponding to $(E-E_1)$ values within a bin of the form $[0.1j-0.05,0.1j+0.05)$, where $j\in \mathbb{Z}$. All $S_{\alpha}$ values are theoretically at most $\log_2(2^{n/2}) = 8$.
  • Figure 2: Actual versus predicted energy spectra for bond dimension (a) $D=50$, (b) $D=100$, and (c) $D=150$. "Actual" denotes the actual spectra of $\ket{\psi(D)}$; "Predict+" denotes the predicted spectra when \ref{['eq:gs_overlap_bound']} is used; "Predict" denotes the predicted spectra when \ref{['eq:gs_overlap_bound']} is unused; All spectra are broadened by Gaussians with width $0.1$.

Theorems & Definitions (10)

  • Definition 1: Stable rank
  • Definition 2: Stable Schmidt rank
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 3
  • proof : Proof of \ref{['prop:energy_min']}.