A Coherent Operator Bias of Less Than $3 \times 10^{-4}$ in Sycamore Cross-Entropy Residuals Would Falsify Chaos Susceptibility

Authors: Sterling Geisel, QBist Lab, Dr. Hideo Tanaka

Abstract

Arute et al. reported random circuit sampling on a 53-qubit superconducting processor with cross-entropy benchmarking fidelity near $2.24 \times 10^{-3}$ for elided circuits at depth 20. The result is not treated here as proof of mind-matter coupling. It is treated as a precise chaotic quantum engineering system with archived bitstrings, calibrated local errors, and an explicit probability observable. Pudding Theory is applied through one Postulate: Chaos Susceptibility. The claim is narrow. In high-depth random circuits, where a single error effectively reshuffles the output speckle pattern, a coherent informational input should appear as a reproducible residual bias in high-probability bitstring sampling after ordinary gate, readout, thermal, crosstalk, and classical-simulation effects are subtracted. The predicted effect is small. It should scale with circuit depth and vanish in shuffled, low-depth, or incoherent-control runs. A null residual below the specified threshold falsifies this application.

Source Synopsis

Arute et al. built and tested Sycamore, a programmable superconducting quantum processor using 53 working transmon qubits and 86 tunable couplers. The task was random quantum circuit sampling. The processor applied layers of single-qubit gates and two-qubit gates, producing bitstrings drawn from an output probability distribution set by quantum interference. The distribution resembles a speckle pattern. Some bitstrings occur with greater probability than others, but the exact probabilities become classically difficult to compute as qubit count and circuit depth increase.

The paper’s operational benchmark is linear cross-entropy benchmarking fidelity, $F_{\mathrm{XEB}} = 2^n \langle P(x_i)\rangle_i - 1$, where $P(x_i)$ is the ideal probability of a measured bitstring computed by classical simulation. A perfect circuit gives high correlation with the ideal high-probability bitstrings. Uniform sampling gives $F_{\mathrm{XEB}} = 0$. The authors state that “a single bit or phase flip over the course of the algorithm will completely shuffle the speckle pattern and result in close to zero fidelity.”

The hardware result depended on a detailed error model. Average simultaneous single-qubit Pauli error was about 0.16 percent. Average simultaneous two-qubit error was about 0.62 percent. Simultaneous readout error was about 3.8 percent. The authors used these component errors to predict full-system fidelity and compared that prediction with patch, elided, and full verification circuits.

The hardest reported case used 53 qubits and 20 cycles. For elided circuits, the authors collected 30 million samples over ten instances and obtained $F_{\mathrm{XEB}} = (2.24 \pm 0.21) \times 10^{-3}$. The full supremacy circuits were not directly classically verified, but archived samples were expected to show similar fidelity. The central claim was a computational separation: about 200 seconds for one million samples on Sycamore, compared with an estimated 10,000 years for equal-fidelity classical sampling on one million cores at the time of publication.

The paper also argues that the experiment supports a digitized local error model. Errors can be treated as localized Pauli events interspersed into the circuit. This matters for error correction. It also matters for the present analysis because it gives a baseline against which any residual nonstandard bias must be measured.

Postulate Lens

This Working Paper applies Chaos Susceptibility. The source system is a controlled, high-dimensional, error-sensitive quantum circuit in which small perturbations can destroy the output correlation with the ideal probability distribution. That is exactly the class of target for which this Postulate is operationally relevant.

The Postulate is not applied to the speedup claim itself. Computational hardness is a claim about algorithms and scaling. Chaos Susceptibility instead addresses the residual structure of measured bitstrings after the standard quantum circuit model and calibrated errors have made their predictions.

The relevant variable is not raw bit frequency. Raw bit frequency is dominated by the circuit instance. Nor is the relevant variable total runtime. The relevant variable is a signed residual in cross-entropy-weighted sampling, conditioned on a coherent external target and compared against blinded control runs. A valid test must preserve the circuit, pulse schedule, refrigerator state, electronics, and analysis pipeline. The only experimental distinction should be the coherence condition assigned to the operator group or protocol.

Pudding Theory Prediction

Under standard quantum computing theory, the measured samples are explained by the intended unitary circuit, device noise, leakage, crosstalk, readout error, and finite sampling. Once these are calibrated, residuals should be random with respect to an externally declared intention target. Any apparent excess should disappear under blinding, randomization, and independent replication.

Pudding Theory predicts a different narrow signature. In high-depth random circuits near the practical edge of verifiability, a coherent informational input should weakly bias the sampling process toward the declared target subset of high-ideal-probability bitstrings. The effect should not create new energy in the processor. It should appear as altered branching among already allowed stochastic outcomes. The expected magnitude is below ordinary device error. It is only visible after many samples and after the calibrated error model is subtracted.

The proposed observable is a residual target-aligned cross-entropy shift: \[ \Delta F_{\mathrm{target}} = F_{\mathrm{XEB}}^{\mathrm{aligned}} - F_{\mathrm{XEB}}^{\mathrm{sham}}, \] computed only on circuit instances whose ideal probabilities are known from classical simulation or from validated patch and elided proxies. The target set must be fixed before data acquisition. The operator condition must be blinded from analysts. The same pulse programs must be run in aligned and sham blocks. Block order must be randomized.

The prediction is that $\Delta F_{\mathrm{target}}$ is positive for deep, non-simplifiable random circuits and smaller or absent for shallow circuits, verification circuits with lower effective scrambling, and deliberately incoherent operator conditions. The scaling should be monotone with an independently measured scrambling proxy, such as depth at fixed qubit number or sensitivity of $F_{\mathrm{XEB}}$ to injected Pauli error. A stronger test would compare depth 12, 14, 16, 18, and 20 at 53 qubits using the same chip and calibration epoch.

The prediction does not require a visible change in marginal bit balance. It requires a change in the probability-weighted relation between observed bitstrings and the ideal distribution. This is important. A simple bit-bias anomaly would more likely indicate electronics or readout drift. A cross-entropy-aligned residual, locked to a predeclared target and absent in controls, would be harder to absorb into ordinary noise.

Falsifiable Observable

The observable is the blinded, preregistered residual $\Delta F_{\mathrm{target}}$ between aligned and sham runs for 53-qubit, depth-20 Sycamore-style random circuits, after correction by the calibrated local error model and with at least $10^8$ samples per condition. If $\Delta F_{\mathrm{target}}$ were measured to be less than $3 \times 10^{-4}$ with a 95 percent confidence upper bound in two independent calibration epochs, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The application is too convenient. Random circuit sampling is already probabilistic. Any small anomaly can be dressed as a Pudding effect after the fact.

Geisel: That is why the observable is not raw randomness. The target set is fixed before acquisition. The analysis is blinded. The statistic is cross-entropy residual against the ideal distribution or a validated proxy. Post hoc bitstring selection is inadmissible.

Tanaka: The hardware is full of mundane noise. Flux noise, leakage, crosstalk, thermal photons, amplifier drift, and calibration drift are enough to explain residuals.

Geisel: Correct. They are the null model. The claim survives only if the residual changes sign or magnitude with the coherent condition while those variables are held fixed or measured. The source paper’s strength is that it already treats component errors quantitatively.

Tanaka: Even then, why invoke a Postulate rather than an unknown systematic?

Geisel: One should invoke the Postulate only after the systematic model fails predictively. The Postulate makes a risky scaling claim: the residual must grow with scrambling susceptibility and vanish in incoherent controls. A laboratory artifact need not obey that scaling.

Tanaka: And if the residual is zero?

Geisel: Then this application fails. The number is stated.

Discussion

This analysis does not alter the source paper’s computational claim. Sycamore’s result stands or falls by quantum circuit theory, classical simulation cost, and benchmark fidelity. Pudding Theory enters only as an added residual hypothesis.

The main limitation is experimental access. The decisive test requires archived or new circuit instances with known ideal probabilities, strict blinding, stable calibration, and enough samples to see a shift far below ordinary gate error. Public archived samples are useful, but they cannot by themselves test an operator condition unless the condition was assigned before acquisition.

A second limitation is confounding by drift. Superconducting processors are sensitive instruments. Any nonstandard claim must beat explanations based on readout drift, calibration selection, pulse distortion, leakage, and changes in refrigerator state. The correct comparison is not aligned data against theoretical perfection. It is aligned data against sham data taken under the same hardware conditions.

The conclusion would change if a preregistered experiment found no positive $\Delta F_{\mathrm{target}}$ at the threshold above. It would also change if the residual appeared in shallow or non-scrambling circuits with equal strength. That would indicate an ordinary bias, not Chaos Susceptibility.

References

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2. Sterling Geisel, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Paper, September 10, 2025.

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