A Symmetry-Enabled Direct Quantum Protocol for Many-Body Green's Functions

Problem Statement

Many-body Green's functions are fundamental tools for characterizing strongly correlated quantum systems, encoding critical information about density of states, band structures, and phenomena like superconductivity and magnetism. While experimental techniques (e.g., ARPES, RIXS) can extract this information, computational approaches face significant bottlenecks. Classical methods such as exact diagonalization are limited by the exponential growth of the Hilbert space, and Quantum Monte Carlo methods struggle with the dynamical sign problem and ill-conditioned analytic continuation required to recover real-time dynamics from imaginary time.

On the quantum computing side, existing algorithms for estimating Green's functions often rely on resource-intensive primitives. Methods based on phase estimation or variational approaches typically require ancilla qubits and controlled time evolution, which drastically increase circuit depth and error rates, making them unsuitable for near-term Noisy Intermediate-Scale Quantum (NISQ) devices. Other ancilla-free approaches, such as Linear Response Frameworks (LRF) or Quantum Imaginary Time Evolution (QITE), often involve time-dependent Hamiltonians, perturbative approximations, or complex dissipative evolutions. Direct measurement schemes relying on mid-circuit measurements (MCMs) face challenges due to the noise and latency of current MCM hardware.

Methodology: Tailored Quench Spectroscopy (TQS)

The authors propose Tailored Quench Spectroscopy (TQS), a symmetry-enabled, ancilla-free protocol that directly estimates both the real and imaginary parts of two-point time correlators, $C(A, B, t) = \text{tr}[\rho A B(t)]$, from which Green's functions can be reconstructed.

Core Principles:

1. Symmetry Exploitation: The protocol targets systems where the Hamiltonian $H$ commutes with a parity operator $P$ ($[H, P] = 0$), and the observables of interest $A$ and $B$ anti-commute with $P$ ($\{A, P\} = \{B, P\} = 0$). This condition holds for broad classes of models, including the Fermi–Hubbard and Heisenberg models.
2. Tailored Quench Operators: Instead of standard quench operators, the protocol utilizes two specific unitaries:
   • Imaginary Part: $U_{\text{Im}} = \frac{I + iA}{\sqrt{2}}$. This recovers the imaginary part of the correlator via standard quench spectroscopy relations.
   • Real Part: $U_{\text{Re}} = \frac{P + A}{\sqrt{2}}$. By exploiting the parity symmetry, this operator allows the extraction of the real part of the correlator without controlled unitaries.
3. Measurement Scheme: The protocol involves preparing an initial state $\rho$, applying a quench operator, evolving the system under the native time-independent Hamiltonian $e^{-iHt}$, and measuring in the eigenbasis of observable $B$. The full complex correlator is reconstructed from a linear combination of these experimentally accessible expectation values.
4. Finite-Temperature Extension: For thermal states $\rho_\beta$, which generally do not have a definite parity, the authors introduce parity-resolved thermal states ($\rho_S$ and $\rho_A$). They propose a tailored parity-preserving quantum sampler based on dissipative dynamics (Davies generators) that prepares these symmetric and antisymmetric Gibbs states without mid-circuit measurements. The real part of the thermal correlator is then reconstructed as a weighted sum of measurements on these parity-resolved states.
5. Generalization: The framework is extended to Out-of-Time-Ordered Correlators (OTOCs), demonstrating that the same symmetry-based measurement primitives can access higher-order correlators relevant for diagnosing quantum chaos.

Key Contributions

• Ancilla-Free and Noise-Robust Protocol: The TQS protocol eliminates the need for ancilla qubits and controlled unitaries, relying solely on native time evolution and straightforward measurements. This significantly reduces circuit depth and error susceptibility compared to phase estimation or controlled-rotation methods.
• Exact Recovery of Real and Imaginary Parts: Unlike perturbative LRF or asymptotic QITE methods, TQS provides an exact, non-perturbative recovery of both real and imaginary components of time correlators by leveraging parity selection rules.
• Parity-Preserving Thermal State Preparation: The authors develop a method to prepare symmetric and antisymmetric thermal states using parity-preserving Lindbladian dynamics. This enables finite-temperature Green's function estimation within the same symmetry-enabled framework.
• Error Suppression via Symmetrization: Theoretical analysis shows that applying a symmetrization channel to the initial state can suppress parity-breaking preparation errors quadratically (from $O(|\mu|)$ to $O(|\mu|^2)$), enhancing the protocol's robustness against state-preparation noise.
• Extension to OTOCs: The work demonstrates that the symmetry-based primitives naturally extend to the estimation of OTOCs, broadening the applicability of the protocol to quantum chaos diagnostics.

Results

The authors validate the protocol through numerical simulations on two distinct models:

1. Fermi–Hubbard Model (2D, 2×3 lattice): Simulations were performed for both ground states and finite-temperature thermal states ($\beta=1$) across strongly correlated ($U=6$) and weakly interacting ($U=0.1$) regimes. The results show that the reconstructed real and imaginary parts of the correlators remain stable under moderate initial-state preparation errors and Trotterization errors (using first-order Suzuki-Trotter decomposition). The noise-subspace correlation function (MUSIC algorithm) successfully resolved spectral features, matching exact results closely.
2. XXZ Spin Chain (1D, 8 sites): The protocol was applied to estimate OTOCs in a gapless Luttinger liquid phase. The simulations confirmed the robustness of the TQS estimator for OTOCs against Trotter errors and state preparation imperfections.

In all cases, the classical signal processing (MUSIC) successfully reconstructed the Lehmann representation of the Green's function from the noisy time-domain data, demonstrating the viability of the approach for extracting spectral information.

Significance and Claims

The paper claims to provide a practical route to estimating symmetry-resolved dynamical correlators on near-term and early fault-tolerant quantum hardware. By relying only on native time evolution and avoiding ancilla-assisted interference, TQS circumvents the primary overheads that currently limit quantum algorithms for Green's functions.

The authors emphasize that their approach:

• Requires only time-independent Hamiltonian evolution, avoiding the complexity of simulating time-dependent dynamics.
• Offers exact, non-perturbative recovery of correlators, mitigating systematic errors found in linear-response or imaginary-time methods.
• Is compatible with both digital quantum computers and analog simulators, where quench-style protocols are natural.

The work positions symmetry-enabled direct measurement as a promising alternative to existing methods, particularly for systems where parity symmetry is a natural property, such as condensed matter and quantum chemistry models. The authors conclude that while further work is needed on sample complexity guarantees and efficient thermal state preparation, TQS establishes a foundational step toward probing finite-temperature dynamics of strongly correlated quantum matter on quantum hardware.