Estimating ground-state properties in quantum simulators with global control

TL;DR

The paper addresses the challenge of extracting precise ground-state properties from imperfectly prepared quantum states in analog quantum simulators that operate under global control. It introduces the GENTLE protocol, which uses measurements of the Loschmidt echo $\mathcal{L}(T_G)$ (and a short-time $\mathcal{L}(T_s)$) on an approximate ground state, combined with energy moments $\langle H\rangle$ and $\langle H^2\rangle$, to recover individual eigenenergies via a nonlinear reconstruction. The approach leverages compressed sensing and nonlinear fitting to extract energy differences $\omega_p=|E_n-E_m|$ and overlaps $p_n$, maps these to a consistent energy spectrum, and solves the GENTLE equations to obtain $\{E_n\}$ and $\{p_n\}$; it further extends to order parameters using the Hellmann-Feynman theorem and demonstrates robustness against noise with echo-verification techniques and bootstrapping for error bars. The method shows orders-of-magnitude improvements over direct measurements, scales to hundreds of modes, and applies to non-integrable models like the 2D Ising and Fermi-Hubbard ladders, offering a practical route for high-precision spectral and observable estimation on globally controlled quantum simulators.

Abstract

Accurately determining ground-state properties of quantum many-body systems remains one of the major challenges of quantum simulation. In this work, we present a protocol for estimating the ground-state energy using only global time evolution under a target Hamiltonian. This avoids the need for controlled operations that are typically required in conventional quantum phase estimation and extends the algorithm applicability to analog simulators. Our method extracts energy differences from measurements of the Loschmidt echo over an initial ground-state approximation, combines them with direct energy measurements, and solves a set of equations to infer the individual eigenenergies. We benchmark this protocol on free-fermion systems, showing orders-of-magnitude precision gains over direct energy measurements on the initial state, with accuracy improving rapidly with initial-state fidelity and persisting for hundreds of modes. We further demonstrate applicability to the 2D Ising and Fermi-Hubbard models and show that the approach extends naturally to other observables such as order parameters. Finally, we analyze the effect of experimental imperfections and propose error-mitigation strategies. These results establish a practical route to compute physically relevant quantities with high precision using globally controlled quantum simulators.

Figures (10)

• Figure 1: (a) Scheme of the protocol. Using the quantum simulator, the energy of the target Hamiltonian $\langle H \rangle$ is measured on an approximate ground state $\ket{\psi}$, together with the Loschmidt echo $\mathcal{L}(T)$ up to two different evolution times, $T_\text{s}$ and $T_\text{G}$, with $T_\text{s} \ll T_\text{G}$. From the first two quantities, $\langle H^2 \rangle$ is estimated through a Taylor expansion. A signal-processing analysis of $\mathcal{L}(T_\text{G})$ yields the associated spectrum, from which the energy differences between the eigenstates of $H$ are obtained. Together with $\langle H \rangle$ and $\langle H^2 \rangle$, these provide a system of equations $f(\{E_n,|c_n|^2\})=0$ whose solution allows the estimation of individual energies. (b) Quantum circuit illustrating the measurement of $\mathcal{L}(T)$. (c) Classical post-processing involves fitting the measured signals $\mathcal{L}(T_\text{G})$ and $\mathcal{L}(T_\text{s})$ and applying a short-time expansion to extract $\langle H^2 \rangle$ from the latter. (d) Applying the protocol to the perturbed Hamiltonian $H_\mu = {H} + \mu O$ provides access to the ground-state energy as a function of $\mu$, $E_0(\mu)$. The ground-state expectation value of $O$ is obtained from it using the Hellmann-Feynman theorem.
• Figure 2: (a) Sketch of the energy spectrum of the 1D TFIM \ref{['eq:H_Ising']} as a function of $g/J$, showing the energy gap $\Delta = E_1 - E_0$ closing at the critical point (red dot). The arrow indicates the adiabatic ramp, of duration $gT_{\mathrm{a}}$, used to prepare the state $\ket{\psi}$ from a product state. (b) Residual energy $\varepsilon/g$ obtained using the GENTLE protocol applied to $\ket{\psi}$ as a function of $gT_{\mathrm{G}}$, for different preparation times $gT_{\mathrm{a}}$. We consider a chain of $N = 160$ spins and use echo time steps of $g\Delta T_{\mathrm{G}} = 0.1$. The values shown at $gT_{\mathrm{a}} = 0$ (triangles) correspond to the energy of the initial state. (c) Residual energy $\varepsilon/g$ before (initial state) and after applying the GENTLE protocol (with fixed $gT_{\mathrm{G}} = 24$) as a function of $gT_{\mathrm{a}}$ for different system sizes $N$. (d) Low-energy local density of states for two states prepared with different $gT_{\mathrm{a}}$ in a chain of $N = 160$ spins. (e) Total evolution time $gT^{\mathrm{cut}}$ required to reach an energy accuracy $|\tilde{E_0} - E_0|/g < 0.025$ as a function of $N$, comparing cases with and without the GENTLE protocol.
• Figure 3: (a) The TFIM can be implemented using neutral atoms in optical tweezer arrays (red), whereas (b) the FH model is naturally realized with fermionic atoms in optical lattices (blue). (c) Residual energy $\varepsilon$ obtained for a $4\times4$ TFIM at $J/g=-1$ as a function of the initial preparation time $gT_\mathrm{a}$, comparing the energy measured on the initial state (triangles) and after applying the GENTLE protocol (circles). During the state preparation, a small staggered field ${H}_\mathrm{s}=h\,\sum_{x,y} (-1)^{x+y}{\sigma}^{z}_{(x,y)}$, with $h = 0.0025$, is added to lift the ground-state degeneracy. We compute the echo up to $gT_{\mathrm{G}}=40$. (d) Analogous results for a $2\times6$ FH model with eight electrons at $U/t=8$. We also compute the echo up to $t T_{\mathrm{G}}=40$. (e) Error in the staggered magnetization ${m}_{\mathrm{s},x}$, averaged along $L$ rows in the $y$ direction, for the TFIM after a state preparation with $gT_\mathrm{a}=20$, illustrating the improvement achieved by applying the modified GENTLE protocol to ${H}(\mu)$ (see main text). (f) Same comparison for the FH ladder, showing the error in the average density along the $y$ direction for $tT_\mathrm{a}=5.6$.
• Figure 4: (a) Loschmidt echo as a function of $gT_{\mathrm{G}}$ for a $5\times5$ TFIM, starting from an initial state prepared via an adiabatic ramp of duration $gT_{\mathrm{a}} = 12$ ending at $J/g = 1$. A global depolarizing channel $\mathcal{M}_t$ with strength $\gamma/g = 10^{-2}$ is applied during both the state preparation and the GENTLE evolution, and $M = 600$ measurements are performed per data point. We display the measured noisy Loschmidt echo, the value recovered using the echo-verification technique introduced in the text, and the exact noiseless result. (b) Residual energy $\varepsilon$ as a function of $M$ in the Loschmidt echo measured up to $g\,T_{\mathrm{G}}=31$, for different values of the depolarizing strength $\gamma$, and after applying the echo-verification procedure.
• Figure 5: (a) Loschmidt echo for the 2D TFIM considered in Fig. \ref{['fig:3']} of the main text, starting from an initial state prepared with $gT_a = 10$ and using $M = 10^3$ measurements per point. (b) Estimated energies $\tilde{E}_{0,k}$ obtained by applying the protocol to each time $t_k$. White points indicate values randomly discarded in a single bootstrap iteration. (c) Probability density obtained via KDE applied to all data in (b); the maximum, representing the most probable energy, is marked with a grey dot. (d) Final $\tilde{E}_0$ after successive postprocessing steps. Bootstrapping with 70% of the data and $N_{\mathrm{boot}} = 10^4$ iterations is used to determine the error bars. The exact value of the energy is shown as a black dashed line.