Adiabatic echo protocols for robust quantum many-body state preparation

TL;DR

The paper introduces the adiabatic echo protocol, a general, interference-based approach to robustly prepare many-body quantum states in the presence of static perturbations. By combining analytical insights with GRAPE-driven optimal control, it shows that echo-like control profiles arise naturally when crossing symmetry-breaking or topological phase transitions in Ising chains, square and ladder Rydberg arrays, and quantum spin liquids. The core idea is to engineer destructive interference among leading-order perturbative pathways, suppressing infidelity scaling with perturbation strength. This framework yields practical, platform-spanning strategies for reliable adiabatic state preparation in current quantum devices, with demonstrated robustness to disorder, long-range tails, and even some time-dependent noise scenarios.

Abstract

Entangled many-body states are a key resource for quantum technologies. Yet their preparation through analog control of interacting quantum systems is often hindered by experimental imperfections. Here, we introduce the adiabatic echo protocol, a general approach to state preparation designed to suppress the effect of static perturbations. We provide an analytical understanding of its robustness in terms of dynamically engineered destructive interference. By applying quantum optimal control methods, we demonstrate that such a protocol emerges naturally in a variety of settings, without requiring assumptions on the form of the control fields. Examples include Greenberger-Horne-Zeilinger state preparation in Ising spin chains and two-dimensional Rydberg atom arrays, as well as the generation of quantum spin liquid states in frustrated Rydberg lattices. Our results highlight the broad applicability of this protocol, providing a practical framework for reliable many-body state preparation in present-day quantum platforms.

Figures (14)

• Figure 1: Many-body adiabatic echo protocol. (a) Spectrum of a many-body Hamiltonian $\hat{H}_0(s)$ with $\mathbb{Z}_2$ symmetry breaking. The initial and target states lie in the trivial and ordered phases, separated by a critical point $s_c$. Solid (dashed) lines indicate eigenenergies in the even (odd) $\mathbb{Z}_2$ sectors. A perturbation $\hat{V}$ breaks the symmetry and couples these sectors, affecting the dynamics for $s > s_c$, when the gap $\delta E(s)$ is exponentially small with system size. (b) Illustration of the adiabatic echo protocol. Contributions to the infidelity $\mathcal{I}$ from regions I and III vanish, while those from regions II and IV destructively interfere. (c–e) Applications where the echo protocol mitigates the effect of the perturbation $\hat{V}$: (c) Ising chain with transverse and longitudinal fields, (d) square lattice Rydberg array with positional disorder, (e) ruby lattice Rydberg array with long-range interactions.
• Figure 2: Adiabatic echo in the quantum Ising chain. (a) Optimal cost function \ref{['eq:cost_f']} vs total preparation time, shown for various values of the maximal longitudinal field strength $\sigma$. For $\sigma > 0$, the optimization averages over $N_{\rm s} = 30$ samples. Dashed colored lines show the averaged infidelity when evaluated on the time-optimal protocols obtained for $\sigma = 0$. The colored dots in (a) indicate the time $T^*$ beyond which the optimization for $\sigma > 0$ yields a solution that differs from the time-optimal one. (b) Optimized profiles $s(t)$ for $T=14$ (vertical red line in (a)) and varying $\sigma$. The optimization for $\sigma > 0$ takes the form of an echo protocol akin to the one sketched in \ref{['fig1:b']}. (c) Preparation infidelity for the profiles in (b), plotted against the longitudinal field strength $h$.
• Figure 3: GHZ state preparation in Rydberg arrays. (a,b) Optimal cost function \ref{['eq:cost_f']}, on the square lattice (a) and on the ladder (b), vs total preparation time, for $N_{\rm s} = 30$ samples, each consisting of one displacement $\delta \mathbf{x}_j$ per atom, gaussian distributed with standard deviation $\sigma=0.01 a$. In the shaded red region the optimal protocols acquire a profile typical of the echo mechanism. (c,d) Optimal control profiles on the square lattice (c) and on the ladder (d) for $\Omega_0 T=20$ (standard adiabatic) and $\Omega_0 T = 38$ (adiabatic echo). The insets compare the robustness of the two against the standard deviation of the positional disorder. The shaded purple region marks the finite-size critical regime, estimated from two independent ground state diagnostics SupMat.
• Figure 4: RVB state preparation in Rydberg arrays. (a) Optimal RVB preparation fidelity vs total preparation time on the ruby lattice with $24$ atoms, truncating ($U_{3 a} = 0$) and including ($U_{3a} > 0$) the van der Waals potential. For $U_{3a} > 0$ the optimization is also performed including positional disorder via the cost function \ref{['eq:cost_f']}, averaged over $N_{\rm s} = 30$ gaussian distributed samples (green line). (b) Optimal control profiles for $\Omega_0 T = 24$ (vertical red line in (a)). The optimal protocol exhibits the structure of the adiabatic echo only when $U_{3a} > 0$.
• Figure S1: GHZ state preparation infidelity $\mathcal{I}$ in the Ising chain under different perturbations, as a function of the perturbation strength $\epsilon$. (a) The standard adiabatic time-optimal protocol (black line in Fig. 2b of the main text) is robust against symmetry-preserving perturbations ($\hat{V}^X$ and $\hat{V}^{XX}$), but sensitive to symmetry-breaking perturbations ($\hat{V}^Z$ and $\hat{V}^{ZZZ}$). (b) The adiabatic echo protocol (colored lines in Fig. 2b of the main text) remains robust against both $\hat{V}^Z$ and $\hat{V}^{ZZZ}$.