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Experimental Detection of Dissipative Quantum Chaos

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper reports the first experimental detection of dissipative quantum chaos, seen as a donut-shaped complex-spacing-ratio distribution in circuits run on a superconducting processor.

desk verdict A credible first experiment for dissipative quantum chaos; the integrable-branch claim needs a prespecified data-exclusion rule before I'd fully trust the T=5 peak. read the letter →

arxiv 2506.04325 v1 pith:ZK5AHXEM submitted 2025-06-04 quant-ph cond-mat.dis-nncond-mat.stat-mechnlin.CD

Kristian Wold , Zitian Zhu , Feitong Jin , Xuhao Zhu , Zehang Bao , Jiarun Zhong , Fanhao Shen , Pengfei Zhang
show 8 more authors
Hekang Li Zhen Wang Chao Song Qiujiang Guo Sergey Denisov Lucas Sá H. Wang Pedro Ribeiro
This is my paper · ORCID
classification quant-phcond-mat.dis-nncond-mat.stat-mechnlin.CD MSC 81Q5015B5281P68 PACS 03.65.Yz05.45.Mt03.67.Lx
keywords dissipativequantumchaoscomplexspacingratioschannelssuperconductingprocessorrandommatrixtheoryintegrability-to-chaoscrossoveropensystemsnoise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports the first experimental detection of dissipative quantum chaos and integrability, performed on a five-qubit superconducting processor. The authors build a four-qubit open quantum system by running a five-qubit unitary circuit and discarding one ancilla qubit, then reconstruct the resulting quantum channel from measurements and study the complex spacing ratios (CSRs) of its eigenvalues. They find the predicted 'bitten donut' CSR distribution for a chaotic circuit, signalling level repulsion of the class-AI type, and a sharp peak at the origin for a free-fermion integrable circuit, signalling uncorrelated eigenvalues. They also find that increasing circuit depth from 5 to 50 layers drives the integrable circuit's statistics through a crossover to fully chaotic behavior, which they attribute to the processor's intrinsic noise acting as a dissipative chaotic process. The result matters because it turns quantum computers, usually used for unitary simulations, into testbeds for open-system many-body physics.

What carries the argument

The central object is the quantum map $\Lambda$ built by running a five-qubit unitary circuit and tracing out one ancilla qubit, so the remaining four-qubit dynamics are dissipative; the unitary is partitioned into $2\times2$ blocks and the first column supplies the Kraus operators. The diagnostic is the complex spacing ratio (CSR), $z_i = (\lambda_i - \lambda_i^{NN})/(\lambda_i - \lambda_i^{NNN})$, formed from each complex eigenvalue and its two nearest neighbours, which measures level repulsion in the complex plane. The argument is carried by comparing measured CSR distributions against two ensembles computed for the same 4+1 geometry: AI4,1, built from a Haar-random five-qubit unitary and expected for chaotic maps in symmetry class AI, and FF4,1, built from free-fermion matchgate unitaries with $U(1)$ particle-number symmetry. A gradient-based process-tomography routine, with SPAM errors included, retrieves each channel from a few thousand Pauli measurement modes, and bootstrap resampling converts sampling noise into error bars. This chain, from unitary circuit to partial trace to tomographic retrieval to CSR statistics, is what lets a small quantum processor act as a probe of dissipative spectral universality.

What would settle it

Tomographically retrieve the T=10 chaotic circuit's full unitary and test whether its eigenvalue statistics match those of a Haar-random 5-qubit unitary; a measurable mismatch would mean the observed donut cannot be taken as evidence for universal dissipative chaos in symmetry class AI.

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Extended reading notes

Core claim

The paper's central claim is that the spectral fingerprints of dissipative quantum chaos and integrability can be measured directly on existing hardware. Running a depth-10 brickwork circuit of random Z- and Y-rotations and CZ gates produces a channel whose complex eigenvalue spacing distribution is a 'bitten donut', with density suppressed at the origin and along the positive real axis, matching the finite-size random-matrix ensemble AI4,1. The same analysis on a free-fermion (matchgate) circuit at depth 5 yields the sharp central peak expected for uncorrelated eigenvalues, once the particle-number symmetry sector is isolated. Increasing the free-fermion circuit's depth to 20, 30, 40, and 50 layers, without sector resolution, moves the distribution progressively from the integrable peak through a flat disk to the chaotic donut, which the authors read as evidence that the processor's intrinsic noise is itself a dissipative chaotic process.

Load-bearing premise

The chaotic reading depends on an earlier paper's claim, accepted without re-testing here, that the ten-layer circuit is effectively a random unitary; if that is false, the donut could belong to this circuit family rather than to universal dissipative chaos.

Editorial extensions

If this is right

  • The donut-shaped CSR distribution is a measurable, universal signature of dissipative quantum chaos in a finite-size open system, matching random-matrix predictions for symmetry class AI.
  • Integrable free-fermion dissipative circuits show a sharp CSR peak at the origin, so the same statistic cleanly separates chaos from integrability on real hardware.
  • Intrinsic hardware noise is itself a chaotic dissipative process: at sufficient depth it breaks engineered symmetries and drives the spectrum to AI-class statistics.
  • Quantum processors can be repurposed as programmable simulators of open many-body physics, with circuit depth as the control knob for dissipation.
  • A depth-swept CSR measurement gives a crossover time that could serve as a noise benchmark, since it is shorter than both coherence and relaxation times.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the same tomography-plus-CSR pipeline could be pointed at a disordered or driven open circuit to test for dissipative many-body localization, a regime the paper names only as future work.
  • Inference: running the chaotic circuit at intermediate depths T=6 to 9 would map the full onset curve of the donut, testing whether the Haar-convergence rate, not just the final statistics, controls the crossover.
  • Inference: if the T=10 Haar-equivalence premise fails under direct spectral scrutiny, the donut may be specific to this circuit family; re-deriving that equivalence from the measured spectrum would separate universal chaos from finite-family effects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript reports the first experimental observation of dissipative quantum chaos and integrability in engineered open many-body quantum systems. A 5-qubit superconducting processor executes either a free-fermion (integrable) or a random-gate (chaotic) unitary circuit on four system qubits plus one ancilla; after tracing out the ancilla, the induced completely positive trace-preserving map is reconstructed by gradient-based quantum process tomography with SPAM correction. The complex spacing ratio (CSR) distribution of the retrieved map is compared with finite-size random-matrix predictions: a "bitten donut" for the chaotic map (AI class, AI4,1) and a peak at the origin for the integrable map (FF4,1). For the integrable circuit, increasing depth T from 5 to 50 produces a crossover from integrable to chaotic CSR statistics, which the authors attribute to intrinsic hardware noise acting as a dissipative chaotic process.

Significance. If the claims hold, this is a notable first: it moves dissipative quantum chaos from theory to experiment and demonstrates that noisy quantum processors can serve as testbeds for open-system spectral statistics. The paper's strengths include the use of finite-size exact predictions rather than thermodynamic-limit distributions, the explicit synthetic benchmarks of the retrieval protocol in SI SII3, and the bootstrap error estimates in SI SII2. The CSR diagnostic is already established, so the novelty lies in the experimental realization. The main risks are not the chaotic donut—which is stark and supported by benchmarks—but the integrable-branch data selection and the reliance on a self-cited claim of Haar randomness for the chaotic circuit.

major comments (2)
  1. [SI SIII4, Fig. S8] The integrable T=5 result in Fig. 2c rests on the post hoc exclusion of runs whose particle-number sectors overlap substantially. The text says "for a few runs, there can be substantial overlap of the sectors (right panel), which we discarded," but no prespecified criterion, no threshold on the ⟨Q⟩-distribution overlap, and no count of discarded runs are given. Because the discarded runs are exactly those in which hardware noise has strongly mixed the symmetry sectors, keeping only clean runs biases the ensemble toward the ideal free-fermion map. If more than a small fraction of runs were discarded, the observed peak at the origin could be a selection artifact rather than a property of the dissipative circuit. Please specify the exclusion rule a priori, report the number of discarded runs, and show that the conclusion is unchanged when all runs are included (e.g., full-spectrum CSR without sector projection, or a robustness sweep over the threshold).
  2. [Main text, "Integrable and chaotic quantum maps"] The identification of the T=10 chaotic circuit as "effectively indistinguishable from a Haar-random 5-qubit unitary" is cited to Ref. [24], a preprint by three of the present authors, and is not independently tested here. The synthetic benchmarks in SI SII3 use the same T=10 chaotic circuit as the target map and compare the retrieved CSR to the ideal AI4,1 distribution, so they validate the tomography protocol under the assumption that the circuit is already in the AI4,1 universality class; they do not test convergence to Haar. Since the donut shape is the central chaotic signature and the comparison to AI4,1 is quantitative, please provide a direct metric of Haar convergence (e.g., frame potential, Hilbert-Schmidt distance to the CUE, or spectral form factor of the compiled unitary) or explicitly soften the claim to "consistent with" the AI4,1 prediction.
minor comments (5)
  1. [Main text, Fig. 2 caption and "Integrable and chaotic quantum maps"] The Gaussian smearing width δ=0.05 and the histogram truncation at 0.6 are introduced without a sensitivity analysis; please show the raw unsmoothed distributions or demonstrate that the qualitative conclusions are robust to the choice of δ.
  2. [Main text, after Fig. 2] The sentences "level repulsion is clearly absent in the integrable case (Fig. 2d)" and "the depletion near the origin in Fig. 2c" appear to have the figure panels swapped: the integrable case is shown in Fig. 2c and the chaotic case in Fig. 2d.
  3. [Fig. 2c and Fig. 3a] Fig. 2c uses 7 integrable circuits with T=5, while Fig. 3a uses 10 integrable circuits of depth T; please clarify whether these are the same or different experimental datasets, and if different, state why the sample sizes differ.
  4. [SI SIII3 and main text] The paper states that purely real eigenvalues should be excluded from the CSR analysis, but it does not state whether this was done for the experimental spectra and with what distance threshold; please specify this for reproducibility.
  5. [Discussion] The claim that the crossover time "is notably shorter than both the coherence and relaxation times" is not supported by any measured timescale in the paper; please either provide the relevant data or remove the quantitative comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: measured CSR distributions are compared with independently computed RMT ensembles, and the self-cited Haar-equivalence claim is backed by in-paper synthetic benchmarks.

full rationale

The paper's derivation chain is not circular. The predictions are the finite-size RMT distributions AI4,1 and FF4,1, which are computed in Supplementary Information SIII1 by explicit CUE sampling and by a free-fermion construction with a U(1) symmetry projection—not by fitting any experimental output. Experimental quantum maps are retrieved by gradient-based tomography (SII1) and their CSR statistics are compared with those precomputed distributions; the tomography is itself benchmarked on synthetic data in SII3 against the same ideal ensembles, showing agreement within error bars. The chaotic circuit's identification as effectively Haar-random at T=10 is cited to Ref. [24] by three present authors, but this is not the sole support: the in-paper synthetic benchmark in SII3 simulates the actual T=10 chaotic circuits and verifies that their retrieved CSR statistics match AI4,1, so the self-citation is not load-bearing in isolation. The only flagged caveat is the data-exclusion step in SIII4, where T=5 runs with substantial overlap of particle-number sectors 'were discarded' without a prespecified quantitative criterion; this is a legitimate reproducibility and selection-bias concern about the integrable-branch evidence, but it is not a circular equivalence because the FF4,1 prediction is not constructed from the retained experimental data. Overall, the central claims compare measurements with independently generated theoretical ensembles rather than reducing to fitted inputs or self-referential definitions.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, or mediators are introduced. The main external inputs are the CSR theory from Ref [19], the tomography and Haar-randomness claims from Ref [24], and the finite-size AI4,1/FF4,1 numerical simulations. The hand-chosen Gaussian smearing and the qualitative run-exclusion criterion are the clearest free parameters in the pipeline.

free parameters (2)
  • Gaussian smearing width delta = 0.05
    Applied to experimental CSR histograms before comparison with theoretical predictions (Fig. 2 caption); the paper does not report sensitivity to this choice, so the visual agreement in Fig. 2 may depend on it.
  • Run exclusion threshold for integrable T=5 data
    Runs with substantially overlapping particle-number sectors were discarded (SI Fig. S8); no numerical threshold or count is given, so this is a hand-chosen selection criterion affecting the integrable CSR measurement.
assumptions (4)
  • domain assumption The CSR distribution P(z) distinguishes chaotic (AI, donut shape) from integrable (flat or peaked) spectra.
    The statistical diagnostic is taken from Ref [19] (Sa, Ribeiro, Prosen) and used as the ground truth throughout the paper.
  • ad hoc to paper The T=10 chaotic circuit is effectively Haar-random.
    Cited to Ref [24] by three current authors; load-bearing for identifying the measured distribution with the AI4,1 ensemble.
  • domain assumption Gradient-based process tomography recovers the true quantum map with acceptable accuracy.
    The protocol from Ref [24] is used to retrieve the 16 by 16 map; synthetic benchmarks in SI SII3 test this on simulated data, not on hardware ground truth.
  • standard math Bootstrap resampling over finite sampling and optimization gives reliable error estimates.
    Used for the error bars in Fig. 3b; assumes the maps fitted to resampled data are representative of hardware and sampling fluctuations.

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Cite this review

Pith. "Pith review of Experimental Detection of Dissipative Quantum Chaos." pith.science (2026). https://pith.science/paper/ZK5AHXEM

@misc{pith2026250604325,
  author       = {Pith},
  title        = {Pith review of: Experimental Detection of Dissipative Quantum Chaos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZK5AHXEM}},
  note         = {Machine review of arXiv:2506.04325}
}
read the original abstract

More than four decades of research on chaos in isolated quantum systems have led to the identification of universal signatures -- such as level repulsion and eigenstate thermalization -- that serve as cornerstones in our understanding of complex quantum dynamics. The emerging field of dissipative quantum chaos explores how these properties manifest in open quantum systems, where interactions with the environment play an essential role. We report the first experimental detection of dissipative quantum chaos and integrability by measuring the complex spacing ratios (CSRs) of open many-body quantum systems implemented on a high-fidelity superconducting quantum processor. Employing gradient-based tomography, we retrieve a ``donut-shaped'' CSR distribution for chaotic dissipative circuits, a hallmark of level repulsion in open quantum systems. For an integrable circuit, spectral correlations vanish, evidenced by a sharp peak at the origin in the CSR distribution. As we increase the depth of the integrable dissipative circuit, the CSR distribution undergoes an integrability-to-chaos crossover, demonstrating that intrinsic noise in the quantum processor is a dissipative chaotic process. Our results reveal the universal spectral features of dissipative many-body systems and establish present-day quantum computation platforms, which are predominantly used to run unitary simulations, as testbeds to explore dissipative many-body phenomena.

Figures

Figures reproduced from arXiv: 2506.04325 by the authors.

Figure 1
Figure 1. Dissipative quantum chaos and integrability in infinite and finite systems. a, Quantum map Λ obtained by evolving the system and ancilla qubits with a unitary circuit U, after which the ancilla qubits are discarded. The system qubits are prepared in an initial state ρ and evolve into a final state ρ ′ , while the ancilla qubits are initialized in a fixed pure state. b, Computation of the complex spacing ratios (CSRs… view at source ↗
Figure 2
Figure 2. Measured complex spacing ratio (CSR) distributions for finite-size dissipative quantum chaos and integrability. A 4-qubit quantum map is generated by applying a 5-qubit circuit with T layers and tracing out an ancilla qubit. In the experiments, the circuits are further compiled into two-qubit CZ gates and single-qubit gates that can be natu￾rally realized on the processor (see Supplementary Information SI). a, Integ… view at source ↗
Figure 3
Figure 3. Integrability-to-chaos transition with increasing circuit depth. a, Complex spacing ratio distribution P(z) for an ensemble of 10 integrable circuits of different depth T. The distribution exhibits a characteristic free-fermion peak at the origin for T = 5, evolves into an approximately flat disk by T = 20, and eventually develops the bitten-donut shape that is indicative of quantum chaos at T = 50. Particle number … view at source ↗

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    Experimental circuit optimization Figure 2a–b of the main text shows the theoretical circuits of dissipative quantum chaos and integrability. In our experiment, we further transpile them into cir- cuits that only contain single-qubit and CZ gates using Qiskit [34]. For example...

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    Map retrieval procedure To retrieve quantum maps that model noisy quantum circuits, we utilize the gradient-based quantum process tomography protocol introduced in Ref. [24]. The proto- col implements a parameterized Kraus map Tr(ρ;θ) = rX s=1 Ks(θ)ρK † s (θ), rX s=1 K † s (θ)...

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    In particular, it is im- portant to establish how the errors propagate to statistics related to the CSRs, as this is a metric particularly sen- sitive to perturbations

    Bootstrap Error Estimation We employ a bootstrap resampling procedure for esti- mating the combined error of the map retrieval protocol and finite sampling on hardware. In particular, it is im- portant to establish how the errors propagate to statistics related to the CSRs, as...

    2000

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    We then tomo- graphically retrieve the maps from this data and compare them to the known original maps, on the level of their CSR distributions

    Synthetic benchmarks To benchmark the accuracy of the tomography method, we generate synthetic data sampled from known SPAM error models and quantum maps. We then tomo- graphically retrieve the maps from this data and compare them to the known original maps, on the level of th...

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    This suggests that the retrieval protocol is slightly biased at the level of CSR when re- trieving strongly integrable dynamics

    Also some overestimation happens around θ=±πandθ= 0. This suggests that the retrieval protocol is slightly biased at the level of CSR when re- trieving strongly integrable dynamics. Calculating the mean of these marginal distributions, we find, for the chaotic circuit,⟨r⟩= 0.7...

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    particle-number

    Numerical implementation of dissipative quantum circuits LetUbe a unitary evolution operator onLqubits. We divide theLqubits intonsystem qubits, which are not traced out (i.e., erased by the environment), ande environment(orancilla)qubits, whichare. Ifweinitialize all the envi...

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    The results are shown in Fig

    Finite-size effects in the CSR distribution Following the procedure described above, we computed the CSR distribution ofΛfor FFn,e with different values ofnande. The results are shown in Fig. S7. We see that, asnandeincrease, the distribution approaches the flat Poisson distri...

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    particles

    Symmetries in FF circuits When performing a level statistics analysis, all symme- tries need to be resolved before the statistics (say, CSRs) are computed. That is, if the quantum channelΛhas a symmetryQ,[Λ,Q] = 0, then, in the eigenbasis ofQ, Λis block diagonal and the eigenv...

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    However, if the noise is weak and the circuit is sufficiently shallow, the symmetry is only weakly broken and states in different sectors hybridize little

    Approximate Symmetries While the circuit that is theoretically implemented might have a given symmetry, the intrinsic noise of the hardware will not respect it, thus mixing different sec- tors. However, if the noise is weak and the circuit is sufficiently shallow, the symmetry...

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Reviewed August 7, 2026 · model on record in the stance chip above.