Stabilizer-Accelerated Quantum Many-Body Ground-State Estimation

TL;DR

This work addresses how to describe quantum many-body collectivity while preserving symmetries and keeping classical efficiency. It introduces a stabilizer-plus-magic framework that partitions the Hamiltonian into a stabilizer part $H_{stab}$ and a residual $W$, enabling efficient preparation of a stabilizer ground state via graph states and stabilizer tableaux. Applied to the Lipkin-Meshkov-Glick model, the stabilizer ground state reproduces key entanglement features in the collective regime, with entangled stabilizer states capturing large-scale bipartite and multipartite entanglement and a transition around $ar{v}_x oughly 2$ where magic peaks. The paper further investigates injecting non-stabilizerness through variational imaginary-time and full quantum imaginary-time propagation (QITE/QITP), showing accelerated convergence to the exact ground state when starting from stabilizer states, and discusses limitations of ADAPT-VQE in collective settings. Overall, the approach offers a symmetry-preserving, efficiently preparable starting point for simulating strongly correlated systems and a pathway to combine stabilizer methods with classical and quantum techniques for larger, more realistic models.

Abstract

We investigate how the stabilizer formalism, in particular highly-entangled stabilizer states, can be used to describe the emergence of many-body shape collectivity from individual constituents, in a symmetry-preserving and classically efficient way. The method that we adopt is based on determining an optimal separation of the Hamiltonian into a stabilizer component and a residual part inducing non-stabilizerness. The corresponding stabilizer ground state is efficiently prepared using techniques of graph states and stabilizer tableaux. We demonstrate this technique in context of the Lipkin-Meshkov-Glick model, a fully-connected spin system presenting a second order phase transition from spherical to deformed state. The resulting stabilizer ground state is found to capture to a large extent both bi-partite and collective multi-partite entanglement features of the exact solution in the region of large deformation. We also explore several methods for injecting non-stabilizerness into the system, including ADAPT-VQE, and imaginary-time evolution (ITE) techniques. Stabilizer ground states are found to accelerate ITE convergence due to a larger overlap with the exact ground state. While further investigations are required, the present work suggests that collective features may be associated with high but simple large-scale entanglement which can be captured by stabilizer states, while the interplay with single-particle motion may be responsible for inducing non-stabilizerness. This study motivates applications of the proposed approach to more realistic quantum many-body systems, whose stabilizer ground states can be used in combinations with powerful classical many-body techniques and/or quantum methods.

Figures (10)

• Figure 1: Panel a): graph $G$ associated with the state in Eq. \ref{['eq:graph_LMG']} for $N=8$. Panel b): complementary graph $G^c$ (see more details in appendix \ref{['sec:app_stabil_Hami']}).
• Figure 2: From bottom to top panel: relative energy difference $\varepsilon$, fidelity, entanglement, and stabilizer $2$-Rényi entropy $\mathcal{M}_2$, in a system with $N=8$ spins, as a function of the interaction strength $\bar{v}_x$. The exact solution is shown with black curves, the unentangled (non-interacting) stabilizer state $\ket{\Psi_{s,1}}^{(N)} = \ket{1}^{\otimes N}$ is shown with green curves and the entangled stabilizer state $\ket{\Psi_{s,2}}^{(N)}$ is shown with purple curves. In the entanglement panel, the von Neumann entropy is displayed with plain lines while the $N$-tangle $\tau_N$ is shown with dashed lines. The black dashed vertical line denotes the critical point between normal (spherical) and parity-broken (deformed) phases, while the red dotted vertical line denotes the transition from unentangled to entangled stabilizer ground state.
• Figure 3: From bottom to top panel: relative energy difference between exact and stabilizer ground-state energy, fidelity of the stabilizer ground state, stabilizer 2-Rényi entropy $\mathcal{M}_2$ of the exact ground state, entanglement entropy of both exact and stabilizer ground states. The results are shown for various system sizes and have been obtained with $\chi= -1$. The stabilizer ground state is taken to be $\ket{\Psi_{s,1}}^{(N)}$ for $\bar{v}_x < 2$ and $\ket{\Psi_{s,2}}^{(N)}$ for $\bar{v}_x > 2$.
• Figure 4: Relative energy difference $\varepsilon$, fidelity, von Neumann entanglement entropy $S_1^{(N)}$, $N$-tangle $\tau_N$, and SRE $\mathcal{M}_2$ in a system with $N=8$ spins, and $\chi=-1$. The exact solution appears in black. The unentangled and entangled stabilizer states, $\ket{\Psi_{s,1}}^{(N)}$ and $\ket{\Psi_{s,2}}^{(N)}$, are shown in green and purple curves, respectively. The state obtained via Eq. \ref{['eq:var_QITP_Jz']} is shown with red curves. For comparison, the deformed HF without and with projection are shown with cyan and blue curves, respectively.
• Figure 5: Same as Fig. \ref{['fig:mag_Jzop_chi-1']} for $\chi=-0.1$.