Investigating a Quantum-Inspired Method for Quantum Dynamics

TL;DR

This work develops a measurement-assisted tensor-network framework (SEBD with entangled measurements) to simulate real-time 1D quantum dynamics by interleaving unitary evolution with mid-circuit measurements along causal light cones, thereby suppressing entanglement growth and extending the feasible simulation window beyond conventional TEBD. It introduces an entangled-measurement estimator that leverages intermediate pre-measurement states to dramatically reduce sampling variance, enabling efficient estimation of local observables and both equal-time and unequal-time correlation functions. The method is validated on the kicked Ising and Heisenberg models, showing that SEBD achieves longer time per sample with bond-dimension reductions $ ext{χ} obreak o obreak e^{S_{ ext{vN}}}$ suppressed relative to TEBD, and that EM further lowers the required number of samples by orders of magnitude. The results provide a classical benchmark for quantum hardware protocols employing mid-circuit measurements and offer practical avenues for extending classical simulations to regimes relevant for holographic quantum dynamics and non-equilibrium quantum transport.

Abstract

Building on recent advances in quantum algorithms which measure and reuse qubits and in efficient classical simulation leveraging projective measurements, we extend these frameworks to real-time dynamics of quantum many-body systems undergoing discrete-time and continuous-time Hamiltonian evolution, and find improvements that significantly reduce sampling overhead. The approach exploits causal light-cone structure by interleaving time and space evolution and applying projective measurements as soon as local subsystems reach the target physical time, suppressing entanglement growth. Comparing to time-evolving block decimation, the method reaches longer times per sample for the same resources. We also gain the ability to study dynamics of entanglement that would be occurring on quantum hardware when following similar protocols, such as the holographic quantum dynamics simulation framework. We show how to efficiently obtain local observables as well as equal-time and time-dependent correlation functions. Our findings show how optimizations for quantum hardware can benefit classical tensor network simulations and how such classical methods can yield insights into the utility of quantum simulations.

Figures (16)

• Figure 1: Schematic tensor network representation of the SEBD framework for real-time dynamics. (a) Layout for the kicked Ising model, shown for $N = 12$ sites for visual clarity, evolved to time $t=3$ Floquet periods using a Trotter step $\Delta\tau = 1$. All numerical simulations in the main text are performed on significantly larger systems. The time evolution is decomposed into a single left light cone and a sequence of diagonal cones, each corresponding to a causal patch associated with a two-site unit cell. This decomposition preserves causal structure and is compatible with both finite and half-infinite systems; a finite chain is shown here for illustration. (b) Illustration of the SEBD- and holoQUADS-inspired method for time evolution and estimation of local observables used in this work, and discussed further in Section \ref{['sec:combining']}.
• Figure 2: Benchmark of SEBD with entangled measurements against TEBD for real-time dynamics of the 1D kicked Ising model. (a)-(d) Time evolution of the local spin expectation value $\langle S^{x}_{\ell=51}(t) \rangle$, computed using SEBD (dashed lines) and TEBD (solid symbols) at increasing bond dimension cutoffs $\chi^{\mathrm{max}} = 50, 100, 150,$ and $200$. The solid blue line denotes the converged TEBD reference at $\chi^{\max} = 2000$, with convergence verified via bond-dimension extrapolation. SEBD simulations are performed on a chain of $N=100$ sites, using a truncation threshold $\epsilon=10^{-8}$ and averaging over $N_{s}=8000$ stochastic trajectories, making the statistical SEBD error bars smaller than the plotting linewidth.
• Figure 3: Suppression of entanglement growth via projective measurements in SEBD. Panels (a)-(e) show sample-averaged spatial profiles of the von Neumann entanglement entropy $S_{\mathrm{vN}}(B_{\ell})$ at representative times, comparing SEBD (before and after applying projective measurements) to TEBD. Entropy is computed across bond $B_{\ell}$, which partitions the chain between sites $\ell$ and $\ell+1$. Projective measurements are applied to sites $\ell=49$ and $\ell=50$ immediately prior to sampling in SEBD, resulting in a sharp drop in $S_{\mathrm{vN}}$ to zero at $B_{\ell}=49$ and $B_{\ell}=50$, consistent with complete local disentanglement of the measured sites. Dashed lines denote TEBD reference profiles at corresponding times. Each panel reports the sample-averaged peak entanglement gap $\overline{\Delta S^{\max}_{\mathrm{vN}}}$ and the corresponding peak bond dimension difference $\overline{\Delta\chi^{\max}}$ between SEBD and TEBD. The inset of panel (e) shows that $\overline{\Delta S^{\max}_{\mathrm{vN}}}$ grows approximately linearly with time. Since the bond dimension scales as $\chi \varpropto e^{S_{\mathrm{vN}}}$, this linear entanglement separation implies an exponential relative late-time advantage for SEBD in the required bond dimension per sample. Further analysis is provided in Appendix \ref{['appendix:ising_chi']}.
• Figure 4: Time dependence of maximum entanglement and bond dimension in SEBD versus TEBD. (a) Sample-averaged maximum von Neumann entanglement entropy $\overline{S^{\max}_{\mathrm{vN}}}$ and (b) sample-averaged maximum bond dimension $\overline{\chi^{\max}}$ as a function of time, computed using SEBD and TEBD under varying truncation thresholds $\epsilon$. At fixed $t$, SEBD yields substantially lower entanglement and bond dimension across all values of $\epsilon$, even though both methods exhibit asymptotic exponential scaling of bond dimension with time. The entanglement gap $\overline{\Delta S^{\max}_{\mathrm{vN}}}$ and the corresponding bond-dimension gap $\overline{\Delta \chi^{\max}}$ grow monotonically with time, underscoring the increasing late-time advantage of SEBD in terms of computational efficiency. Simulations are performed on a kicked Ising chain of length $N = 100$, initialized in a Néel state, with integrability broken by a longitudinal field $h=0.2$.
• Figure 5: Entanglement suppression in the Heisenberg model via SEBD. Spatial profiles of von Neumann entanglement entropy $S_{\mathrm{vN}}(B_{\ell})$ at (a) $t=5$ and (b) $t=12$, computed using SEBD (solid circles) and TEBD (dashed lines) for the 1D spin-$1/2$ Heisenberg model. SEBD simulations proceed from left to right, following the entanglement light-cone structure. Entropies are recorded immediately before projective measurements at representative unit cells $(\ell, \ell + 1)=(1, 2), (151, 152), (301, 302)$. SEBD suppresses entanglement growth by disentangling measured sites and yields a widening gap in peak entanglement relative to TEBD over time. Simulations are performed on a chain of $N = 400$ sites, initialized in a Néel state, with $N_{s}=1000$ sample trajectories, Trotter step size $\Delta\tau = 0.1$, and truncation threshold $\epsilon = 10^{-6}$.