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Supplementary information for "Quantum supremacy using a programmable superconducting processor"

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus R. Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandra, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, John M. Martinis

arXiv:1910.11333v2quant-ph

TL;DR

The work addresses how to benchmark large programmable quantum systems when full classical computation and tomography become impractical. It develops scalable XEB-based benchmarking methods and reports a nonzero fidelity of (2.24 ± 0.21) × 10^-3 for 53-qubit, 20-cycle circuits, rejecting the null hypothesis at 6σ.

  • Problem

    Benchmarking large quantum systems is difficult because full tomography and classical XEB computation scale exponentially with system size.

  • Method

    The paper develops scalable XEB benchmarking using patch circuits, circuit variations, and fidelity-error decomposition.

  • Results

    (2.24 ± 0.21) × 10^-3: mean fidelity for 10 random 53-qubit, 20-cycle circuits, with the null hypothesis rejected at 6σ.

  • Takeaways & Limitations

    The reported benchmark provides evidence of a nonzero XEB fidelity for the tested 53-qubit, 20-cycle circuits.

  • Takeaways & Limitations

    Full-circuit benchmarking requires classical resources that grow with qubit number, motivating modified circuits that provide approximate performance predictions.

Abstract

from arXiv · show

This is an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature. The main article is freely available at https://www.nature.com/articles/s41586-019-1666-5. Summary of changes since arXiv:1910.11333v1 (submitted 23 Oct 2019): added URL for qFlex source code; added Erratum section; added Figure S41 comparing statistical and total uncertainty for log and linear XEB; new References [1,65]; miscellaneous updates for clarity and style consistency; miscellaneous typographical and formatting corrections.

I. DEVICE DESIGN AND ARCHITECTURE … A. XEB of a small number of qubits

The supplementary information describes Sycamore’s tunable-coupling architecture, 54-qubit layout, synchronized control and multiplexed readout systems, and the XEB framework used to estimate circuit fidelity and gate errors. It connects random-circuit averaging and exponential decay to depolarization fidelity while allowing cycle-polarization estimates independent of SPAM errors.

  • I. DEVICE DESIGN AND ARCHITECTURE: Sycamore targets 0.1% two-qubit-gate errors for error correction, while 0.3-0.6% error rates suffice for quantum supremacy.Short gates motivate tunable transmons with direct, tunable coupling; coupling is minimized when inactive to reduce residual control errors.
  • I. DEVICE DESIGN AND ARCHITECTURE: Tunable qubits require only small frequency excursions, reducing 1/f flux-noise dephasing and enabling coupling to be turned off during measurement.The architecture balances strong active coupling for fast gates against weak inactive coupling for control and measurement crosstalk suppression.
  • II. FABRICATION AND LAYOUT: The processor contains 142 transmons: 54 individually controlled and read-out qubits plus 88 adjustable couplers operated in their ground state.Nearest-neighbor coupling combines direct capacitive and tunable indirect channels, with a flux bias enabling net-zero coupling.
  • A. Control: Control uses 54 microwave signals, 54 qubit flux lines, 88 coupler flux biases, and additional microwave readout signals with phase-sensitive receivers.Custom 14-bit, 1 GS/s DAC modules form a >250-channel phase-synchronous waveform generator with 20 ps interchannel jitter across four chassis.
  • B. Readout: Dispersive readout infers qubit state from resonator phase shifts, using cryogenic amplification, FPGA demodulation, and frequency multiplexing of nine six-qubit groups.The weak probe signal is amplified through an IMPA and cryogenic HEMT chain before IQ processing and digitization.
  • IV. XEB THEORY: XEB estimates single- and two-qubit gate errors and large-circuit fidelity by comparing measured bitstrings with simulated random-circuit probabilities.Errors destroy the random-state speckle pattern, while randomized circuits connect the polarization parameter εm to depolarization fidelity.
  • A. XEB of a small number of qubits: For two qubits, the predicted circuit depolarizing fidelity closely matches the averaged squared overlap across swap-angle errors and circuit depths.The comparison supports the depolarizing-channel model for sufficiently randomized single-qubit gates, including systematic two-qubit errors.
  • A. XEB of a small number of qubits: Linear XEB uses f(ps(q)) = Dps(q) −1, while standard XEB uses f(ps(q)) = log(ps(q)); fitting polarization estimates pm c decay yields pc independently of SPAM errors.Multiple circuits with the same cycle count are combined, and the resulting estimates are fit as an exponential decay in m.

B. XEB of a large number of qubits

For large-qubit circuits, linear and logarithmic XEB estimate fidelity from experimentally sampled bitstrings under a concentration-of-measure assumption. Logarithmic XEB has lower variance above F = 0.32, while linear XEB is better below that threshold; the HOG-based estimator is noisier than XEB.

  • Fidelity estimation: For n ≫1, XEB estimates circuit fidelity F by mapping sampled output bitstrings through functions of their ideal probabilities, with sampling accuracy 1/√Ns.The method assumes error-induced probabilities are typically uncorrelated with the ideal circuit’s chaotic speckles, requiring statistical fluctuations ϵ ≪F.
  • Estimator comparison: Porter-Thomas-distributed measurement probabilities yield explicit linear- and logarithmic-XEB fidelity estimators whose sampling variances are compared experimentally using elided circuits.Figure S8 compares the estimates from observed bitstrings, with standard deviations smaller than the markers.
  • Estimator comparison: For F > 0.32, logarithmic XEB has smaller standard deviation and is best near F ≈1; for F < 0.32, linear XEB is better and optimal when F ≪1.Linear XEB also relates to the maximum likelihood estimator in the low-fidelity regime.
  • HOG-based estimator: The HOG-based fidelity estimator counts measured bitstrings with Dps(q) ≥log(2), but its standard deviation [log−2(2) −F 2]/Ns is always larger than XEB’s.This estimator is related to the HOG test and a definition of quantum volume.

C. Two limiting cases

The section examines two limiting cases for the linear XEB fidelity estimator. Uniform sampling gives zero fidelity, while ideal Porter–Thomas sampling gives FXEB = 1; depolarizing errors follow by convex combination.

  • Uniform sampling: Uniformly sampled bitstrings have FXEB = 0, so the estimator correctly returns zero fidelity for a maximally mixed state.Every bitstring has sampling probability 1/D.
  • Porter–Thomas sampling: Porter–Thomas sampling from a random circuit’s theoretical output distribution yields FXEB = 1.The result follows by substituting the average probability of a sampled bitstring into the main-paper formula.
  • Depolarizing errors: The general depolarizing-error case is obtained by taking a convex combination of the two limiting cases.The two endpoints are uniform sampling and ideal Porter–Thomas sampling.

D. Measurement errors · V. QUANTIFYING ERRORS

The supplementary analysis separates measurement fidelity from circuit fidelity and explains how preparation and measurement errors affect fidelity estimates. It also develops a depolarizing-error framework connecting single- and two-qubit measurements, XEB, RB, and Pauli error metrics.

  • D. Measurement errors: Uncorrelated measurement errors are modeled by bit-dependent probabilities for flipping measured 0s and 1s, with multi-qubit result probabilities given by products of corresponding factors.For large n, the Hamming weight of the actual bitstring is approximated by n/2.
  • D. Measurement errors: Assuming erroneous results are uncorrelated with ideal probabilities, the complete fidelity combines circuit fidelity with measurement fidelity.Measurement fidelity can be obtained independently by preparing and immediately measuring random bitstrings, allowing circuit fidelity to be inferred.
  • D. Measurement errors: State-preparation errors can be incorporated analogously by combining measurement fidelity with a corresponding state-preparation fidelity factor.The treatment assumes a single preparation error produces an uncorrelated resulting distribution.
  • V. QUANTIFYING ERRORS: The error-quantification framework predicts XEB fidelity from simpler single- and two-qubit error measurements using a depolarizing model.The model assigns Pauli error eP and probability eP/3 to each erroneous X, Y, or Z operation after a gate.
  • V. QUANTIFYING ERRORS: For many qubits and operations, depolarization treats no-error probabilities as products across gates because bit- or phase-flip errors effectively decorrelate the state.This assumption is used for randomized benchmarking and XEB.
  • V. QUANTIFYING ERRORS: Repeated cycles isolate the per-cycle polarization decay, after which Pauli error is the recommended dimension-independent metric rather than dimension-corrected RB fidelity or entanglement fidelity.Table I summarizes translations among the measured decay constant p and the related error metrics, including representative comparisons for 0.1% Pauli error.
  • V. QUANTIFYING ERRORS: Kraus operators provide a general description of control errors and decoherence, enabling calculation of total and component error budgets, while random-circuit depolarization connects Pauli errors to XEB in the large-D regime.The explicit Pauli-to-depolarization assumption is needed for small D measurements, whereas large-D XEB requires only probabilistic accumulation after Pauli errors are measured.

VI. METROLOGY AND CALIBRATION … 4. Optimizing qubit operating frequencies

The supplementary information presents an automated, bootstrapped calibration framework for Sycamore and a Snake optimizer for selecting operating frequencies. Together, these methods address calibration complexity and improve full-system performance while balancing multiple error mechanisms.

  • A. Calibration overview: Calibration automates experiments that determine optimal analog control parameters despite pulse-shaping errors, qubit-to-qubit variation, parameter drift, and bootstrapping requirements.The approach abstracts calibration complexity and enables systematic comparison of strategies for time, performance, and reliability.
  • 1. Device registry: >100 parameters per qubit are stored in the device registry to achieve high fidelity across qubit operations, motivating automated calibration.The registry contains operating frequencies, control biases, gate parameters, durations, amplitudes, and circuit-model parameterizations.
  • B. Calibration procedure: The calibration procedure progresses from coarse experiments to precise metrology for single-qubit gates, two-qubit gates, and readout.The sequence bootstraps control and system parameters through increasingly refined experiments.
  • 2. Scheduling calibrations: “Optimus”: Optimus represents each calibration as a node in a directed acyclic graph, with directed edges encoding bootstrapping dependencies between calibration experiments.This formulation makes calibration scheduling well-defined when system information is incomplete.
  • 1. Device configuration: Three device configurations—root, single qubit, and grid—progressively increase system knowledge and culminate in high-fidelity calibration across a user-defined qubit grid.The grid may span any desired size up to the entire chip.
  • 3. Single-qubit config: procedure: Single-qubit calibration isolates each qubit, then tunes resonance, state preparation, π pulses, readout, coherence, and frequency-control response before frequency placement.The isolated-qubit data provide circuit and coherence information for selecting operating frequencies in full-processor operation.
  • 2. Root config: procedure: Root calibration uses frequency-domain experiments to map qubit and coupler responses to flux bias, including resonator spectroscopy and coupler-bias measurements.These experiments estimate frequency-versus-bias curves and identify coupler settings with small, relatively bias-insensitive coupling.
  • 4. Optimizing qubit operating frequencies: Snake optimizes idle, interaction, and readout frequencies using an algorithm- and gate-dependent objective that combines physical error mechanisms; locally optimal solutions suffice for state-of-the-art performance.As additional mechanisms are enabled, frequency solutions transition from structured patterns to unstructured configurations.

5. Grid config: procedure … 4. Speckle purity benchmarking (SPB)

The supplementary sections describe grid calibration, tunable fSim-gate control, XEB-based unitary metrology, agreement with randomized benchmarking, and SPB for estimating purity from raw XEB data. Together, these methods support high-fidelity gates and scalable error budgeting.

  • 5. Grid config: procedure: Grid calibration adds qubit–qubit coupling-off calibrations to isolated-qubit procedures, then restores target frequencies and calibrates the entangling gate.Coupling is minimized with resonant-swap experiments for nearby-frequency pairs and conditional-phase experiments for more-detuned pairs.
  • C. Two-qubit gate metrology: Tunable frequencies and interactions provide flexibility for implementing high-fidelity two-qubit gates.The section presents a control and metrology strategy tailored to the system.
  • 1. The natural two-qubit gate for transmon qubits: The natural transmon interaction yields fSim gates, with θ ≊90◦ and φ ≊30◦ selected for the supremacy experiment because they are easy to calibrate and intrinsically high fidelity.The iSWAP-like interaction is preferred, while small angle deviations remain viable.
  • 2. Using cross entropy to learn a unitary model: Two-qubit XEB alternates single- and two-qubit gates, measures output bitstring probabilities, and optimizes a unitary model against classical cross-entropy data.The procedure uses approximately 10-20 circuit instances and tomography rotations to separate fidelity decay from purity decay.
  • 2. Using cross entropy to learn a unitary model: XEB learns control parameters across gate pairs, including partial-iSWAP angles near 90 degrees and conditional phases near 30 degrees.Three Z-rotations suffice to uniquely define the operation in the control model.
  • 3. Comparison with randomized benchmarking: 0.59% measured XEB cycle error agrees with the 0.57% prediction from randomized benchmarking for the same two-qubit gate.The XEB cycle combines one single-qubit gate on each qubit with one two-qubit gate.
  • 3. Comparison with randomized benchmarking: Single-qubit RB errors exceed XEB-extracted errors because conventional RB includes π pulses, whose fidelities are worse than π/2 pulses.Using only π/2 pulses in single-qubit RB brings the extracted error close to XEB.
  • 4. Speckle purity benchmarking (SPB): SPB estimates state purity from raw XEB probabilities by comparing their variance with the Porter-Thomas variance under a depolarizing-channel model.It requires no knowledge of the specific gate sequence and uses exponentially fewer pulse sequences than full state tomography when sufficient Hilbert-space randomization is present.

5. “Per-layer” parallel XEB … B. Overview and technical requirements

The supplementary sections describe parallel entangling-gate benchmarking, readout calibration, and processor-level readout performance. They also explain random-circuit use cases and the technical tension between classically hard sampling and experimentally measurable fidelity.

  • 5. “Per-layer” parallel XEB: Per-layer parallel XEB uses four separate device-wide experiments to benchmark entangler layers under simultaneous operation; parallel operation increases optimized XEB error by roughly 0.003.The increase is primarily attributed to purity error from unintended interactions with other qubits.
  • D. Grid readout calibration / 1. Choosing qubit frequencies for readout: Readout calibration dynamically biases qubits to optimized frequencies while avoiding qubit-qubit resonances, swapping transitions, and unwanted resonator-resonator photon exchange.Frequency selection scans readout fidelity versus qubit and resonator-drive frequencies while accounting for detuning and TLS-related low T1 regions.
  • 2. Single qubit calibration: Single-qubit readout calibration uses 1 µs drive pulses and demodulation windows, targets separation error below 0.3%, and achieves median identification errors of 0.97% for |0⟩ and 4.5% for |1⟩.The procedure selects drive frequency, drive power, demodulation weights, and the discrimination line.
  • 3. Characterizing multi-qubit readout: In simultaneous 53-qubit readout, 13.6% of trials correctly identified prepared 150-state bitstrings, with 3000 trials per state.The multiqubit analysis decomposes overall fidelity into per-qubit simultaneous-readout errors and correlated state-identification behavior.
  • E. Summary of system parameters: Aggregate processor parameters summarize single-qubit metrics over 53 qubits and two-qubit metrics over 86 pairs.A complete per-qubit parameter table is provided in supporting online materials.
  • VII. QUANTUM CIRCUITS: Random quantum circuits serve both quantum-supremacy sampling and experimental-fidelity estimation, motivating a circuit family with variable qubit count n and cycle depth m.The demonstration uses n = 53 qubits and m = 20 cycles.
  • A. Background: Large n hinders Schrödinger simulation while large m impedes tensor-network simulation, making the Schrödinger-Feynman algorithm the most competitive simulator for the hardest circuits.Schrödinger-Feynman simulation sums paths generated by cross-partition gates and their Schmidt decompositions.
  • B. Overview and technical requirements: Quantum supremacy requires circuits that are classically hard to simulate, whereas performance evaluation requires estimating fidelity for circuits that cannot be directly simulated.XEB additionally requires theoretical cross-entropy; the Porter-Thomas approximation is valid for these circuits when depth exceeds 12.

C. Circuit structure … 1. Decomposition of CZ into fSim gates

The supplementary section specifies the RQC circuit architecture, seeded randomness, native gate construction, and universality of the Sycamore gate set. It also shows that controlled-phase gates, including CZ, can be synthesized from two fSim gates and single-qubit rotations, with the decomposition verified across all 86 device fSim gates.

  • C. Circuit structure: For most RQCs below 51 qubits, the qubits can be partitioned into similarly sized blocks connected by five couplers; the 51-qubit RQC uses seven couplers along its most promising cut.This partitioning is relevant because SFA simulation cost grows exponentially in the number of connecting couplers.
  • C. Circuit structure: Each RQC contains m full cycles and one half cycle, with single-qubit gates followed by two-qubit gates arranged in the repeating neighbor-interaction sequence ABCDCDAB.Different qubit pairs interact in different cycles.
  • D. Randomness: The PRNG seed s determines the single-qubit gates, so matching seeds produce matching gates wherever qubit and cycle positions overlap, while two-qubit gates are independent of s.The seed is the third parameter in the RQC family.
  • E. Quantum gates: Single-qubit gates are selected from three π/2 equatorial rotations, with subsequent choices excluding the preceding gate, yielding 3n2nm possible random choices for an n-qubit, m-cycle RQC.The exclusion rule prevents simplifications in some SFA simulation paths.
  • E. Quantum gates: The randomized single-qubit set includes two Clifford gates and one non-Clifford gate, while two-qubit gates preserve excitation number and are specified by five real parameters.The two-qubit family has block structure 1×1, 2×2, and 1×1, and is related to fractional iSWAP and controlled phase.
  • E. Quantum gates: The experiment tunes two-qubit gates near θ ≈π/2 and φ ≈π/6, infers all five parameters per qubit pair using XEB, and leaves corrective Z rotations unapplied.The omitted corrections increase interactions within the circuit-depth budget and add implicit non-Clifford single-qubit operations.
  • F. Programmability and universality: The supremacy gate set is universal because CZ can be composed from two fSim gates and single-qubit rotations, while X1/2 and W1/2 can realize SU(2).Universality follows from the known universality of CZ together with SU(2).
  • 1. Decomposition of CZ into fSim gates: All 86 device fSim gates were numerically verified to yield the CZ decomposition, with the target controlled-phase gate obtained at δ = π.The construction implements a broad set of controlled-phase gates except those very close to the identity, and matches the target up to single-qubit Z rotations.

2. Universality for SU(2)

The universality proof for H and T gates is adapted to X1/2 and W1/2 gates by constructing single-qubit rotations whose angles are irrational multiples of π. The W1/2 construction has the same minimal-polynomial justification as the original rotation, so the remainder of the universality argument applies.

  • Construction: The H-and-T universality argument is adapted to X1/2 and W1/2 by identifying suitable single-qubit rotations with irrational angle-to-π ratios.The construction uses T ≡ RZ(π/4) followed by HTH ≡ RX(π/4) for one rotation, and W1/2 ≡ RX+Y(π/2) followed by X1/2 ≡ RX(π/2) for the other.
  • Irrationality proof: The first rotation angle is irrational relative to π because the monic minimal polynomial of eiα is not cyclotomic.The argument invokes Theorem B.1 and notes that the polynomial fails the cyclotomic criterion because not all coefficients are integers.
  • Irrationality proof: The second rotation angle β is also an irrational multiple of π because eiβ has the same monic minimal polynomial as eiα.The shared polynomial is identified as (75).
  • Conclusion: The remaining steps of the H-and-T universality proof therefore apply to the X1/2 and W1/2 gate construction.The passage explicitly states that the rest of the universality argument carries over.

G. Circuit variants … E. Understanding system performance: error model prediction

The supplementary analysis develops circuit variants that reduce classical simulation cost while preserving fidelity, then uses patch and elided circuits to benchmark larger systems. These methods, together with unitary-model comparisons and error-model tests, support scalable fidelity estimation and the digital error model.

  • G. Circuit variants; 1. Gate elision; 2. Wedge formation: Circuit variants control simulation cost through gate elision and coupler-pattern changes that reduce bond dimension while primarily preserving gate-dependent experimental fidelity.Each circuit is specified by n, m, PRNG seed, the number of elided gates, and the coupler activation sequence.
  • 1. Gate elision; 2. Wedge formation: Eliding cross-partition gates reduces bond dimension by factors of two or four, with complete elision producing disconnected patch circuits and no elision defining full circuits.Wedge-forming EFGH sequences enable efficient SFA simulation, whereas ABCDCDAB prevents wedge formation and is used for supremacy circuits.
  • VIII. LARGE SCALE XEB RESULTS; A. Limitations of full circuits: Large-scale XEB uses patch and elided circuit variations because full-circuit analysis becomes exponentially more expensive as system size increases.Full-circuit experiments sampled output bitstrings 500k times per circuit and used Schrödinger, hybrid Schrödinger-Feynman, and supercomputer simulations across increasing system sizes.
  • B. Patch circuits: a quick performance indicator for large systems: Patch-circuit XEB estimates full-system fidelity by multiplying fidelities of two non-interacting, roughly half-sized subsystems, omitting entanglement between patches.Its exponentially reduced computational cost enables rapid day-to-day performance estimates through 53 qubits and in the quantum-supremacy regime.
  • C. Elided circuits: a more rigorous performance estimator for large systems: Elided circuits remove only early boundary gates, retaining later interactions so entanglement, control effects, and readout crosstalk more closely resemble full-circuit behavior at lower computational cost.Against full circuits, the average elided-to-full fidelity ratio was 1.01 with a standard deviation of 5%, establishing a systematic relative uncertainty of 5%.
  • D. Choice of unitary model for two-qubit entangling gates: Simultaneous pair XEB unitaries gave the best full-system fidelity at every tested size, while alternative calibration models differed by less than a factor of 2 at 50 qubits.The reported fidelity changed from 9 × 10^-3 to 5 × 10^-3 at 50 qubits.
  • E. Understanding system performance: error model prediction: Multiplying constituent single- and two-qubit gate entanglement fidelities with simultaneous-measurement fidelities predicted patch-circuit fidelities within deviations of up to only 10-20%.The readout fidelities also include state-preparation errors, and the agreement held for circuits containing tens of qubits and ∼1000 quantum gates.
  • E. Understanding system performance: error model prediction: Agreement among full, patch, and elided fidelities, together with gate-based predictions, supports the digital error model and indicates that uncorrelated quantum noise is achievable in existing processors.The digital error model assumes no space or time correlations between quantum-gate errors, with implications for quantum error correction.

F. Distribution of bitstring probabilities · G. Statistical uncertainties of XEB measurements

The bitstring-probability distributions agree with theoretical PDFs, while Kolmogorov–Smirnov tests reject zero-fidelity and uniform-random hypotheses with high confidence. Bootstrap and combined-circuit analyses validate the Gaussian statistical-uncertainty estimates for both XEB fidelities, with linear XEB giving the smaller uncertainty.

  • F. Distribution of bitstring probabilities: The analysis compares sampled bitstring probabilities with theoretical linear- and log-XEB distributions, first in a non-supremacy circuit and then in supremacy circuits.For 53-qubit circuits, gate elisions make classical probability estimation feasible while minimizing their fidelity effect.
  • F. Distribution of bitstring probabilities: The 20-qubit experiment agrees well with theory, while the uniform-random null hypothesis is rejected with very high confidence.The test compares experimental probabilities against estimated-fidelity and zero-fidelity theoretical distributions using Kolmogorov–Smirnov statistics and p-values.
  • F. Distribution of bitstring probabilities: For every 53-qubit circuit, zero fidelity is rejected better than a 95% confidence level, while estimated-fidelity p-values range from 0.18 to 0.98 for linear XEB and 0.33 to 0.98 for log XEB.These ranges indicate consistency between empirical and theoretical cumulative distributions at the estimated fidelity.
  • F. Distribution of bitstring probabilities: For the combined 30-million-bitstring sample, the zero-fidelity null has p-value < 2.2 × 10^-16 from R and is rejected with much higher confidence than for individual circuits.The table reports the more conservative R value, despite scipy giving p-value = 3 × 10^-24.
  • G. Statistical uncertainties of XEB measurements: Bootstrap fidelity distributions are consistent with Gaussian fits, with Kolmogorov–Smirnov p-values of 0.99 for F̂_l and 0.41 for F̂_c.The bootstrap uses 4000 samples, and finite variances permit verification of standard error-on-mean uncertainty estimates.
  • G. Statistical uncertainties of XEB measurements: For the example circuit, the three uncertainty measures are 5.78, 5.78, 5.78 (×10^-3) for σ̂_Fl and 7.40, 7.46, 7.46 (×10^-3) for σ̂_Fc, with relative differences below 1%.The measures are the estimated statistical uncertainty, bootstrap standard deviation, and Gaussian-fit σ parameter.
  • G. Statistical uncertainties of XEB measurements: The combined fidelities are F̂_l = (2.24±0.18)×10^-3 and F̂_c = (2.34 ± 0.23) × 10^-3; theoretical uncertainties of 1.8 × 10^-4 and 2.3 × 10^-4 agree with experiment.Inverse-variance weighting combines 10 random circuits, and the linear-XEB estimator has the smaller statistical uncertainty as predicted.
  • G. Statistical uncertainties of XEB measurements: Across 10 elided circuit instances, fidelity variations match finite-sample statistical noise, and averaging over instances yields smaller uncertainties at each depth.Figure S38 reports linear-XEB fidelities with 5σ statistical uncertainties for each depth.

H. System stability and systematic uncertainties · I. The fidelity result and the null hypothesis on quantum supremacy · IX. SENSITIVITY OF XEB TO ERRORS

The sections quantify XEB’s stability and uncertainties, establish a final fidelity of (2.24 ± 0.21) × 10−3 that rejects F ≤10−3 at 6σ, and test XEB’s response to discrete and continuous errors. Systematic fluctuations contribute a 4.4% relative uncertainty, while inserted Pauli errors reduce the XEB estimate by almost 100-fold and indicate error probabilities on the order of 1%.

  • H. System stability and systematic uncertainties: A 17.4-hour, 53-qubit stability measurement found fidelity degradation and fluctuations beyond the statistical 1σ band, with χ2/degree of freedom = 26.3/11 and p = 0.0058.The excess variance was attributed to systematic fluctuation in addition to degradation.
  • H. System stability and systematic uncertainties: 4.4% is the estimated relative systematic uncertainty, obtained from the maximum observed σF/F across measured fidelities.This estimate was used in subsequent analysis.
  • I. The fidelity result and the null hypothesis on quantum supremacy: The final benchmark fidelity is (2.24 ± 0.10(syst.) ± 0.18(stat.)) × 10−3 for ten 53-qubit, 20-cycle circuits.The estimate combines the linear-XEB fidelity and statistical uncertainty with the 4.4% relative systematic drift uncertainty.
  • I. The fidelity result and the null hypothesis on quantum supremacy: The combined fidelity is (2.24 ± 0.21) × 10−3, rejecting the null hypothesis F ≤10−3 with a significance of 6σ.The null hypothesis is motivated by the stated classical cost of sampling at F = 10−3.
  • I. The fidelity result and the null hypothesis on quantum supremacy: Systematic fluctuations change XEB magnitude multiplicatively and do not create a false positive at zero fidelity; additive statistical fluctuations are the relevant false-positive mechanism.This distinction determines which uncertainty contributes to evaluating the supremacy claim.
  • IX. SENSITIVITY OF XEB TO ERRORS: XEB estimates fidelity by comparing experimentally sampled bitstrings with ideal probabilities from reference circuits, and errors can be tested by inserting gates into those circuits.The method evaluates how well the processor realizes circuits of specified qubit number and depth.
  • IX. SENSITIVITY OF XEB TO ERRORS: A single inserted Pauli error reduced the XEB fidelity estimate by almost 100-fold, implying an error probability on the order of 1%, comparable to e2c/2 ≃0.5%.The authors note that multiple gate failures may manifest as the same Pauli error, and continuous rotations were also examined.

X. CLASSICAL SIMULATIONS · A. Local Schr¨odinger and Schr¨odinger-Feynman simulators · B. Feynman simulator

The supplementary information describes local state-vector, hybrid Schrödinger–Feynman, and tensor-network simulators for classical circuit simulation. qFlex achieves large Sycamore simulations on Summit but becomes difficult at 16 cycles and beyond because tensor sizes and runtime scale exponentially with depth.

  • A. Local Schr¨odinger and Schr¨odinger-Feynman simulators: qsim computes all 2^n amplitudes by repeatedly applying gate matrices to the full state vector.Gate fusion, single-precision arithmetic, AVX/FMA vectorization, and OpenMP enable 38-qubit simulations on one Google cloud node.
  • A. Local Schr¨odinger and Schr¨odinger-Feynman simulators: qsimh cuts the lattice into two parts and uses Schmidt decompositions, generating r^g paths when each cut gate has Schmidt rank r.All paths must be summed for unit fidelity, while prefixes allow remaining paths to be computed without restarting the simulation.
  • B. Feynman simulator: qFlex computes selected output amplitudes by summing Feynman paths through tensor-network contractions, following a Feynman approach to circuit sampling.It was adapted to GPUs for Summit and is open source.
  • B. Feynman simulator: Preselected bitstrings are evaluated with frugal rejection sampling, making the sampled subset indistinguishable from quantum-computer samples and the tensor-network cost linear in output count.This makes tensor-network methods more competitive for small sets of output bitstrings.
  • B. Feynman simulator: qFlex reduces distributed-contraction overhead by slicing tensor networks into locally independent paths and uses GPU-efficient contraction strategies for high arithmetic intensity.Its TAL-SH backend supports asynchronous GPU operations, fast transposition, and out-of-core contractions exceeding GPU memory.
  • B. Feynman simulator: 1.29 hours suffices for qFlex to sample 1M Sycamore bitstrings at 12 cycles with fidelity close to 0.5% on Summit.At 14 cycles, the estimated time is 68 days for 1M bitstrings at approximately 0.5% fidelity, and 1.1 years for 3M bitstrings near 1.0% fidelity.
  • B. Feynman simulator: Out-of-core contractions across multiple GPUs achieve efficiency close to 90% at arithmetic intensity about 3000 over three GPUs, versus 1000 for one GPU.This feature allows tensors exceeding GPU memory to be contracted over multiple GPUs on the same node.
  • B. Feynman simulator: Sampling Sycamore random circuits is difficult for tensor-network simulators at 16 cycles and beyond because runtime and tensor sizes grow exponentially with circuit depth.The resulting large tensors require many cuts, creating impractical computational demands.

C. Supercomputer Schr¨odinger simulator · D. Simulation of random circuit sampling with a target fidelity · 1. Optimality of the Schmidt decomposition for gates embedded in a random circuit

The supplementary sections describe exact and approximate Schrödinger simulators, low-fidelity classical sampling via selected Feynman paths, and the optimality of Schmidt decompositions for gates in random circuits. They also report simulator limitations, XEB consistency, and the conditions under which these approximations apply.

  • C. Supercomputer Schr¨odinger simulator: JUQCS-E computes exact output probabilities, cumulative distributions, and samples bitstrings, while also selectively saving probabilities for a specified set of bitstrings.The simulator operates in double-precision floating point and supports a user-defined set Q of M bitstrings.
  • C. Supercomputer Schr¨odinger simulator: For D = 2^39 random-state outputs with pA(j) ≈ O(10^-12), summing 2^39 small probabilities requires care even with approximately 16-digit double precision.JUQCS-A uses adaptive two-byte encoding and is therefore an approximate simulator without a general guarantee that pA(j) ≈ pU(j).
  • C. Supercomputer Schr¨odinger simulator: The results support Porter–Thomas output distributions, accurate finite-sample XEB estimates, and consistency between full experimental XEB fidelities and probabilistic, patch, and elided XEB estimates.The approximate simulator’s αA,U increasingly deviates from one despite αA,A ≈ 1, while αX,U and bαX,U remain consistent for X = A, M, C.
  • D. Simulation of random circuit sampling with a target fidelity: Low-fidelity sampling accelerates classical simulation by considering only a fraction of Feynman paths after Schmidt decomposing selected two-qubit gates, with speedups of at least 1/FXEB.The method assumes the different paths produce orthogonal states and has computational cost proportional to FXEB.
  • D. Simulation of random circuit sampling with a target fidelity: Uniform random sampling would yield FXEB = 0, whereas experimental samples at 53 qubits and m = 20 retain FXEB ≥ 0.1%.This limits the adequacy of approximating noisy output distributions solely by uniform sampling.
  • 1. Optimality of the Schmidt decomposition for gates embedded in a random circuit: Under maximal mixing of the gate’s two qubits in a sufficiently scrambling random circuit, the embedded-gate optimization reduces to optimizing the gate as a standalone operator.For sufficiently large depth, the residual term can be ignored; in one dimension ε ≤ (4/5)^D, and faster decay is expected in two dimensions.
  • 1. Optimality of the Schmidt decomposition for gates embedded in a random circuit: The optimal rank-1 product replacement of Vab is given by its largest Schmidt singular value λ1, with Ma = Ra,1 and Nb = Sb,1.The operator Schmidt singular values are ordered λ1 ≥ λ2 ≥ . . .; the rank-k generalization uses the leading Schmidt components.

2. Classical speedup for imbalanced gates · 3. Verifiable and supremacy circuits · E. Treewidth upper bounds and variable elimination algorithms

The supplementary information analyzes classical simulation speedups from imbalanced Schmidt spectra, fused gate structures, and projected-variable treewidth methods. It finds an upper bound of 25.4 for a representative decomposition, wedge fusion reducing paths from 42 to 4, and projected-variable methods reaching practical limits for deeper supremacy circuits.

  • 2. Classical speedup for imbalanced gates: The geometric mean of λmax is about 1.047, giving an upper bound of 1.0472^g and 25.4 for decomposing g = 35 gates.This bound applies to the largest decomposition considered with the SFA simulator on m = 20 cycles.
  • 2. Classical speedup for imbalanced gates: For g = 35 gates at m = 20 cycles and fidelity F ≈0.2%, the estimated practical speedup is well below an order of magnitude.The estimate assumes typical |δθ| values of about 0.05 radians.
  • 3. Verifiable and supremacy circuits: Fusing wedge sequences acting on three qubits produces 4 paths instead of 42 paths from separate gate decompositions, providing a speedup factor of 4 per wedge.Wedges are consecutive two-qubit gates whose fused unitary has lower effective rank than the product of individual ranks.
  • 3. Verifiable and supremacy circuits: Verifiable circuits contain many wedges and are classically simulatable in reasonable time, enabling full XEB with perfect fidelity computations through m = 14.Supremacy circuits contain only a small number of similar sequences, with larger fused regions often impractical to construct explicitly.
  • E. Treewidth upper bounds and variable elimination algorithms: Variable elimination computes amplitudes through tensor contractions whose cost depends on the elimination ordering, with treewidth equal to the minimum contraction width.Projecting p variables creates 2^p independently contractible subgraphs, enabling embarrassingly parallel computation.
  • E. Treewidth upper bounds and variable elimination algorithms: The projected-variable workflow uses QuickBB for an initial ordering, selects variables minimizing projected contraction cost, repeats until resources are reasonable, and reruns QuickBB to refine the ordering.The procedure is designed to lower contraction cost after projection.
  • E. Treewidth upper bounds and variable elimination algorithms: Projecting between 8 and 63 variables reduces supremacy-circuit contraction width to 28 or below, where a tensor with 28 binary indexes consumes 2 GB.The projection count depends on circuit depth.
  • E. Treewidth upper bounds and variable elimination algorithms: Sampling from supremacy circuits at m = 16 and beyond is out of reach for this algorithm, although verifiable-circuit simulation shows some speedup at m = 16.The comparison is reported in the variable-elimination estimates for Sycamore circuits.

F. Computational cost estimation for the sampling task … H. Energy advantage for quantum computing

The supplementary analysis estimates classical simulation cost, runtime scaling, memory requirements, and energy use for quantum sampling and verification. It identifies SFA as the most efficient simulator for the hardest circuits and finds exponential growth in classical energy consumption with circuit depth.

  • F. Computational cost estimation for the sampling task: SFA is the most efficient simulator for the hardest circuits, using 1000 machines to estimate simulating a 53-qubit, 20-cycle circuit.The cluster uses n1-standard-2 machines; the circuit has 35 cross-cut gates, and perfect fidelity requires simulating all 431 × 24 paths.
  • F. Computational cost estimation for the sampling task: A proposed decomposition could reduce the presented computation time by a factor of 16.
  • G. Understanding the scaling with width and depth of the computational cost of verification: The runtime analysis compares distributed Schrödinger and hybrid Schrödinger-Feynman algorithms for computing exact amplitudes needed for XEB on Sycamore circuits.The analysis assumes a supercomputer with 1M cores and studies scaling with circuit width n and depth m.
  • 1. Runtime scaling formulas: SA runtime scales with circuit depth and width through the wave-function size, while SFA scales with cross-gate paths and patch-simulation cost.For SFA, cross-gates generate path factors, the number of cross-gates scales with √n and m, and patch simulation scales exponentially with n/2.
  • 2. Assumptions and corrections: Runtime estimates rely on optimistic assumptions, including perfect SA scaling to 1M cores and hypothetical wave-function storage beyond existing supercomputer memory limits.Finite-size effects can cause discrepancies of around an order of magnitude, while core-hour costs vary across CPU types.
  • 3. Fitting constants: For SFA, fitted runtimes at n = 53 and m = 14 are 5 hours for verifiable circuits and 4 years for supremacy circuits on 1M cores.The fitting uses B = 1/4 and separately fitted constants for the two circuit classes.
  • 4. Memory usage scaling: Using 2-byte complex-number encoding, SA requires 2n+1 bytes for the wave function, while 1M-core estimates remain below 3 PB only for sufficiently smaller systems.State-of-the-art supercomputers have less than 3 PB of memory, and single- or double-precision storage increases requirements by factors of 4 or 8.
  • H. Energy advantage for quantum computing: Quantum-apparatus power is approximately 26 kW and is largely independent of processor activity and circuit depth, requiring ∼5×106 J (∼1 kWh) for 1M samples.The dilution refrigerator consumes ∼10 kW directly plus 10 kW or more for chilled-water cooling, while supporting electronics use nearly 3 kW; classical energy consumption grows exponentially with circuit depth.

XI. COMPLEXITY-THEORETIC FOUNDATION OF THE EXPERIMENT … C. Computational hardness of unbiased-noise sampling

The foundation defines quantum supremacy through a precise computational task and a scalable quantum–classical runtime gap, then argues that unbiased noise preserves sampling hardness up to a linear fidelity penalty. It formalizes the task, states complexity limitations, and identifies remaining finite-size lower-bound questions.

  • XI. COMPLEXITY-THEORETIC FOUNDATION OF THE EXPERIMENT: Quantum supremacy requires a well-defined computational task and a runtime separation that makes classical solution impractical as problem size grows.The framework emphasizes that asymptotic formal proofs do not directly establish concrete costs for fixed-size experiments.
  • 2. Programmable computational device: Random circuit sampling satisfies these requirements, but experimental claims must compare against highly optimized classical algorithms on state-of-the-art supercomputers.Known classical methods include exact simulation with cost exponential in circuit treewidth and approximate simulation with cost reduced linearly in global fidelity F.
  • 2. Programmable computational device: Complexity results are asymptotic and cannot directly provide concrete lower bounds for circuits with a fixed number of qubits and depth.Fine-grained complexity results provide several finite-size bounds, while approximate-sampling hardness relies on less-studied assumptions than exact-sampling hardness.
  • A. Error model: The error model assumes global depolarizing noise, with experimental fidelity F in the range 10^-2–10^-3.This model is used for the complexity argument but is not assumed elsewhere for cross-entropy validation or classical-algorithm comparisons.
  • B. Definition of computational problem: Circuit Sampling asks for samples from pU(x) := |⟨x|U|0⟩|2, while Random Circuit Sampling restricts this task to most circuits in a circuit ensemble.Under the assumption that the polynomial hierarchy does not collapse, efficient classical solutions are ruled out for several circuit classes and for random circuits.
  • B. Definition of computational problem: Because the achieved approximation error is far from zero, the analysis uses Unbiased-Noise F-Approximate Random Circuit Sampling, sampling from rU,F defined by Eq. (112).The experiment can also be interpreted as solving b-Heavy Output Generation with b = 1 + F.
  • C. Computational hardness of unbiased-noise sampling: 10T/F: Theorem 1 shows that a classical sampler running in time T for rU,F enables an AM[LT + 2Lm] protocol, with L = c/F, to estimate a transition probability.Thus global white noise causes no more than a linear decrease in classical simulation time with fidelity F, and this scaling is optimal for the cited method.
  • C. Computational hardness of unbiased-noise sampling: The theorem equates noisy sampling hardness with transition-amplitude estimation up to a linear reduction in F, but does not itself prove a transition-amplitude lower bound.Existing results give #P-hardness for random circuits at additive error 2^-poly(n), while finite-size bounds under SETH apply to some circuits; analogous random-circuit bounds remain open.

D. Proof of Theorem 1 · ERRATUM

The proof reduces efficient classical sampling of rU,F to an AM protocol that estimates the ideal output probability, then removes unbiased noise to estimate |⟨0|U|0⟩|2. The erratum clarifies that Figure 4 error bars show statistical uncertainty only, while both statistical and systematic uncertainties were included in the analysis.

  • D. Proof of Theorem 1: A sampler for rU,F running in time T yields an AM protocol estimating rU,F(0) with classical verification cost LT.Because rU,F(0) = F|⟨0|U|0⟩|2 + (1−F)/2^n, unbiased noise can then be subtracted to estimate |⟨0|U|0⟩|2.
  • D. Proof of Theorem 1: For every θ and L, Lemma 1 distinguishes rU,F(0) ≥ θ(1 + 2/L) from rU,F(0) ≤ θ(1 − 2/L).The resulting AM protocol has verification time LT + 2Lm.
  • D. Proof of Theorem 1: The protocol uses random 2-universal linear hash functions, with Merlin providing an Lm-bit witness w and integer s.Arthur checks the witness conditions and applies a threshold test based on the hashed solution count.
  • D. Proof of Theorem 1: In the YES case, hashing produces a valid witness w with high probability, so Arthur accepts with high probability.The completeness argument applies Lemma 2 to the set of ML solutions and uses the interval [(1/2)ML/2^s, 2ML/2^s].
  • D. Proof of Theorem 1: In the NO case, the hashed constraint has no solution, so no Merlin witness can make Arthur accept.This establishes soundness and completes Lemma 1.
  • D. Proof of Theorem 1: Setting θ = Fλ + (1−F)/2^n transfers the distinguishing protocol for rU,F(0) to the theorem’s AM protocol for |⟨0|U|0⟩|2.The proof concludes by relating the noisy output probability to the ideal probability through the fidelity parameter F.
  • ERRATUM: Figure 4’s error bars represent statistical uncertainty, not both statistical and systematic uncertainty.Both uncertainty types were included in the analysis, and the conclusions remain intact.
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