Exploiting emergent symmetries in disorder-averaged quantum dynamics

TL;DR

Disorder-averaged quantum dynamics in many-body systems are computationally challenging due to Hilbert-space growth and symmetry breaking in individual realizations. The authors exploit emergent average symmetries to construct the disorder-averaged dynamical map $\Lambda_t$ directly within the relevant symmetry sector, enabling polynomial-scale simulations in $N$. They introduce two perturbative schemes—short-time Lindbladian expansion and weak-disorder expansion—together with regularization strategies to control long-time behavior, and benchmark on a transverse-field Ising model with all-to-all Gaussian couplings (and connections to Sherrington-Kirkpatrick-like models). The results demonstrate substantial efficiency gains and robustness across system sizes, highlighting a general, symmetry-driven framework for simulating disorder-averaged dynamics in quantum many-body systems with potential experimental relevance.

Abstract

Symmetries are a key tool in understanding quantum systems, and, among many other things, can be exploited to increase the efficiency of numerical simulations of quantum dynamics. Disordered systems usually feature reduced symmetries and additionally require averaging over many realizations, making their numerical study computationally demanding. However, when studying quantities linear in the time-evolved state, i.e. expectation values of observables, one can apply the averaging procedure to the time evolution operator, resulting in an effective dynamical map, which restores symmetry at the level of superoperators. In this work, we develop schemes for efficiently constructing symmetric sectors of the disorder-averaged dynamical map using short-time and weak-disorder expansions. To benchmark the method, we apply it to an Ising model with random all-to-all interactions in the presence of a transverse field. After disorder averaging, this system becomes effectively permutation-invariant, and thus the size of the symmetric subspace scales polynomially in the number of spins allowing for the simulation of large systems.

Figures (6)

• Figure 1: (a) The time evolution averaged over different realizations $\lambda_i$ of a disordered system, each with their respective states $\rho_{\lambda_i}$, can be understood as the action of an effective dynamical map $\Lambda_t$. If the ensemble of realizations is invariant under a symmetry (here permutations), then the average state is also compatible with the symmetry even though each shot by itself breaks the symmetry. (b) Representing the density operators as coefficient vectors with respect to an appropriate, i.e. symmetry-adapted, operator basis reveals the potential reduction in computational complexity arising due to the average symmetry. The dynamics remains confined within each symmetry sector, which is exponentially smaller than the full space in the case of average permutation invariance.
• Figure 2: Time expansion ansatz. Solving equation \ref{['eq.:Lindblad_defining_property2']} for $\rho_{0}=\ket{\uparrow\dots\uparrow}\bra{\uparrow\dots\uparrow}$ and $\mathcal{L}_{t}$ expanded up to $\mathcal{O}(t^3)$. Exact results have been obtained averaging over $1000$ disorder shots. The corresponding statistical error is indicated by the width of the gray line (smaller than the default line width here). Panel (a) shows the average total $Z$-magnetization $\langle \Sigma_{(0,0,1)}\rangle$ over time in units of $h^{-1}$. Panel (b) shows its variance. In the regime $t\ll\epsilon_{max}^{-1}$ the accuracy increases systematically with the approximation order.
• Figure 3: Short-time expansion ansatz. Solving equation \ref{['eq.:Lindblad_defining_property2']} for an expanded Lindbladian truncated after order $\mathcal{O}(t)$. Simultaneously the exponential regularized expansion of the dynamical map up to order $\mathcal{O}(t^2)$ (blue) and exact results (black) are plotted against time in units of $h^{-1}$ for a vanishing mean coupling strength $\bar{J}=0$. Exact results were obtained using 1000 disorder shots resulting in noticeable statistical fluctuations (gray) in the variance. Panels (a), (b) and (c) show the average total $Z$-magnetization $\langle \Sigma_{(0,0,1)}\rangle$ for different disorder strengths $\sigma$, while in panels (d) to (f) the variances in $Z$ are displayed. First deviations of the approximations from the exact results on the order of the corresponding variance are marked and compared to the estimate region $t_{bound}$ proving reliability of the expansion in predictable regimes (vertical lines).
• Figure 4: Weak-disorder expansion of transverse-field Ising model. Exponential (orange) and inverse (blue) regularization schemes are plotted together with exact calculations (black) for different system sizes. Panels (a) to (c) show the total $Z$-magnetization for $N=4,\ 8$ and $12$, while panels (e) to (g) illustrate the corresponding variances. In panels (d) and (h) results for $N=40$ are given for the exponential regularized weak-disorder expansion (blue). The deviation at $t\approx\sigma^{-1}$ is clearly observable, showcasing the power of the exponential regularization.
• Figure 5: Weak-disorder expansion of transverse field Ising model. Exponential (orange) and inverse (blue) regularization schemes are plotted together with exact calculations (black) for $N=12$ spins. Panel (a) shows the total $X$-magnetization and panel (b) the corresponding variances for a $Z$-polarized initial state calculated for $10000$ shots.