Demonstration of low-overhead quantum error correction codes

THEORETICAL DETAILS

• qLDPC codes are motivated by potential high encoding efficiency and efficient resource scaling, and this section covers BB-code construction, stabilizer measurement circuits, and decoding [33] [34].The paper situates these codes against the resource overhead of traditional codes such as the surface code [53].Source: THEORETICAL DETAILS · p. 12

Bivariate bicycle code construction

• A stabilizer code uses an Abelian subgroup of the n-qubit Pauli group, whose common +1 eigenspace encodes k logical qubits when generated by n − k independent stabilizers.The code space has dimension 2^k.Source: Bivariate bicycle code construction · p. 12
• BB codes are CSS-type codes defined from two polynomials A(x, y) and B(x, y) over binary matrices, with HXH⊤Z = AB + BA = 0 (mod 2) ensuring commutation [34].Each check row has six nonzero entries, so each check acts on six data qubits.Source: Bivariate bicycle code construction · pp. 12–13
• The experimental bivariate bicycle code uses l = m = 3 and A = x + 1 + y^2, B = y + 1 + x^2, giving parameters [[18, 4, 4]] and encoding rate 1/9.The matrices are defined by x = S3⊗I3 and y = I3⊗S3; the code parameters are computed from the minimum-weight nontrivial vector in ker(HX) outside rs(HZ).Source: Bivariate bicycle code construction · p. 13
• The [[18, 4, 4]] code has rank-seven X- and Z-check matrices, so two redundant checks of each type are removed; its Tanner graph has a two-dimensional toric layout but nonlocal checks.The implementation removes X2 and X8, and Z3 and Z4; each check acts on six data qubits with at least two long-range couplings.Source: Bivariate bicycle code construction · p. 13, Figure S2 · p. 15
• Removing checks X5 and Z5 from the [[18, 4, 4]] code yields a [[18, 6, 3]] code, increasing encoded logical qubits from four to six while reducing distance from four to three.The six-logical-qubit code uses a selected set of logical Pauli-X and Pauli-Z operators.Source: Bivariate bicycle code construction · p. 14

Syndrome measurement circuit

• The efficient syndrome measurement circuit simultaneously extracts all stabilizer values to detect and correct physical errors.It periodically measures the stabilizers and records their corresponding error syndromes.Source: Syndrome measurement circuit · p. 16
• The full syndrome cycle for the [[18, 4, 4]] BB code uses seven layers of non-overlapping CZ gates, a layer count that remains constant as BB codes scale up.Seven CZ layers are the minimal configuration across alternative syndrome measurement circuit designs [34].Source: Syndrome measurement circuit · p. 16
• The circuit replaces the CNOT gates in the design proposed in ref. [34] with CZ and Hadamard gates because CZ is directly implementable in the experiment.Source: Syndrome measurement circuit · p. 16

Decoding

• BP-OSD infers likely data-qubit errors from observed detections; decoding succeeds when the inferred error differs from the actual error only by check-operator products, and fails when the difference is a nontrivial logical operator [34].The decoder is applied to experimental syndromes after a preparation step that simulates single faulty operations in the Clifford syndrome circuit.Source: Decoding · pp. 16, 18
• The circuit noise model applies independent Pauli noise to gates, idles, dynamical decoupling, check measurements, and final data readout, using experimentally characterized rates and uniform rates for each operation type.Initialization is not modeled separately because its effects are absorbed into Hadamard- or CZ-gate error probabilities.Source: Decoding · p. 16
• With leakage probability η = 0.05 and computational-basis probability parameter β = 0.475, the estimated check-measurement error pM is 4.03 × 10^-2 and final-readout error pF is 3.29 × 10^-2.The estimates follow leakage rejection and normalization of the remaining outcomes to a two-state confusion matrix.Source: Decoding · p. 18
• Simulated faulty circuit realizations are weighted by operation-specific error rates; identical syndrome triples are merged by summing their probabilities, and X- and Z-type errors are decoded independently for the CSS code.The discussion focuses on decoding X-type errors when a logical-Z basis state is prepared.Source: Decoding · p. 18
• For experimental or simulated syndromes, BP-OSD seeks the solution with the highest reliability using a greedy algorithm [41].The preparation simulation tracks the resulting syndromes and errors for single faulty operations.Source: Decoding · pp. 18–19

Device Information

• The Kunlun processor contains 32 frequency-tunable transmon qubits and 84 tunable couplers.Each qubit has a readout resonator for dispersive measurement and a dedicated microwave and flux-tuning control line.Source: Device Information · p. 19
• The square lattice uses 18 data qubits and 14 check qubits for implementing the BB code.Data qubits are labeled R/L and check qubits Z/X.Source: Device Information · p. 19
• Each check qubit connects to six data qubits through tunable couplers spanning approximately 1 mm to 6.5 mm.Air-bridges enable coupler crossovers to realize the BB-code connectivity topology in two dimensions.Source: Device Information · p. 19

Multi-length couplers

• Tunable transmon couplers mediate interactions between distant qubits, extending beyond nearest-neighbor connectivity while minimizing unwanted crosstalk.The coupler lengths are varied to establish a capacitance gradient.Source: Multi-length couplers · p. 19
• Coupler nonlinearities span −10 MHz to −170 MHz, while engineered qubit-coupler capacitances and Josephson junction resistances compensate for capacitance differences to maintain consistent tunability.Each coupler is a frequency-tunable transmon connected capacitively to two distant qubits.Source: Multi-length couplers · p. 19
• Air bridges form quasithree-dimensional structures where couplers intersect and connect grounds separated by couplers and control lines, maintaining a uniform ground reference.Multiple air bridges are incorporated into the coupler capacitance.Source: Multi-length couplers · p. 19

Measurement setup

• Six readout lines multiplex five or six resonators each at approximately 100 MHz spacing for low-crosstalk simultaneous measurement, with Purcell filters on every line.Check-qubit and data-qubit coupling strengths are approximately 200 MHz and 80 MHz, respectively; the filters suppress Purcell-effect decay while maintaining a high signal-to-noise ratio.Source: Measurement setup · p. 19

Quantum operations

• A full syndrome-measurement cycle uses seven layers of two-qubit CZ gates interspersed with Hadamard and echo gates, then measures the check qubits.The passage identifies operation performance as critical to implementing quantum error correction codes.Source: Quantum operations · p. 19
• Heatmaps and cumulative distributions summarize idle frequency, anharmonicity, relaxation time, and Hahn-echo and CPMG pure-dephasing times.Dashed lines indicate mean values.Source: Figure S4 · p. 20
• The single-qubit characterization also reports Pauli error for simultaneous gates, shown with heatmaps and cumulative distributions.Dashed lines indicate mean values.Source: Figure S4 · p. 20

Single- and two-qubit gates

• Arbitrary single-qubit gates combine 30-ns resonant XY microwave pulses with virtual-Z gates, while idle frequencies are optimized and active microwave compensation suppresses crosstalk.Parallel-gate optimization accounts for decoherence times and parasitic couplings between qubits [54].Source: Single- and two-qubit gates · p. 20
• CZ gates bring |11⟩ near resonance with |02⟩ or |20⟩, activate coupling for 95 ns, and use two 5-ns buffers at the beginning and end.The interaction frequencies are selected for each qubit pair.Source: Single- and two-qubit gates · p. 20
• The BB-code syndrome cycle uses seven CZ layers, with interaction frequencies optimized for layers of 7 or 14 parallel gates and assessed using simultaneous XEB.Optimization accounts for coherence times, pulse shaping, and parasitic couplings; Fig. S5 presents settings for the first and third layers and parallel-gate error characterization.Source: Single- and two-qubit gates · p. 21, Figure S5 · p. 21

• 520-ns check-qubit and 890-ns data-qubit readout pulses, followed by a 400-ns photon-decay buffer, are paired with a mode-matched filter.The filter is based on time-resolved measurements of the resonator response during readout-pulse demodulation [44].Source: Readout · p. 21
• Flux pulses optimize qubit frequencies during readout to prevent AC-Stark-induced frequency collisions and unwanted crosstalk during simultaneous mid-circuit readout.The optimized readout qubit frequencies, resonator frequencies, and dispersive shifts χ are shown in Fig. S6.Source: Readout · p. 21, Figure S6 · p. 22
• Readout assignment errors are characterized by randomly preparing each qubit in |0⟩, |1⟩, or |2⟩; Fig. S6 reports the errors for all three states.The figure also shows qubit frequencies during readout, readout resonator frequencies, and measured dispersive shifts χ.Source: Readout · p. 21, Figure S6 · p. 22

Characterization of stabilizer measurements

• Pauli gates inserted between adjacent CZ gates mitigate dephasing during two-qubit gates while preserving the stabilizer-measurement circuit identity.The resulting circuit is illustrated in Fig. 2c of the main text.Source: Characterization of stabilizer… · p. 22
• For each of 14 weight-6 stabilizers, six data qubits were prepared in all 64 basis states, with 3,000 repetitions per basis.The measured stabilizer-value averages are shown for the input bases in Fig. S8.Source: Characterization of stabilizer… · p. 22, Figure S8 · p. 23

Dephasing error mitigation

• Echo pulses around CZ gates and ten Pauli X dynamical-decoupling pulses during 920-ns readout each reduced detection probabilities; combining them performed best.Dephasing caused more pronounced performance degradation in X-basis than Z-basis experiments.Source: Dephasing error mitigation · p. 23

Leakage rejection

• The leakage probability was 0.127(3) for any of 18 data qubits and 0.082(6) for any of 14 check qubits; per-cycle leakage rates were 0.007(5) and 0.006(1), respectively.Data qubits operated at higher frequencies during CZ gates, which the authors attribute to their slightly higher leakage rate; Tab. S2 lists retained fractions by syndrome-cycle count.Source: Leakage rejection · p. 23, Table S2 · p. 24

Converting measurement outcomes into error detection results

• Because check qubits are not reset, stabilizer values must be recovered from their measurement outcomes across cycles.For intermediate cycles, the preceding check-qubit outcome is used to determine the actual stabilizer value.Source: Converting measurement outcomes into… · p. 24
• When the preceding check-qubit outcome is 0, the current stabilizer value equals the current measurement outcome.The passage states y_j = x_j for this case.Source: Converting measurement outcomes into… · pp. 24–25
• When the preceding check-qubit outcome is 1, the current stabilizer value is the current outcome minus 1 modulo 2.This conditional rule is given for cycles other than the first and final cycles.Source: Converting measurement outcomes into… · pp. 24–25
• In the first and final cycles, stabilizer values are given directly by measurement outcomes.The first cycle has no preceding cycle, while final-cycle parity outcomes come directly from data-qubit measurements.Source: Converting measurement outcomes into… · p. 25, Figure S12 · p. 26
• An error-detection event is recorded when a stabilizer value differs from its value in the preceding cycle.The result is z_j = (y_j - y_{j-1}) mod 2.Source: Converting measurement outcomes into… · p. 25, Figure S12 · p. 26
• Check-qubit outcomes are converted into stabilizer values and then into error-detection events, as illustrated in Fig. S12.The process has two steps: first determine each stabilizer value, then compare it with the preceding cycle.Source: Converting measurement outcomes into… · pp. 24–25, Figure S12 · p. 26