Spectral Gaps via Imaginary Time

Technical Summary: Spectral Gaps via Imaginary Time

Problem Statement
The spectral gap, defined as $\Delta E = E_1 - E_0$ (the energy difference between the ground state and the first excited state), is a critical parameter in diverse physical contexts, ranging from fluorescence wavelengths and excitonic energy transfer in materials to adiabatic run times and state-transfer fidelities in quantum information. Determining this gap is computationally challenging; in the worst case, deciding if a local Hamiltonian is gapped is undecidable, and estimating low-lying spectra is generally NP-hard. Existing numerical methods, such as Fermionic Quantum Monte Carlo (suffering from the sign problem) and Tensor-Network methods (limited by bond dimension), face significant tradeoffs regarding system size, basis size, and measurement overhead. While Imaginary-Time Evolution (ITE) is a robust classical tool for finding ground states, extracting specific spectral gaps without resolving the full density of states or performing quantum phase estimation remains a specific challenge.

Methodology
The authors propose a method to estimate the spectral gap $\Delta E$ directly from the ratio of two expectation values of nested commutators evaluated along an imaginary-time trajectory.

1. Theoretical Framework:
   For a finite-dimensional Hamiltonian $H$ with spectral decomposition $H = \sum E_n \Pi_n$, and an initial state $|\phi_0\rangle$ with support on both the ground ($\Pi_0$) and first excited ($\Pi_1$) subspaces, the system evolves in imaginary time $\tau$ as $|\psi(\tau)\rangle = e^{-\tau H}|\phi_0\rangle$.
   The method utilizes a "coordinating" observable $O$ that couples the ground and first excited subspaces. The authors define the $M$-th nested commutator $[H, O]_M$.
   The core result (Theorem 1) demonstrates that as $\tau \to \infty$, the ratio of expectation values converges to the square of the energy gap:
   $$\frac{\langle [H, O]_{M+2} \rangle_\tau}{\langle [H, O]_M \rangle_\tau} = (E_0 - E_1)^2 + \mathcal{O}(e^{-\tau(E_2 - E_1)})$$
   This estimator is agnostic to degeneracies in higher excited levels because it relies on projector formulations rather than individual eigenvectors.

2. Logarithmic Estimators:
   The paper also derives logarithmic estimators to extract $E_0 + E_1$ (from the slope of $\ln|\langle [H, O]_M \rangle_\tau|$ vs. $\tau$) and the second gap $E_2 - E_1$ (from the residual of the ratio fit), providing alternatives when numerical division of small numbers is unstable.

3. Quantum Implementation (ITQDE):
   To implement this on quantum hardware, where unitary evolution is native but imaginary time is not, the authors utilize Imaginary-Time Quantum Dynamical Emulation (ITQDE). They employ a Hubbard-Stratonovich transformation to express the Euclidean kernel $e^{-\tau H^2}$ as a Gaussian average of real-time unitaries. This kernel is discretized using Gauss-Hermite quadrature, approximating the evolution as a weighted sum of unitaries.
   The expectation values are then reduced to weighted sums of real parts of overlaps of the form $\langle \phi_0 | U_j^\dagger P_\alpha U_k | \phi_0 \rangle$, where $P_\alpha$ are Pauli strings. These overlaps can be measured using standard Hadamard tests. The authors note that due to the symmetry of the kernel, only the real parts of these overlaps are required.

Key Results
The authors benchmarked the method using classical simulations (via the QuSpin library) on two standard models:

• 1D Transverse-Field Ising Model (TFIM): For system sizes $L=4$, the relative error in the estimated gap decreased exponentially with $\tau$ (reaching $\sim 10^{-8}$) before increasing again at very large $\tau$ due to the suppression of the first excited state below numerical noise.
• 1D Fermi-Hubbard Model: Similar exponential decay in error was observed for a half-filled chain ($L=4$), with relative errors reaching $\sim 10^{-4}$. The method successfully handled the interacting fermionic setting with non-trivial spin and charge structures.
• Quantum Simulation: A proof-of-principle simulation was conducted on the TFIM using Qiskit's Aer backend with $10^6$ shots. While the method was compatible with the ITQDE framework, the minimum error achieved was $\sim 10^{-1}$. The authors attribute this to the "bandwidth tradeoff": the Gaussian filter required to suppress higher states also drives the relevant overlaps toward the shot-noise floor if the imaginary time $\tau$ and quadrature order $K$ are not carefully tuned.

Significance and Claims
The paper claims to offer a "modest but genuine structural advantage" for spectral estimation.

• Simplicity and Scope: The method provides a way to estimate low-lying gaps without resolving individual eigenstates, performing quantum phase estimation, or reconstructing the full density of states. It converts spectral questions into questions about relaxation along a one-parameter flow.
• Robustness: The estimator is robust against degeneracies in higher energy levels and applies to both classical sparse linear algebra and quantum sum-of-terms Hamiltonians.
• Modesty: The authors explicitly state that this approach does not bypass the worst-case complexity of the local Hamiltonian problem. It is not a replacement for full spectral reconstruction or phase estimation when detailed density of states is required.
• Future Outlook: The work serves as a proof of principle. The authors note that making the method competitive on quantum hardware requires importing the full bandwidth analysis from prior work to optimize parameters ($\tau, K$), designing better coordinating observables, and employing more measurement-efficient overlap estimators. The paper positions this gap-focused perspective as a tool that dovetails with broader programs using effective Hamiltonians to solve differential equations, where relaxation rates and gaps are often the primary quantities of interest.