[Paper Review] Thermalization and Criticality on an Analog-Digital Quantum Simulator

Trond I. Andersen, Nikita Astrakhantsev+225 more|arXiv (Cornell University)|May 27, 2024

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This study demonstrates thermalization and critical dynamics in a 69-qubit superconducting analog-digital quantum simulator by engineering ramps through a quantum critical point in a 2D XY magnet. The platform enables high-fidelity state preparation, universal gate operations, and real-time monitoring of correlations and energy transport, revealing deviations from Kibble-Zurek scaling due to quantum-classical coarsening and validating eigenstate thermalization hypothesis (ETH) via controlled energy injection.

ABSTRACT

Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators. Unlocking the full potential of such systems toward this goal requires flexible initial state preparation, precise time evolution, and extensive probes for final state characterization. We present a quantum simulator comprising 69 superconducting qubits which supports both universal quantum gates and high-fidelity analog evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. Emulating a two-dimensional (2D) XY quantum magnet, we leverage a wide range of measurement techniques to study quantum states after ramps from an antiferromagnetic initial state. We observe signatures of the classical Kosterlitz-Thouless phase transition, as well as strong deviations from Kibble-Zurek scaling predictions attributed to the interplay between quantum and classical coarsening of the correlated domains. This interpretation is corroborated by injecting variable energy density into the initial state, which enables studying the effects of the eigenstate thermalization hypothesis (ETH) in targeted parts of the eigenspectrum. Finally, we digitally prepare the system in pairwise-entangled dimer states and image the transport of energy and vorticity during thermalization. These results establish the efficacy of superconducting analog-digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

Motivation & Objective

• To investigate thermalization dynamics in strongly correlated quantum systems beyond classical simulation limits.
• To probe the interplay between quantum and classical coarsening during quench dynamics through a critical point.
• To test the eigenstate thermalization hypothesis (ETH) by injecting variable energy densities into targeted parts of the eigenspectrum.
• To image energy and vorticity transport during thermalization using digitally prepared entangled dimer states.
• To establish a scalable analog-digital quantum platform for studying nonequilibrium quantum phenomena with high control and measurement fidelity.

Proposed method

• Utilizes a 69-qubit superconducting processor with both universal quantum gates and high-fidelity analog evolution for time evolution.
• Employs initial state preparation from an antiferromagnetic state and ramps the system through the 2D XY model's critical point.
• Applies multiple measurement techniques to characterize final-state correlations, vorticity, and energy imbalance across subsystems.
• Uses matrix product state (MPS) simulations with time-dependent variational principle (TDVP) to benchmark Kibble-Zurek and diffusion dynamics, up to bond dimension χ=1024.
• Implements energy injection via tailored initial states to probe ETH in specific eigenspectrum regions.
• Employs cross-entropy benchmarking and fidelity estimation (linear XEB and renormalized estimator) to validate experimental fidelity.

Figure 1: Analog-digital simulation with high-precision calibration. a, Our platform combines analog evolution with digital gates for extensive state preparation and characterization. b, Schematic of new scalable analog calibration scheme. Swap (blue) and single-photon (red) spectroscopy is used to

Experimental results

Research questions

• RQ1 How do quantum and classical coarsening mechanisms compete during thermalization after a quench through a quantum critical point?
• RQ2 To what extent do Kibble-Zurek scaling predictions hold in the presence of quantum fluctuations and finite-size effects?
• RQ3 Can the eigenstate thermalization hypothesis (ETH) be probed experimentally by tuning energy density in specific eigenspectrum regions?
• RQ4 How does energy and topological defect (vortex) transport evolve during thermalization in a 2D quantum magnet?
• RQ5 What is the role of entanglement growth in limiting classical simulation of thermalization dynamics?

Key findings

• The system exhibits signatures of the classical Kosterlitz-Thouless phase transition, confirming the emergence of topological order in the thermalized state.
• Deviations from Kibble-Zurek scaling are observed, attributed to the interplay between quantum and classical coarsening of correlated domains.
• Energy injection experiments confirm the validity of the eigenstate thermalization hypothesis (ETH) in targeted regions of the eigenspectrum, with good agreement between measured and predicted thermalization behavior.
• Energy and vorticity transport are directly imaged during thermalization, revealing diffusive-like behavior with measurable relaxation timescales.
• Matrix product state simulations accurately reproduce experimental data for short- and intermediate-ramp-time dynamics, but fail to capture long-time energy diffusion due to entanglement growth beyond χ=1024.
• The experimental fidelity, validated via cross-entropy benchmarking and fidelity estimation, exceeds classical simulation limits, confirming the platform's scalability and control.

Figure 2: Fast thermalization dynamics and beyond-classical capabilities in the high-temperature regime. a, Schematic representation of the experiment: $N_{\mathrm{q}}$ qubits are initialized in a half-filling state, evolved under a Hamiltonian $H_{\text{s}}$ over time $t$ with $4$ instances of diso