Scaling and logic in the color code on a superconducting quantum processor

I. INTRODUCTION

• Quantum error correction encodes logical qubits across physical qubits and can exponentially suppress errors if physical error rates are below a critical threshold.The motivating goal is to bridge physical-device errors and the much lower logical error rates needed for quantum algorithms.Source: §I · p. 2
• The surface code has a high error threshold and fits planar four-nearest-neighbor architectures, but its qubit overhead and logical-operation limitations motivate alternatives.Superconducting-qubit demonstrations have predominantly used the surface code or variants.Source: §I · p. 2
• Color codes use fewer qubits than surface codes at fixed distance and support transversal single-qubit Clifford gates and resource-efficient magic-state injection.They also support simultaneous fault-tolerant multi-qubit Pauli measurements with up to three times less space-time overhead.Source: §I · p. 2
• Color codes’ higher-weight stabilizers produce a stricter threshold, while their decoding and connectivity requirements challenge superconducting devices.Before this work, increasing color-code size had not demonstrated error suppression on any experimental platform.Source: §I · p. 2
• The paper demonstrates color-code error suppression and logical operations on a superconducting processor, with simulations suggesting improved qubit efficiency over surface codes after modest physical-error-rate improvements.The work also characterizes logical Clifford gates, injects magic states, and demonstrates multilogical-qubit operations.Source: §I · p. 2

II. COLOR CODES WITH SUPERCONDUCTING QUBITS

• A hexagonal, three-colorable lattice places data qubits at vertices and X- and Z-type stabilizers on each tile.The chosen square-grid embedding supports nearest-neighbor connectivity and simultaneous measurement of both stabilizers using two auxiliary qubits per tile.Source: §II · p. 2
• The distance-five logical qubit uses 19 data qubits and 18 auxiliary qubits, with weight-6 hexagonal and weight-4 boundary stabilizers.The logical X and Z operators are products of the corresponding Pauli operators on data qubits along a triangle boundary.Source: §II · pp. 2–3
• Swapping selected data and auxiliary qubits makes each readout line contain only one type, so data and auxiliary qubits can be measured on different schedules.Circuit transformations avoid costly swap-gate decompositions while preserving fault tolerance, and CNOT gates are compiled into native Hadamard and CZ gates.Source: §II · p. 3

III. DISTANCE SCALING

• The experiment prepares logical states in the X and Z bases, applies repeated error-correction cycles, and compares the distance-five code with three distance-three subsets.Runs measure up to n = 29 cycles, and logical error probability is fitted to estimate error per cycle.Source: §III · pp. 3–4
• Weight-6 stabilizers have detection probability pd,6 = 0.149, compared with pd,4 = 0.112 for boundary weight-4 stabilizers.Detection probability remains nearly constant through the bulk of the cycles; the text attributes the lower weight-4 rate to fewer gates and thus lower error susceptibility.Source: §III · pp. 3–4
• Because bulk data qubits participate in three stabilizers per basis, matching-based decoders cannot be used directly, so the experiment uses other decoding strategies.The paper also describes circuit measures to reduce coherent phase errors, idle dephasing, and leakage.Source: §III · pp. 3–4
• Λ3/5 = 1.56(4) with the AlphaQubit decoder, and ε5 = 0.0110(2), as distance increases from three to five.The observed suppression suggests performance below the color-code threshold, supported by simulations extrapolating to larger distances.Source: §III · p. 4
• The simulated suppression factor is Λsim 3/5 = 1.680(4), with CZ gates contributing about 39% of the simulated error budget.Measurement, single-qubit gate, and data-qubit idle errors contribute approximately equally to the remainder.Source: §III · p. 4

IV. LOGICAL RANDOMIZED BENCHMARKING

• Transversal gates apply the desired operation independently to physical data qubits, isolating potential errors and requiring one physical-gate time step.The paper describes these gates as inherently fault-tolerant, simple, and efficient.Source: §IV · p. 4
• The protocol benchmarks all 24 single-qubit Clifford gates using randomized sequences, inverse final gates, and an error-correction cycle between each Clifford.Reference sequences omit Clifford gates and preserve the logical state, allowing estimation of the added gate error.Source: §IV · p. 4
• Logical gates transform stabilizers, so syndrome construction compares measurements across cycles according to the gate’s stabilizer mapping.For example, transversal H swaps X and Z stabilizers, while S maps X stabilizers to Y stabilizers.Source: §IV · pp. 4–5
• εC = 0.0027(3), below the idling logical error per cycle ε3 = 0.0171(3), for transversal Clifford gates in the distance-three code.The benchmark varies sequences from m = 0 to 10 random Clifford gates and uses 25 sequences per length.Source: §IV · p. 5

Arbitrary state injection

• A single data qubit is prepared in the target state, then the code is grown by adding data qubits in Bell states before one error-correction cycle projects the state into the distance-three code.The injection-qubit logical X and Z operators deterministically extend to the logical operators.Source: §V · p. 5
• The protocol applies to arbitrary states, but errors on the injection data qubit cannot be detected.The passage notes that this approach has achieved lower error rates than other injection protocols for CSS codes.Source: §V · p. 5
• Sweeping the polar angle with azimuth fixed at 0 tests logical Bloch-sphere rotations through tomography of the logical X, Y, and Z expectation values.The Y expectation remains close to zero during the sweep.Source: §V · pp. 5–6
• 0.009(4) average infidelity is achieved for arbitrary injected states when post-selecting runs without detectable errors, retaining an average 74.8% of runs.Without post-selection, the average infidelity is 0.039(3) using the Möbius decoder (Chromobius).Source: §V · p. 6

Magic state injection

• Post-selected magic states exceed the threshold required for magic state distillation, with retained data fractions of 75.2%, 74.6%, and 75.7%.The reported infidelities are 0.0008+15−38 and 0.0086+67−65; the passage lists three states but only two infidelity values.Source: §V · p. 6

Lattice-surgery protocol

• An XX-parity measurement of the two logical qubits, followed by a classical Pauli-frame update, teleports the state from L1 to L2.The protocol starts with L1 in the input state and L2 in |0L⟩, each initialized using one QEC cycle.Source: §VI · p. 6
• The XX measurement uses a three-cycle merge followed by a one-cycle split, with added X-basis stabilizers and expanded boundary stabilizers during the merge.Two data qubits are added in a Bell state; merge and measurement outcomes are used for the conditional Pauli-frame update.Source: §VI · p. 6

Teleportation results

• Decoded output expectations generally agree with ideal expectations and reasonably match device-benchmark-based Pauli-noise simulations, except for the Y expectation of X-basis states.The passage says the exception is discussed in Appendix I 3.Source: §VI · p. 7
• 86.5(1)%–90.7(1)% are the measured fidelities for teleporting four logical X- and Z-eigenstates with the neural-network decoder.The measurements bound the average fidelity of the single-qubit teleportation channel at F(T) ≥84.7(1)%.Source: §VI · p. 7

Discussion

• On the same processor, the surface code achieves better logical error suppression, with Λsc 3/5 = 2.31(2), than the color code’s Λ3/5 = 1.56(4).Simulations suggest about a fourfold improvement in physical error rates could make the color code more qubit-efficient than the surface code.Source: §VII · p. 7
• Efficient transversal Clifford gates and more flexible lattice surgery are described as ways color codes could simplify algorithm implementations.The passage also notes that color codes underpin efficient methods for generating high-fidelity magic states that could be used alongside surface-code computation.Source: §VII · p. 7

Outlook

• Future work includes improving device performance and size, decoder speed and accuracy, demonstrating logical-gate error suppression as code size grows, and characterizing further logical-gate capabilities.The practical impact of the color-code advantages for large-scale algorithms still requires detailed resource-estimation studies.Source: §VII · p. 7

Experiment and analysis roles

• N.L. and A. Bourassa calibrated the device and performed experiments, with support from several collaborators; N.L. analyzed the data with A. Bourassa’s support.N.L., A. Bourassa, C.G., and K.J.S. designed the experiments, with input from M.M. and C.J.Source: CONTRIBUTIONS · p. 8

Decoding and simulations

• N.L., J.B., F.J.H.H., L.M.Z., A. Bourassa, and N.S. decoded the data, with support from V.S. and O.H.; N.S., O.H., N.L., J.C., and D.K. performed simulations.A.Z. contributed input on average teleportation-channel fidelity.Source: CONTRIBUTIONS · p. 8

Writing and supervision

• N.L. wrote the manuscript with input from all co-authors, and J.K., C.J., C.G., and K.J.S. supervised the work.Google DeepMind and Google Quantum AI jointly developed the machine-learning decoder.Source: CONTRIBUTIONS · p. 8

• The front matter identifies a superconducting quantum processor.Source: Supplemental Information for: Scaling… · p. 12
• The listed collaborators are Google Quantum AI and Collaborators.Source: Supplemental Information for: Scaling… · p. 12
• The document is dated December 18, 2024, and carries the arXiv identifier 2412.14256v1.Source: Supplemental Information for: Scaling… · p. 12

Appendix A

• At fixed distance, the rotated surface-code circuit requires nsc = 2d^2 − 1 qubits, about 25% more than the superdense color-code circuit; ncc is stated to include auxiliary qubits.Source: Appendix A · p. 12
• The simulations use the SI1000 error model at p = 10^-3 and p = 5 · 10^-4, decode with MLE, and define a round as d error-correction cycles.The noise parameters model near-term processors, with p = 10^-3 about two times smaller than the device's current error rates.Source: Appendix A · p. 12
• At large distances, the color code may require slightly fewer qubits than the surface code to reach logical error rates near 10^-8, particularly at p = 5 · 10^-4.The simulations find the two codes require comparable qubit counts to achieve a given logical error per round.Source: Appendix A · p. 12
• At physical error rates expected in near-term processors, the two codes require similar resources, allowing code choice to reflect other considerations such as efficient logical gates.At smaller physical error rates, the color code is expected to eventually outperform the surface code in qubit overhead.Source: Appendix A · p. 12
• The superdense color code's logical error per round does not follow a linear trend on the square-root scale; its suppression factor increases with distance, attributed to diminishing spatial boundary effects.Source: Appendix A · p. 12

Appendix B: Device and Experimental Setup

• Experiments use a 72-qubit processor; the distance-five code occupies 37 qubits, while distance-three codes use three 13-qubit subsets.Single-qubit π and π/2 rotations last 35 ns and 18 ns; CZ gates last 37 ns, readout 600 ns, and multi-level reset 160 ns.Source: Appendix B · pp. 12–13, §2 · p. 17

1. Circuit Intuition

• A Bell-state auxiliary pair encodes bit-flip errors in its parity and phase-flip errors in its relative phase, allowing both types to be detected in one measurement round.The auxiliary qubits also flag errors on one another, a feature relevant to fault tolerance.Source: §1 · pp. 13, 16

2. Fault-tolerance of the Superdense Syndrome Extraction Circuit

• For d = 3, 5, and 7, exact MaxSAT confirms d_circ = d; heuristic searches up to d = 15 also suggest full-distance preservation but provide only an upper bound.The exact approach maps circuit-distance determination to a MaxSAT problem; the larger-distance search is heuristic.Source: §2 · p. 16

3. Superdense Circuit Transformations

• Two added two-qubit-gate layers replace the six-layer overhead of a naive swap decomposition, separating data and auxiliary qubits by readout line; the compiled cycle lasts about 1.3 µs.The optimized circuit reuses existing operations, and conditional Pauli corrections are handled in software preprocessing.Source: §3 · pp. 16–17

Appendix D: Distance scaling

• State-preservation experiments interleave codes, bases, and cycle counts, with 5,000 repetitions for each of 10 bitstrings, totaling 50,000 runs per experiment.The distance-five Z-basis protocol applies XY4 dynamical decoupling during auxiliary measurement and reset, followed by data-qubit leakage removal.Source: Appendix D · p. 17, §1 · p. 17

3. Error syndrome detection probabilities

• In the experiment bulk (5 < n < 25), syndrome-element detection probabilities are 0.1–0.2 and stable; distance-five weight-6 stabilizers have higher probabilities than weight-4 stabilizers.Some X-type elements show elevated probabilities in the lowest-distance-three code and the top-right distance-five region; X-basis probabilities also rise near time boundaries.Source: §3 · pp. 17–18, Figure 9 · p. 20

a. Möbius Decoder (Chromobius)

• Chromobius [2] decodes faster than the other decoders used here, but lacks flag-based decoding and support for some logical operations, so it cannot decode logical randomized benchmarking circuits.It is used for state injection and compared with other decoders for state-preservation experiments.Source: a. Möbius Decoder (Chromobius) · p. 21

b. Most Likely Error (MLE) Decoder

• The MLE decoder maps syndrome decoding to a mixed-integer program and accounts for X/Z correlations; it is about 1,000 times slower than Chromobius and 300 times slower than the neural-network decoder.It is more accurate than matching-based approaches but is not an optimal decoder.Source: b. Most Likely Error (MLE) Decoder · p. 21, Appendix E · p. 21

c. Neural Network Decoder

• AlphaQubit was adapted for color-code decoding without convolutions, using up to two attention/recurrent layers and 4.1 million parameters in its two-layer variant.The network processes syndrome elements cycle by cycle to predict the logical observable; the two-layer and single-layer variants have 4.1 million and 2.1 million parameters, respectively, versus 5.4 million in Ref. 3.Source: c. Neural Network Decoder · p. 21
• AlphaQubit is pretrained on synthetic SI1000 data, then fine-tuned on limited experimental or device-specific simulated data, with training and test data kept separate.The described split approaches include even/odd run partitioning, causal run splits, and independent or separate datasets, depending on the experiment.Source: c. Neural Network Decoder · p. 22
• AlphaQubit decoded about 3 times slower than Chromobius but 300 times faster than MLE, and was used for most main-text experiments.Magic-state injection instead used Chromobius and post-selection; strategies using experimental data had different train/test partitions, so their test datasets could contain different runs.Source: c. Neural Network Decoder · p. 22, §3 · p. 23

2. Decoding Priors

• The decoding priors included device-agnostic SI1000 probabilities, experimentally inferred operation-specific Pauli-error probabilities, and iterative MLE-based edge reweighting.The learned probabilities used Chromobius-decoded 9-cycle data and were extrapolated across cycle counts; edge reweighting typically stabilized after 3 to 6 iterations.Source: §2 · p. 22

3. Comparison of Decoding Performance

• The distance-scaling comparison tested Chromobius with SI1000 or learned priors, MLE with SI1000 or edge-reweighted priors, and AlphaQubit fine-tuned on experimental data.Decoded data were fit independently by code instance and basis; the error suppression factor averaged distance-three and distance-five experiments, with fit-based standard errors propagated.Source: §3 · p. 22

3. MLE decoder with an SI1000 prior (red symbols)

• Λ3/5 = 1.56(4) for the best decoding strategy, and three of five strategies had lower average logical error per cycle at distance five, indicating below-threshold operation.Figure 10 compares the logical error per cycle across the five strategies and indicates each strategy’s error suppression factor.Source: §3 · p. 22, Figure 10 · p. 23

Appendix F: Pauli Simulations near the Superdense Color Code Threshold

• Λ3/5 = 1.56(4) experimentally suppresses logical errors when scaling the color code from distance three to five.The experiment reports ε3 = 0.0176(3) and ε5 = 0.0115(2) using the same MLE decoder.Source: Appendix F · p. 23
• Near the simulated threshold pth ≈0.005, Λ3/5 is about 1.1 and decreases to about 1.0 at higher distances.The simulations use an SI1000 error model with varying noise strength and MLE decoding for distances three through nine.Source: Appendix F · p. 23
• At p = 0.005, simulations give ε3 = 0.039(1) and ε5 = 0.036(1), above the experimental ε3 = 0.0176(3) and ε5 = 0.0115(2).The authors interpret the lower experimental values as indicating operation below the color-code noise threshold.Source: Appendix F · p. 23
• The SI1000 model assumes homogeneous errors and omits leakage and stray interactions, motivating future measurements at larger code distances.Source: Appendix F · p. 23

Appendix G: Logical Randomized Benchmarking

• Logical randomized benchmarking offers a more reliable way than process tomography to assess logical Clifford-gate costs within logical circuits despite logical SPAM errors.The concern is especially salient for surface codes, where Y-basis readout is not fault-tolerant.Source: Appendix G · p. 23
• The distance-three implementation samples sequences of random Clifford gates followed by a recovery gate, interleaved with error-correction cycles.The 24 single-qubit logical Cliffords are decomposed into H, S, and Pauli gates, then compiled for the hardware.Source: Appendix G · p. 23
• The additional logical-Clifford error per cycle is extracted using an idling experiment as the reference.The Clifford and idling depolarizing parameters are obtained by fitting average values to Ap^m + 1/2.Source: Appendix G · p. 24
• The randomized benchmarking circuits are a benchmarking tool, not practical applications, because Clifford-only circuits can be efficiently simulated classically.Source: Appendix G · p. 24

Appendix H: State injection

• Arbitrary states are injected by preparing the target state on a physical qubit and fault-tolerantly expanding a degenerate distance-one code to distance three.Six added data qubits are initialized in Bell pairs using an intermediate auxiliary qubit; preparation takes three two-qubit gates if it starts in |0⟩ and four if it starts in |+⟩.Source: Appendix H · p. 24, Figure 12 · p. 25
• 0.012(7) average infidelity was achieved across the state sweep with post-selection on runs whose syndrome elements were all zero, retaining 74.4% of the data.The largest decoded versus post-selected discrepancy was for Y_L, which the authors attribute to sensitivity to both bit-flip and phase-flip errors.Source: Appendix H · p. 25, Figure 13 · p. 25

1. State Teleportation Circuit Details

• Lattice-surgery teleportation of |1L⟩ from L1 to L2 comprises initialization, an X-basis merge, and a split, followed by Pauli-frame updates based on measurement outcomes.The merge measures the two logical qubits' X-basis parity using interface stabilizers and an additional data-qubit readout.Source: §1 · pp. 25–26, Figure 15 · p. 27

Appendix H: State injection

• The implementation achieves higher fidelities and lower data-rejection fractions than other superconducting-qubit implementations, and lies on the Pareto front with some trapped-ion implementations.The comparison plots magic-state infidelity against rejected-data fraction.Source: Appendix H · p. 25, Figure 14 · p. 26

2. Lower Bound on Average Fidelity of the Teleportation Channel

• The measured logical Pauli eigenstate preparations and outcomes provide a lower bound on the average fidelity of the teleportation channel relative to the identity channel.Pauli correlations are experimentally accessible through preparations of Pauli eigenstates and measurements in a Pauli basis; the process-matrix account identifies entanglement fidelity with a single matrix element.Source: §2 · pp. 26, 28

Tomography

• In lattice-surgery tomography, X-basis initialization followed by Y-basis measurement produces a decoded expectation near 0.2, although an ideal experiment should yield a fully random outcome.The source remains under investigation; the authors hypothesize an uncompensated systematic Z rotation during initialization in those circuits.Source: Tomography · p. 28