Hamiltonian spectra in quantum computers through the generalized eigenvalue method

Technical Summary: Hamiltonian Spectra in Quantum Computers through the Generalized Eigenvalue Method

Problem Statement
Determining the energy spectrum of a Hamiltonian is a fundamental objective in quantum physics, particularly for strongly coupled field theories where perturbative methods fail. While classical lattice field theory utilizes Monte Carlo evaluations of Euclidean correlators to extract spectra, this approach faces a "mixed blessing": while the ground state dominates at large imaginary time, excited states are suppressed exponentially, making their extraction difficult due to the degradation of the signal-to-noise ratio.

Quantum computers offer a distinct advantage by naturally generating real-time correlators, where all states contribute with undamped, oscillating weights. However, extracting energy eigenvalues from these oscillatory signals typically relies on Fourier transforms. The resolution of a Fourier transform is limited by the inverse of the total evolution time ($1/T$). Current quantum hardware, constrained by finite coherence times and accumulating gate/Trotter errors, cannot sustain the long evolution times required to resolve nearly degenerate levels or distinguish genuine peaks from ringing artifacts via Fourier analysis.

Methodology
The authors propose adapting the Generalized Eigenvalue Problem (GEVP)—a standard technique in Euclidean lattice field theory—to real-time correlators generated on quantum computers. The core methodology involves:

1. Operator Basis and Correlator Matrix: Instead of analyzing a single correlator, the method constructs a matrix of correlators $C_{ij}(t) = \langle \Omega | O_i(t) O_j^\dagger(0) | \Omega \rangle$, where $O_i$ is a set of imperfect interpolating operators and $|\Omega\rangle$ is an approximate ground state.
2. Theoretical Framework: The authors demonstrate that the eigenvalues $\lambda_k(t)$ of the matrix product $C^{-1}(0)C(t)$ behave as $e^{-iE_k t}$, where $E_k$ are the energy eigenvalues. The effective energy is extracted via $E_k^{\text{eff}}(t) = \frac{i}{t} \log(\lambda_k(t))$. Theoretical analysis shows that corrections to the exact energies decay as $1/t$ (or faster due to destructive interference), provided the operator basis sufficiently spans the relevant subspace of eigenstates.
3. Quantum Implementation: A specific algorithm is detailed for measuring off-diagonal correlators of Pauli operators on a quantum computer using a single ancilla qubit. The procedure involves:
   • Preparing the approximate ground state $|\Omega\rangle$ and an ancilla.
   • Entangling the system with the ancilla via a controlled operation involving the operator $O_j$.
   • Time-evolving the system.
   • Measuring specific observables ($X \otimes O_i$ and $Y \otimes O_i$) to reconstruct the real and imaginary parts of the correlator.
4. Comparison Benchmark: The method is benchmarked against the discrete Fourier transform (DFT) of the trace of the correlator matrix, utilizing a damping factor to mitigate Gibbs phenomena.

Key Contributions and Results
The paper validates this approach using the fuzzy $\sigma$-model (equivalent to the Heisenberg comb), a theory sharing features with physically relevant field theories like asymptotic freedom. The study employs three distinct testing regimes:

• Exact Simulations ($L=8$): Using exact time evolution (no noise or Trotter errors), the method successfully resolved 8, 36, and 276 energy levels using three increasingly complex operator sets. The results achieved sub-percent precision for low-lying states within a maximum evolution time of $T_{\text{max}} = 5.6$. The method correctly reproduced the degeneracy patterns dictated by the model's $O(3)$ symmetry.
• Gate-Based Simulations ($L=4$): The authors analyzed the impact of Trotter errors. By extrapolating results from various time steps ($\Delta t$) to $\Delta t \to 0$, they confirmed that the method is robust, with deviations scaling quadratically with the step size.
• Quantum Hardware ($L=4$ on IonQ Forte-Enterprise): The method was executed on actual quantum hardware. Using a single ancilla qubit and a minimal operator set, the authors resolved four low-lying energy levels. The extracted energies agreed with exact diagonalization within a few percent. The residual discrepancies were attributed to a combination of Trotter errors (due to the coarse $\Delta t = 0.4$) and hardware noise/decoherence, rather than a failure of the GEVP method itself.

Significance and Claims
The paper claims that the GEVP approach is "substantially more efficient" than Fourier transform methods for extracting spectra from real-time quantum simulations. Specifically, it allows for the resolution of low-lying energy levels at time extents where the Fourier transform fails to distinguish peaks from artifacts.

The authors emphasize that the method does not require perfect operators or an exact ground state; it relies only on an operator basis that approximately spans the states of interest and an approximate ground state with good overlap. While demonstrated here using strong-coupling limits to construct the necessary states, the authors note the method is general and applicable to any variational or adiabatic preparation of states.

The primary significance highlighted is the potential for this technique to enable spectroscopy in theories where the "sign problem" obstructs Euclidean approaches, specifically mentioning finite density gauge theories and real-time transport phenomena. The work serves as a proof-of-principle that real-time quantum simulations can yield precise spectral data without the need for the prohibitively long evolution times required by frequency-domain analysis.