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Digital quantum magnetism on a trapped-ion quantum computer

Reza Haghshenas, Eli Chertkov, Michael Mills, Wilhelm Kadow, Sheng-Hsuan Lin, Yi-Hsiang Chen, Chris Cade, Ido Niesen, Tomislav Begušić, Manuel S. Rudolph, Cristina Cirstoiu, Kevin Hemery, Conor Mc Keever, Michael Lubasch, Etienne Granet, Charles H. Baldwin, John P. Bartolotta, Matthew Bohn, Justin J. Burau, Julia Cline, Matthew DeCross, Joan M. Dreiling, Cameron Foltz, David Francois, John P. Gaebler, Christopher N. Gilbreth, Johnnie Gray, Dan Gresh, Alex Hall, Aaron Hankin, Azure Hansen, Nathan Hewitt, Craig A. Holliman, Ross B. Hutson, Mohsin Iqbal, Nikhil Kotibhaskar, Elliot Lehman, Dominic Lucchetti, Ivaylo S. Madjarov, Karl Mayer, Alistair R. Milne, Steven A. Moses, Brian Neyenhuis, Gunhee Park, Abigail R. Perry, Boris Ponsioen, Michael Schecter, Peter E. Siegfried, David T. Stephen, Bruce G. Tiemann, Maxwell D. Urmey, James Walker, Andrew C. Potter, David Hayes, Garnet Kin-Lic Chan, Frank Pollmann, Michael Knap, Henrik Dreyer, Michael Foss-Feig

arXiv:2503.20870v3quant-phcond-mat.str-el

TL;DR

Digital quantum simulations must control both hardware noise and digitization errors to reproduce continuous-time many-body dynamics rather than Floquet heating. This paper uses Quantinuum’s H2 to simulate digitized Ising models, revealing prethermal thermalization, diffusive hydrodynamics, and frustrated-lattice dynamics beyond reliable classical reach.

  • Problem

    Quantum computers face simultaneous hardware-noise and digitization-error constraints, while the error scale needed to preserve continuous-time thermal physics remains unclear.

  • Method

    The paper uses Quantinuum’s H2 to simulate digitized transverse-field Ising dynamics with sufficiently fine Trotter steps, alongside classical-method comparisons and error-mitigation procedures.

  • Results

    The simulations observe Floquet-prethermal thermalization, emergent hydrodynamics with a computed diffusion constant, and frustrated-lattice dynamics consistent with emergent gauge and topological constraints.

  • Takeaways & Limitations

    Digital quantum computers can study effectively continuous-time many-body dynamics in regimes where available classical methods are not simultaneously efficient and trustworthy.

Abstract

from arXiv · show

Digital quantum matter -- realized when discrete quantum gates approximate continuous time evolution -- is susceptible to heating into chaotic, structureless states. If digitization errors are adequately suppressed, a long-lived transient regime of approximately energy-conserving dynamics can be observed on gate-based quantum computers. Conservation of energy, in turn, enables the exploration of a wide variety of complex behaviors observed in equilibrium systems, ranging from the nontrivial microscopic origins of thermalization itself to the stabilization of effective models hosting exotic emergent properties. Here, we use Quantinuum's system model H2 quantum computer to simulate digitized dynamics of the quantum Ising model, suppressing digitization errors well enough to observe thermalization on timescales that severely challenge classical simulation methods. Relaxation of an inhomogeneous state reveals an emergent hydrodynamics due to approximate energy conservation, and we compute the associated diffusion constant. By reprogramming our simulations to take place on a triangular lattice with periodic boundary conditions, we observe thermalization consistent with emergent gauge and topological constraints resulting from lattice frustration. Our results were enabled by continued advances in two-qubit gate quality (native partial entangler fidelities of $99.94(1)\%$), and establish digital quantum computers as powerful tools for studying (effectively) continuous-time dynamics.

I. INTRODUCTION

Digital quantum computers face a tradeoff between hardware noise and digitization error, but sufficiently fine discretization can preserve thermal physics in a long-lived prethermal regime. Using Quantinuum’s H2, the paper studies effectively continuous-time Ising dynamics and emergent phenomena beyond practical classical simulation.

  • Quantum simulations must simultaneously suppress hardware noise with shallow circuits and digitization errors with deep circuits.
  • Small Trotter steps can produce Floquet prethermalization, whereas large steps drive digitized systems toward chaotic infinite-temperature states.
  • H2 is used to simulate effectively continuous-time transverse-field Ising dynamics and compare them with tensor-network, neural-network, and operator-truncation methods.
  • The study reaches a regime where no known classical methods are both efficient and trustworthy, enabling measurements of emergent hydrodynamics and gauge-theoretic dynamics.

II. DIGITIZED ISING DYNAMICS

The paper digitizes two-dimensional transverse-field Ising dynamics with periodic boundaries using second-order Trotterization. Finely stepped circuits exhibit a long prethermal window governed by a dressed Hamiltonian before eventual Floquet heating.

  • The transverse-field Ising model describes competing nearest-neighbor magnetic exchange and transverse-field quantum fluctuations on two-dimensional lattices.
  • Periodic boundary conditions reduce boundary effects, enable momentum-resolved energy relaxation, and induce topological constraints in thermalization studies.
  • Second-order Trotterization implements each time step with single-qubit X rotations enclosing layers of two-qubit ZZ gates.
  • For sufficiently small dt, observables equilibrate under a dressed Hamiltonian similar to the target Hamiltonian for a heating time τH ∼ exp(1/|Jdt|).
  • At late times, Floquet driving can continuously pump energy into the system, causing local observables to approach infinite-temperature values.

III. QUANTUM THERMALIZATION

The thermalization experiments combine error-mitigated quantum measurements with classical benchmarks across square and frustrated triangular lattices. They observe persistent prethermal behavior, hydrodynamic energy relaxation, and classical-simulation limits at larger entanglement and system size.

  • The initial-state offset Δθ controls the energy density and plays a temperature-like role in late-time observables.
  • For dt|J| = 0.25, quantum data show persistent prethermal order-parameter behavior rather than immediate relaxation to the infinite-temperature value.
  • Intermediate-temperature dynamics produce larger late-time entanglement, making MPS zero-truncation extrapolation ineffective except at short times.
  • Classical extrapolation becomes unreliable with modest system-size increases, while the deepest circuits contain over 2000 two-qubit gates and only about 10% error-free data.
  • The 14×4 quench exhibits diffusive energy relaxation with Γq ≈ Dq^2 and diffusion constant D = 0.38(5).

IV. EMERGENT HYDRODYNAMICS

Approximate energy conservation in the Floquet prethermalization regime produces hydrodynamic relaxation of long-wavelength energy inhomogeneities. The measured relaxation rates follow diffusion, yielding D = 0.38(5).

  • Approximate energy conservation produces transient local thermalization followed by slow hydrodynamic relaxation of long-wavelength energy inhomogeneities.This behavior is expected in chaotic prethermalizing systems when observables overlap with conserved energy density.
  • D = 0.38(5) for the diffusion constant extracted from long-wavelength energy-relaxation rates.The smallest nonzero wavevectors obey Γq ≈ Dq2, while larger-q components decay during initial local thermalization.
  • The q = 0 energy mode quickly saturates, while the smallest nonzero wavevectors relax at rates consistent with the heat equation.The energy density is averaged over the transverse direction and Fourier transformed along the elongated geometry.
  • The results provide direct evidence for an approximately conserved energy current in the Floquet prethermalization regime.The transport-coefficient calculation is relevant to materials simulation, although this particular simulation is not claimed to exceed classical limits.

V. EMERGENT GAUGE THEORY ON A FRUSTRATED LATTICE

On a frustrated triangular lattice, approximate energy conservation manifests as emergent gauge and topological constraints. Dynamics differ sharply between winding sectors: the maximal sector is effectively frozen, while the minimal sector rapidly relaxes within its constrained sector.

  • Emergent gauge structure: Antiferromagnetic interactions on a triangular lattice with periodic boundaries produce dynamics consistent with emergent gauge-theoretic and topological constraints.These constraints arise from approximate energy conservation achieved with sufficiently small Trotter steps.
  • Emergent gauge structure: The frustrated classical ground space maps to dimer coverings of a dual hexagonal lattice, and weak transverse fields produce an effective quantum dimer model.Each ground-state configuration has exactly one ferromagnetic bond per triangle.
  • Emergent gauge structure: The low-energy dimer subspace obeys a local Gauss-law constraint and, with periodic boundaries, conserves topological winding numbers Wx and Wy.The quantum dimer model preserves these constraints, whereas higher-order transverse-field processes can violate the local Gauss law in the full Ising dynamics.
  • Sector-dependent dynamics: The maximal (Wx, Wy) = (18, 6) sector is effectively frozen, whereas the minimal (Wx, Wy) = (0, 0) state quickly relaxes within its winding sector.At late times, the mobile state becomes ergodic while satisfying both Gauss-law and winding-number constraints.
  • Sector-dependent dynamics: The minimal-winding correlations approach classical Markov-chain Monte Carlo values, while late-time spatial correlations remain structured by gauge constraints.Small departures from ideal behavior are attributed to Trotter errors and finite transverse-field corrections beyond the effective model.

VI. OUTLOOK

The work presents stable equilibrium behavior emerging from digitized quantum dynamics at a scale that appears difficult for numerically exact classical simulation. Extending this capability to more general models will likely require larger systems and higher gate fidelities.

  • The work provides first evidence of stable equilibrium behavior from digitized quantum dynamics at a scale that appears difficult for numerically exact classical simulation.
  • Accurately simulating more general and realistic models will likely require improvements in both system scale and gate fidelities.The cited examples include more complex spin-spin interactions and fermionic degrees of freedom.

A. Quantum hardware and data acquisition

The experiments comprise four quench settings on Quantinuum’s H2-1 and H2-2 56-qubit trapped-ion computers. Cycle benchmarking and randomized interleaving estimate two-qubit gate infidelity at each Trotter step, including temporal drift.

  • Four quench experiments cover low- and intermediate-temperature square lattices, hydrodynamics, and a triangular lattice.The geometries are 7 × 8, 14 × 4, and 9 × 6, using H2-1 and H2-2 according to the experiment.
  • Cycle benchmarking circuits learn an error model for the two-qubit gates immediately before and after each quench.
  • Randomly interleaved benchmarking estimates average two-qubit gate infidelity at each Trotter step and tracks drift over time.

B. Error mitigation and data processing

The experiment suppresses hardware errors with circuit-level mitigation and post-processing. These methods include dynamical decoupling, randomized compiling, leakage-aware zero-noise regression, and calibrated extrapolation with bootstrap-derived uncertainties.

  • Dynamical decoupling suppresses coherent memory errors, while randomized compiling converts coherent memory and other errors into incoherent errors.
  • Zero-noise regression estimates the zero-noise limit of detected leakage and two-qubit gate errors instead of discarding all data containing detected leakage.
  • The extrapolation uses two points and an exponential fit, with amplification rates estimated from interleaved benchmarking circuits.
  • Reported error bars are standard errors of the mean obtained by bootstrap resampling raw or post-processed observables.

C. Zero truncation extrapolation of MPS data

Zero-truncation extrapolation estimates observables at unit simulation fidelity from MPS calculations with varying bond dimensions. It is reliable when global state fidelities remain above roughly 80%, but intermediate-temperature data become uncontrolled at longer times for χ ≤4000.

  • Zero-truncation extrapolation estimates ⟨O⟩ at F →1 by relating observables and estimated fidelities across MPS bond dimensions.The procedure linearly extrapolates using the three largest available bond dimensions.
  • Global state fidelities above ∼80% lead to reliable zero-truncation extrapolations in tested evolution times and system sizes.
  • For the low-temperature quench, fidelities above ∼80% are maintained through Trotter steps s ≤20 with maximum bond dimension χ =4000.
  • Intermediate-temperature data are not accurate beyond very short times with χ ≤4000 because the MPS state fidelities are insufficient.

S1. IMPLEMENTATION OF TIME EVOLUTION ON H2

The H2 implementation combines circuit-level techniques to address memory, gate, leakage, and measurement errors during digitized Ising-model evolution. The protocol uses dynamical decoupling, randomized compiling, leakage detection, and zero-noise regression while accounting for their residual costs and limitations.

  • The direct second-order Trotter circuit uses four layers of UZZ gates surrounded by UX gates, but its observables are degraded by memory, gate, and leakage errors.
  • Dynamical decoupling inserts periodic X gates that are logically equivalent to the original circuit and reduce coherent memory-error buildup.
  • Dynamical decoupling increases the number of single-qubit gates by roughly five times, making coherent single-qubit over- and under-rotations a relevant error source.The typical coherent rotation error is δθ ∼5 × 10−3 radians.
  • Randomized compiling is applied over two UZZ layers so residual memory and coherent two-qubit gate errors become incoherent.
  • Leakage detection maps leakage to ancilla measurement outcomes, while zero-noise regression estimates the zero-noise observable without exponential post-selection overhead.
  • False-positive leakage events are reduced by excluding |0⟩q |1⟩a outcomes, because the relevant false-positive mechanism is most likely for H2.The cited passage states this mechanism is approximately twice as likely as one alternative and five times as likely as another.

E. The error-mitigated Trotter circuit

The error-mitigated Trotter circuit combines dynamical decoupling, randomized compiling, leakage detection, and Pauli insertion for zero-noise extrapolation. These randomizations are implemented with compiled and real-time-generated Pauli operations to reduce circuit overhead.

  • The combined circuit uses dynamical decoupling, randomized compiling, leakage detection, and Pauli insertion for zero-noise extrapolation.
  • X Paulis from dynamical decoupling and random Paulis from randomized compiling are compiled together to reduce single-qubit gates and ion transport.
  • Random Paulis for zero-noise extrapolation are generated per shot using a classical pseudo-random number generator and conditional single-qubit gates.

A. Pauli error learning

The paper learns a Pauli error model for the non-Clifford UZZ gates using cycle benchmarking with pseudo-twirling, then uses it for zero-noise extrapolation. Benchmarking tests error-model stability across machines and simulation runs while accounting for leakage separately.

  • Pauli error channels: A Pauli error channel assigns probability pj to each Pauli error Pj, whose channel eigenvalues are called Pauli fidelities.Each Pauli operator is an eigenvector of the channel, with commutation or anticommutation determining the associated eigenvalue.
  • Cycle benchmarking: The UZZ(±2Jdt) error channel is estimated with a non-Clifford cycle-benchmarking protocol that uses pseudo-twirling.The protocol adapts Pauli-group twirling to the non-Clifford UZZ gate.
  • Cycle benchmarking: The protocol measures Pauli expectations after repeated pseudo-twirls to extract symmetrized Pauli fidelities from their dependence on cycle count.The measured expectation follows a form determined by whether the prepared Pauli commutes or anticommutes with ZZ.
  • Error model construction: The resulting Pauli error models are averaged across four gate zones and used for Pauli insertion in zero-noise extrapolation.The symmetrized fidelities are the quantities that can be learned independently of state-preparation-and-measurement errors.
  • Error-model stability: The total error scale can drift, but relative Pauli probabilities remain constant within error bars across measurements before and after simulation experiments.Leakage-detection post-selection excludes leakage errors from the learned model, which are treated separately.

B. Optimization of zero-noise extrapolation

The paper optimizes two-point zero-noise extrapolation by modeling observable decay as exponential in noise and selecting the amplified-noise level and sample allocation. The procedure uses an estimated noise response, assumes κ = 1 for parameter selection, and acknowledges uncertainty in scaling beyond the studied system size.

  • ZNE model: Zero-noise extrapolation amplifies a physical noise level p0 to learn O(p) and extrapolate the observable to O(0).The method treats the observable as an implicit function of noise and uses noisy measurements to estimate its zero-noise value.
  • ZNE model: The computed observable ⟨Z2_tot⟩ responds approximately exponentially to noise over the explored range, enabling two-point extrapolation.Numerical tests find exponential fits generally introduce no more than 2% bias.
  • Parameter optimization: Two-point ZNE estimates use the raw and noise-amplified values to infer the noiseless observable and optimize their sampling variance.The sample fraction r at p0 and the amplified-to-bare noise ratio α = p1/p0 jointly determine the variance.
  • Parameter optimization: The variance optimization depends on κ, an implicit function describing how the variance of Z2 changes with Z2 at fixed sample count.Because κ is difficult to determine a priori, the procedure chooses parameters assuming κ = 1.
  • Parameter optimization: For κ values spanning roughly one order of magnitude above or below unity, parameters optimized at κ = 1 impose little excess standard deviation.This supports ignoring the detailed dependence of estimate variance on noise level in practical parameter selection.
  • Practical implementation: The noise-response exponent is estimated from noisy simulations across system sizes and finite-size extrapolated to the 7×8 lattice.The method assumes the decay exponent scales linearly with circuit depth, ζ = c × s.
  • Practical limitations: For system sizes N > 56, the study cannot reliably determine the saturated noise-response value or how quickly mitigation overhead grows before saturation.This limits extrapolation of the error-mitigation cost beyond the largest studied system size.

C. Zero-noise regression

Zero-noise regression uses leakage detections from every circuit shot to estimate zero-noise observables with reduced statistical uncertainty. The paper also benchmarks extrapolation strategies for classical simulations, finding fidelity-based extrapolation generally more reliable than bond-dimension extrapolation in the studied regimes.

  • Zero-noise regression: Zero-noise regression fits observables against detected leakage errors, using all shots rather than only the zero-error subset.This avoids the exponentially costly post-selection procedure and can reduce uncertainty in the zero-noise estimate.
  • Zero-noise regression: Bootstrap resampling estimates observable error bars by repeatedly sampling the measurement dataset with replacement.The standard deviation across resampled observable values estimates the standard error of the original mean.
  • Thermal references: Imaginary-time purification prepares finite-temperature states, whose converged observables provide thermal reference values for comparison with quantum dynamics.The reported purification results converge by bond dimension χ = 512, and exact-statevector comparisons are used for small systems.
  • Classical extrapolation: Fidelity extrapolation to Fsim →1 is generally more reliable than extrapolation in 1/χ →0 for the studied intermediate-temperature simulations.For the examined data, fidelity extrapolation is more accurate above approximately 0.65 fidelity, while both methods become inaccurate at lower fidelities.
  • Classical extrapolation: Fidelity extrapolation remains reliable through s = 40 Trotter steps when the highest-bond-dimension simulations retain relatively high fidelity.For the intermediate-temperature quench, fidelity decreases faster, and extrapolation becomes substantially less reliable from s ≥ 10.
  • Classical extrapolation: Bond dimensions exceeding 2^16 = 65,536 are likely required for below-5% relative error on ⟨Ztot^2⟩ across s ≤ 20 in the 7 × 8 intermediate quench.This estimate motivates comparison with hardware data because χ = 2^16 was the largest bond dimension identified for comparable simulations.
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