Digital quantum magnetism at the frontier of classical simulations

TL;DR

This work demonstrates digitized quantum dynamics on a 56-qubit gate-based quantum computer to study Floquet prethermalization in a 2D TFIM, revealing a long-lived prethermal regime and emergent hydrodynamics accessible beyond feasible classical simulations. By combining advanced error mitigation (dynamical decoupling, randomized compiling, leakage detection, and zero-noise extrapolation with zero-noise regression) with both hardware experiments and extensive classical benchmarks (MPS, PEPS/PEPO, sparse Pauli dynamics, and neural-network states), the authors establish digital quantum computers as powerful tools for exploring continuous-time dynamics and benchmarking classical heuristics in challenging regimes. The results include a measured diffusion constant consistent with diffusive transport in the prethermal phase and a demonstration of how quantum hardware can outperform or complement state-of-the-art classical methods in regimes of high entanglement and long evolution times. Overall, the work highlights the potential and current limitations of digitized quantum simulations for many-body dynamics and lays groundwork for future quantum-advantaged studies of non-equilibrium quantum matter.

Abstract

The utility of near-term quantum computers for simulating realistic quantum systems hinges on the stability of digital quantum matter--realized when discrete quantum gates approximate continuous time evolution--and whether it can be maintained at system sizes and time scales inaccessible to classical simulations. Here, we use Quantinuum's H2 quantum computer to simulate digitized dynamics of the quantum Ising model and observe the emergence of Floquet prethermalization on timescales where accurate simulations using current classical methods are extremely challenging (if feasible at all). In addition to confirming the stability of dynamics subject to achievable digitization errors, we show direct evidence of the resultant local equilibration by computing diffusion constants associated with an emergent hydrodynamic description of the dynamics. Our results were enabled by continued advances in two-qubit gate quality (native partial entangler fidelities of 99.94(1)%) that allow us to access circuit volumes of over 2000 two-qubit gates. This work establishes digital quantum computers as powerful tools for studying continuous-time dynamics and demonstrates their potential to benchmark classical heuristics in a regime of scale and complexity where no known classical methods are both efficient and trustworthy.

Figures (28)

• Figure 1: Digitizing an Ising quantum magnet: (a) We study quantum quenches in the 2D transverse-field Ising model on a $7\times8$ rectangular lattice with periodic boundary conditions, and compare results from Quantinuum's H2 quantum computer to classical simulation methods. (b) Digitization of the dynamics (required for simulation on a gate-based quantum computer) via a second-order Trotterization of the time evolution operator. (c) Depending on the Trotter step size two outcomes are possible: (Top) For step sizes that are too large the system swiftly heats into a structureless infinite-temperatures state. (Bottom) For step sizes small enough, the system evolves into a long-lived Floquet-prethermalized state (here evidenced in raw shots output by H2 showing persistent ferromagnetic correlations).
• Figure 2: Floquet prethermalization: (a) As the Trotter step size $dt$ is reduced to decrease digitization errors, Floquet heating is suppressed and emergent (pre)thermal physics can be observed, here witnessed in exact numerical simulations by the persistence of the squared magnetization $\overline{\langle Z_{\rm tot}(s)^2\rangle}$ (the overline denotes a moving time average of 6 Trotter steps, used to smooth out strong finite-size oscillations). (b) By tuning the ratio $|h/J|$ and $\Delta\theta$ (energy density) one can map out the prethermal phase diagram of the Floquet 2D TFIM by time integrating observables (here over all $s\leq 30$). Here we show exact numerical simulations for a small $(L_x,L_y)=(4,4)$ system: (Top) The ordered phase is indicated by the persistence of time-averaged squared magnetization $\overline{\langle Z_{\rm tot}(s)^2\rangle}$ after a quench from a symmetry-broken state (white dashed-line is a a guide to the eye denoting the ordered region). (Bottom) Connected correlations $\overline{\langle Z_{\rm tot}(s)^2\rangle-\langle Z_{\rm tot}(s)\rangle^2}$ are peaked near the (finite-size) phase boundary. The quench data presented in Fig. \ref{['fig:main_results']} (a) and (b) are at the parameters indicated by the $\square$ and $\Diamond$ symbol, respectively. (c) $\Delta\theta$ controls the initial energy density and therefore the late-time saturated entanglement entropy $S$ of the time-evolved state. Results computed for several system sizes up to $(L_{x},L_{y})=(4,5)$ with exact numerical simulations [lighter curves for smaller systems, symbols for (4,5)] collapse when normalized by the Page limit entanglement entropy $S_{\rm max}$, indicating that we can reliably infer the saturated entanglement entropy for larger system sizes from these numerics. (d) By comparing exact and MPS-based numerical simulations of the intermediate-temperature quench at $\Delta\theta = 2\pi/9$ for system sizes up to $6\times 6$, we can estimate the bond dimension of an MPS simulation necessary to determine (using extrapolation from the MPS data up to the reported bond dimension) observables for a $7\times 8$ system with a given accuracy.
• Figure 3: Prethermalization dynamics with $56$ qubits. (a) A quench at low energy density $(h/|J|,dt|J|,\Delta\theta)=(2,0.25,0)$, for which the dynamical stability of prethermalization can be assessed classically using MPS methods and their extrapolation to zero truncation error (ZTE). We use this low temperature quench to benchmark the efficacy of our error mitigation methods (ZNE+ZNR) on the quantum data: ZNR removes the effect of qubit leakage errors from the raw (blue) and noise amplified (NA, green) data, which is then extrapolated to the limit of zero two-qubit gate noise to produce the ZNE+ZNR (purple) estimates for $\langle Z_{\rm tot}^2\rangle$. The extrapolated results are statistically consistent with the extrapolated MPS results out to the timescales probed here. (b) At intermediate temperatures, $(h/|J|,dt |J|,\Delta\theta)=(2,0.25,2\pi/9)$, ZTE of the MPS results is only controlled at quite early times. For $s\leq9$ (purple solid line) the $\chi=4000$ MPS fidelity is high enough that, while the extrapolation itself is not necessarily accurate, we can have reasonable confidence about where the exact observable lies (purple shaded region). For $s>9$ (purple dashed line) the $\chi=4000$ MPS fidelity is low enough that it becomes difficult to even quantify how wrong the extrapolation might be. (c) Various classical simulation methods compared to the quantum data. (d,e) Samples of raw data taken at $s=20$ for the low temperature and intermediate temperature quenches demonstrate the late-time stability of (at least short-range) ferromagnetic order, while (f) shows (randomly generated) infinite temperature samples for visual reference.
• Figure 4: Emergent hydrodynamics. Hydrodynamic relaxation of the spin-exchange contribution to energy density for a quench in an $14\times 4$ strip with an inhomogeneous initial state (described in text). (a) Evolution of the y-averaged exchange energy density profiles, $\mathcal{E}(x)$. (b) Dashed lines show a fit of $\log|\tilde{\mathcal{E}}(q)|^2$ to $\log(a e^{-\Gamma_q s}+c\delta_{q,0})$. Fits are restricted to long-wavelengths ($q = 2\pi n/L_x$ for $n=0,1,2,3$), and exclude data points with poor signal-to-shot-noise ratio (unfilled symbols). The extracted decay rates are consistent with the expected diffusive scaling of the heat equation, $\Gamma_q \approx \mathcal{D} q^2$ (inset) with diffusion constant $\mathcal{D}=0.38(5)$.
• Figure A1: (a) This leakage detection gadget either applies $X$ or does nothing on the top (ancilla) qubit depending on whether the bottom (data) qubit has or has not leaked, such that leakage events on the data are mapped to $1$ measurement outcomes for an ancilla initialized in $\ket{0}$. (b) If all available qubits are utilized in a circuit, one must be measured without leakage detection, but the remaining $N-1$ can then be leakage detected prior to measurement using a $\sim\log(N)$ depth circuit by reusing measured qubits. (c) The results of the $Z$-basis measurements and leakage detection results are stored in two separate classical registers, $c$ and $a$, respectively.