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

Abstract

We present a symmetry-enabled direct quantum protocol for computing many-body Green's functions, a central tool for studying strongly correlated quantum systems. Our protocol relies only on native time evolution and straightforward measurements available on current hardware platforms. By exploiting parity symmetry -- satisfied by a broad class of Hamiltonians in condensed matter physics and quantum chemistry, including the Fermi--Hubbard and Heisenberg models -- we introduce a tailored quench spectroscopy scheme that recovers both the real and imaginary parts of two-point time correlators, from which Green's functions can be reconstructed via efficient classical signal processing. We further develop a tailored symmetric quantum Gibbs sampler that prepares parity-resolved (symmetric and antisymmetric) thermal states, enabling finite-temperature extensions within the same framework. Finally, we show that the same symmetry-based measurement primitive extends naturally to out-of-time-ordered correlators (OTOCs). Our results provide a practical route to estimating symmetry-resolved dynamical correlators on near-term and early fault-tolerant quantum hardware.

Figures (9)

• Figure 1: Direct quantum protocol for computing many-body Green’s functions via tailored quench spectroscopy. For an initial thermal state $\rho_\beta$ (with $\beta\to\infty$ corresponding to the zero-temperature limit, so that the thermal state becomes the ground state), the imaginary and real parts of the two-point time correlator $C(A,B,t)$ are obtained by applying the tailored quench operators $(I+\mathrm{i} A)/\sqrt{2}$ and $(P+A)/\sqrt{2}$, respectively, followed by time evolution under $e^{-\mathrm{i} H t}$ and measurements in the eigenbasis of the observable $B$. For a thermal initial state, the symmetric and antisymmetric thermal states $\rho_\mathrm{S}$ and $\rho_\mathrm{A}$ are prepared using a tailored symmetric quantum Gibbs sampler (step 2). For a ground-state initial condition, this thermal-state preparation step is not needed. The full Green’s function is then reconstructed via classical signal-processing routines (e.g., MUltiple SIgnal Classification (MUSIC)).
• Figure 2: Preparation of thermal states via Davies generators. The horizontal axis is the Lindbladian evolution time $t$, and the vertical axis is the $\ell_2$-distance between the vectorized evolved state and the corresponding steady state. The blue solid curve ($\mathcal{L}_D$) shows convergence to a regular thermal state, while the orange dashed curve ($\mathcal{L}_{D,S}$) shows convergence to a symmetric thermal state.
• Figure 3: Quantum circuits for estimating $\mathrm{OTOC}(A,B,t)$ using tailored quench spectroscopy.
• Figure 4: Fermi--Hubbard model on a $2\times3$ lattice. Each site $(x,y)$ has spin-$\uparrow$ and spin-$\downarrow$ orbitals, mapped to indices $j$ as $(x,y,\uparrow)\mapsto 6x+2y$ and $(x,y,\downarrow)\mapsto 6x+2y+1$.
• Figure 5: Ground-state numerical simulations for the 2D Fermi--Hubbard model on a $2\times 3$ lattice with open boundary conditions. The interaction strength is $h_U = 6$. The observable is $A = B = (c_0 + c_0^\dag)/2$. The system is evolved up to time $200\pi$ with a Trotter step of $\pi/20$ (only the region $t\in [0,50\pi]$ is shown). Initial-state preparation error is modeled by perturbing the initial state once as $\widetilde{\rho}=(\rho+\varepsilon\varrho)/\Tr(\rho+\varepsilon\varrho)$ with $\varepsilon = 0.1$ and a fixed random state $\varrho$. In the legend, the first "Exact" denotes an exact initial state, the second "Exact" denotes exact time evolution; "Noisy" corresponds to using the perturbed initial state $\widetilde{\rho}$; and "Trotter" refers to simulation via the TS decomposition. (a) Imaginary part of $C(A,B,t)$ as a function of time. (b) Real part of $C(A,B,t)$. (c) Noise-subspace correlation function $R(\omega)$ for $\omega \in [0,\pi/2]$, constructed from the sampled time signal.

Theorems & Definitions (13)

• Lemma 1
• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Lemma 1
• proof
• proof
• Lemma 2
• proof