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Measuring Spectral Form Factor in Many-Body Chaotic and Localized Phases of Quantum Processors

Hang Dong, Pengfei Zhang, Ceren B. Dag, Yu Gao, Ning Wang, Jinfeng Deng, Xu Zhang, Jiachen Chen, Shibo Xu, Ke Wang, Yaozu Wu, Chuanyu Zhang, Feitong Jin, Xuhao Zhu, Aosai Zhang, Yiren Zou, Ziqi Tan, Zhengyi Cui, Zitian Zhu, Fanhao Shen, Tingting Li, Jiarun Zhong, Zehang Bao, Hekang Li, Zhen Wang, Qiujiang Guo, Chao Song, Fangli Liu, Amos Chan, Lei Ying, H. Wang

arXiv:2403.16935v1quant-ph

TL;DR

Measuring many-body SFF is challenging because exponentially small level spacings are difficult to resolve experimentally. The experiment uses randomized measurements on superconducting processors to estimate SFF and pSFF for Floquet and Hamiltonian dynamics. The pSFF plateau time is roughly independent of subsystem size LA and equals the whole-system plateau time in both Floquet and Hamiltonian models.

Abstract

from arXiv · show

The spectral form factor (SFF) captures universal spectral fluctuations as signatures of quantum chaos, and has been instrumental in advancing multiple frontiers of physics including the studies of black holes and quantum many-body systems. However, the measurement of SFF in many-body systems is challenging due to the difficulty in resolving level spacings that become exponentially small with increasing system size. Here we experimentally measure the SFF to probe the presence or absence of chaos in quantum many-body systems using a superconducting quantum processor with a randomized measurement protocol. For a Floquet chaotic system, we observe signatures of spectral rigidity of random matrix theory in SFF given by the ramp-plateau behavior. For a Hamiltonian system, we utilize SFF to distinguish the quantum many-body chaotic phase and the prethermal many-body localization. We observe the dip-ramp-plateau behavior of random matrix theory in the chaotic phase, and contrast the scaling of the plateau time in system size between the many-body chaotic and localized phases. Furthermore, we probe the eigenstate statistics by measuring a generalization of the SFF, known as the partial SFF, and observe distinct behaviors in the purities of the reduced density matrix in the two phases. This work unveils a new way of extracting the universal signatures of many-body quantum chaos in quantum devices by probing the correlations in eigenenergies and eigenstates.

Supplementary material for “Measuring Spectral Form Factor in Many-Body Chaotic and

The supplementary material accompanies the study of localized phases in quantum processors.

S1. EXPERIMENTAL SETUP AND PROTOCOL FOR THE SFF MEASUREMENT

The experiment uses two superconducting processors and a randomized measurement protocol to estimate SFF and pSFF for programmable Floquet and Hamiltonian dynamics.

EXPERIMENTAL SETUP

  • Two superconducting processors provide controllable nearest-neighbor qubit chains, with experiments using up to L = 5 qubits.Processor I is a 6 × 6 array and processor II a 2 × 20 ladder; both support individual flux control, microwave rotations, and tunable couplers.Source: S1. EXPERIMENTAL SETUP AND PROTOCOL… · p. 8, Figure S1 · p. 8

EXPERIMENTAL PROTOCOL

  • The randomized protocol initializes |0⟩^⊗L, applies local random Clifford gates, evolves under Floquet or Hamiltonian dynamics, applies their adjoints, and measures in the z basis.For each realization, the procedure is repeated 3000 times, with readout-error mitigation used during data collection.Source: S1. EXPERIMENTAL SETUP AND PROTOCOL… · p. 8, Figure S2 · p. 9

MODEL IMPLEMENTATION

SFF AND PSFF ESTIMATION

  • SFF is estimated from measured bit-string probabilities, while pSFF is obtained by projecting those probabilities onto subsystem A.The same randomized-measurement framework supports both quantities, with subsystem measurements used for pSFF.Source: S1. EXPERIMENTAL SETUP AND PROTOCOL… · p. 9

S2. ERROR MITIGATION AND ESTIMATION OF THE PLATEAU TIMES

The supplementary analysis mitigates decoherence effects in measured SFF data and fits corrected Hamiltonian results to estimate plateau times.

DECOHERENCE MODELING

  • Qubit decoherence lowers the experimental SFF plateau relative to theoretical predictions, so noisy Lindblad simulations serve as more suitable references.The noisy simulations closely agree with the experimental SFF and pSFF results.Source: S2. ERROR MITIGATION AND ESTIMATION OF… · pp. 9–10

ERROR MITIGATION

  • Two error-mitigation schemes rescale the experimental SFF to make ramp-plateau and dip-ramp-plateau behavior more visible.The schemes give quite similar results for the Floquet model, and the main text reports the first scheme.Source: S2. ERROR MITIGATION AND ESTIMATION OF… · p. 10

PLATEAU-TIME ESTIMATION

  • For Hamiltonian data, corrected SFF curves are fitted to the GOE prediction to extract plateau times.The analysis compares the normalized corrected data with a fitting curve parameterized by μ and t_H.Source: S2. ERROR MITIGATION AND ESTIMATION OF… · p. 10

S3. EXPERIMENTAL RESULTS OF THE PSFF FOR OTHER SYSTEM SIZES

Additional pSFF measurements for smaller systems show that plateau times are largely subsystem-size independent and that chaotic and prethermal localized regimes remain distinguishable.

PLATEAU-TIME SCALING

  • The pSFF plateau time is representative and roughly independent of subsystem size L_A in both Floquet and Hamiltonian models.It is also equal to the plateau time of the whole system for the reported system sizes.Source: S3. EXPERIMENTAL RESULTS OF THE PSFF… · p. 10

CHAOTIC VERSUS LOCALIZED REGIMES

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