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Spectral Gaps via Imaginary Time

Jacob M. Leamer, Alicia B. Magann, Gerard McCaul, Denys I. Bondar

TL;DR

This work introduces an estimator for the spectral gap ΔE = E1 − E0 of finite-dimensional Hamiltonians by analyzing imaginary-time evolution of a state under nested commutators with a local observable. The key result is that the ratio ⟨[H,O]_{M+2}⟩_τ/⟨[H,O]_M⟩_τ converges to (E0−E1)^2 as τ grows, with corrections governed by the next gap, enabling gap extraction without resolving individual eigenstates. The authors also show how logarithmic fits can yield E0+E1 and E2−E1, and extend the approach to H^2 via ITQDE, linking spectral gaps to relaxation dynamics in both classical and quantum computation settings. Classical benchmarks on TFIM and FH validate rapid convergence within a practical imaginary-time window, while a minimal ITQDE-based quantum implementation demonstrates feasibility and highlights current hardware limitations, outlining concrete paths for optimization. Overall, the method provides a structurally simple, measurement-friendly diagnostic for low-lying spectral structure that integrates seamlessly with imaginary-time and effective-Hamiltonian frameworks, with potential impact on adiabatic runtimes and relaxation analyses.

Abstract

The spectral gap occupies a role of central importance in many open problems in physics. We present an approach for evaluating the spectral gap of a Hamiltonian from a simple ratio of two expectation values, both of which are evaluated using a quantum state that is evolved in imaginary time. In principle, the only requirement is that the initial state is supported on both the ground and first excited states. We demonstrate this approach for the Fermi-Hubbard and transverse-field Ising models through numerical simulation. We then go on to explore avenues for its implementation on quantum computers using imaginary-time quantum dynamical emulation.

Spectral Gaps via Imaginary Time

TL;DR

This work introduces an estimator for the spectral gap ΔE = E1 − E0 of finite-dimensional Hamiltonians by analyzing imaginary-time evolution of a state under nested commutators with a local observable. The key result is that the ratio ⟨[H,O]_{M+2}⟩_τ/⟨[H,O]_M⟩_τ converges to (E0−E1)^2 as τ grows, with corrections governed by the next gap, enabling gap extraction without resolving individual eigenstates. The authors also show how logarithmic fits can yield E0+E1 and E2−E1, and extend the approach to H^2 via ITQDE, linking spectral gaps to relaxation dynamics in both classical and quantum computation settings. Classical benchmarks on TFIM and FH validate rapid convergence within a practical imaginary-time window, while a minimal ITQDE-based quantum implementation demonstrates feasibility and highlights current hardware limitations, outlining concrete paths for optimization. Overall, the method provides a structurally simple, measurement-friendly diagnostic for low-lying spectral structure that integrates seamlessly with imaginary-time and effective-Hamiltonian frameworks, with potential impact on adiabatic runtimes and relaxation analyses.

Abstract

The spectral gap occupies a role of central importance in many open problems in physics. We present an approach for evaluating the spectral gap of a Hamiltonian from a simple ratio of two expectation values, both of which are evaluated using a quantum state that is evolved in imaginary time. In principle, the only requirement is that the initial state is supported on both the ground and first excited states. We demonstrate this approach for the Fermi-Hubbard and transverse-field Ising models through numerical simulation. We then go on to explore avenues for its implementation on quantum computers using imaginary-time quantum dynamical emulation.
Paper Structure (8 sections, 2 theorems, 52 equations, 2 figures, 1 table)

This paper contains 8 sections, 2 theorems, 52 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Results
  4. Transverse-field Ising model
  5. Fermi-Hubbard Model
  6. Towards Quantum Computer Implementations
  7. Conclusion and outlook
  8. Commutator Expansion

Key Result

Theorem 1

Let $H$ be a finite-dimensional Hamiltonian with spectral decomposition where the $\Pi_n$ are orthogonal projectors and $E_0 < E_1 < \dots < E_N$ are distinct eigenvalues. Let be the imaginary-time-evolved state for some initial $\ket{\phi_0}$. Define the $M$-th nested commutator for a coordinating observable $O$. Suppose that: Then, as $\tau \to \infty$, where $\langle[H,O]_{M}\rangle_\tau \

Figures (2)

  • Figure 1: Relative error $\epsilon$ in $\Delta E$ computed via Eq. \ref{['eq:comm_spectral_gap']} plotted versus imaginary time $\tau$ for (a) the 1D transverse-field Ising model and (b) the 1D Fermi–Hubbard model, with $M=1$ (blue) and $M=2$ (orange). In both cases we observe an extended regime of exponential decay in $\epsilon$ before the error eventually increases again as the contribution from the first excited state is quenched.
  • Figure 2: Relative error $\epsilon$ in $\Delta E$ computed via Eq. \ref{['eq:qde_spectral_gap']} plotted versus imaginary time $\tau$ for the 1D transverse-field Ising model with $L=2$, using $M=1$ (blue) and $M=2$ (orange). Calculations are performed using Qiskit’s Aer simulator with $10^6$ shots per Hadamard-test configuration and $K=30$ Gauss–Hermite points.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Theorem 2: Imaginary time under $H^2$