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Removing leakage-induced correlated errors in superconducting quantum error correction
M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. J. Satzinger, C. Quintana, P. V. Klimov, D. Sank, C. Gidney, A. G. Fowler, F. Arute, K. Arya, B. Buckley, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, S. Demura, A. Dunsworth, C. Erickson, B. Foxen, M. Giustina, T. Huang, S. Hong, E. Jeffrey, S. Kim, K. Kechedzhi, F. Kostritsa, P. Laptev, A. Megrant, X. Mi, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Niu, A. Paler, N. Redd, P. Roushan, T. C. White, J. Yao, P. Yeh, A. Zalcman, Yu Chen, V. N. Smelyanskiy, John M. Martinis, H. Neven, J. Kelly, A. N. Korotkov, A. G. Petukhov, R. Barends
TL;DR
The paper addresses leakage in superconducting transmons, where higher-state population complicates error correction and can produce correlated errors. It introduces and evaluates a multi-level reset strategy through leakage-rate and correlation analyses, finding reduced leakage correlations and improved error-suppression behavior in the bit-flip code.
Problem
Higher transmon levels create computational and leakage errors, and leakage can produce correlated error patterns that complicate quantum error correction.
Method
The paper designs a quasi-adiabatic reset pulse using parameters µ and fswap, then models leakage during code operation with rate equations and analyzes error correlations.
Results
Reset increases the effective leakage decay rate, while leakage-induced correlations produce distinctive odd-even patterns in the correlation matrix.
Takeaways & Limitations
The results support using reset and correlation information to reduce leakage-related error structure in stabilizer-code operation.
Abstract
from arXiv · showhide
Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing.
Supplementary information for “Removing leakage-induced correlated errors in
The supplementary information accompanies the paper on removing leakage-induced correlated errors in superconducting quantum error correction.
- The supplementary information is for “Removing leakage-induced correlated errors in superconducting quantum error correction.”
I. RESET GATE PARAMETERS
The reset pulse is designed through quasi-adiabatic control parameters and optimized across initial states |1⟩, |2⟩, and |3⟩. The chosen parameters trade off performance across these states while maintaining acceptable reset at sufficiently long swap durations.
- I. RESET GATE PARAMETERS: The pulse trajectory replaces the physical coupling and resonator frequency with free parameters µ and fswap, which are optimized experimentally.The trajectory is designed to address three relevant resonance conditions associated with the first three excited states.
- I. RESET GATE PARAMETERS: The dimensionless control variable x is converted numerically into physical time t before generating the qubit-frequency trajectory fq(t).The trajectory uses interpolation after analytical calculation of the angle profile and numerical calculation of t(x).
- I. RESET GATE PARAMETERS: At sufficiently long swap durations, setting fswap = fr provides acceptable performance for all initial states, although |2⟩ and |3⟩ favor higher fswap values.For short swaps, the optimum for |1⟩ is near fswap = fr, while the higher states have distinct optima.
- I. RESET GATE PARAMETERS: µ controls pulse adiabaticity by broadening the near-crossing frequency range as it increases, improving higher-state performance while increasing |1⟩ errors.The parameter must balance different transition structures for the three initial states.
- I. RESET GATE PARAMETERS: At long swap and hold durations, computational error dominates because leakage error is approximately 10x lower for all initial states.The supplementary figures separate computational error P1 from leakage error P2 + P3.
II. LEAKAGE ERROR AND SUPPRESSION
The reset analysis distinguishes computational from leakage errors and shows that leakage decreases faster with hold time. At sufficiently long durations, leakage error is about 10x below computational error, while specialized readout separates computational and higher states.
- II. LEAKAGE ERROR AND SUPPRESSION: Computational error is occupation of |1⟩ after reset, whereas leakage error is residual occupation of |2⟩ or |3⟩.Computational errors are preferred because stabilizer codes naturally identify and correct errors within the computational basis.
- II. LEAKAGE ERROR AND SUPPRESSION: Higher initial states produce more leakage error at short swap and hold durations, but leakage decreases faster with hold time than computational error.The faster reduction reflects higher energy-relaxation rates from higher states.
- II. LEAKAGE ERROR AND SUPPRESSION: Leakage error is approximately 10x lower than computational error once swap and hold durations reach the readout floor.This indicates that computational error is the dominant reset error source in that regime.
- II. LEAKAGE ERROR AND SUPPRESSION: Complex readout signals from preparations in |0⟩, |1⟩, |2⟩, and |3⟩ are used to calibrate discrimination between computational and leakage states.Each demodulated I-Q shot is labeled by its prepared state to evaluate readout fidelity.
- II. LEAKAGE ERROR AND SUPPRESSION: The readout floor is estimated by heralding |0⟩ with one measurement and measuring the fidelity of |0⟩ in a second sequential measurement.
III. RATE EQUATIONS FOR LEAKAGE DURING CODE OPERATION
Leakage growth during code operation is modeled with rate equations fitted to measured leakage populations. Applying reset increases the effective leakage decay rate, with the reset-qubit estimate using an assumption for the upward rate.
- III. RATE EQUATIONS FOR LEAKAGE DURING CODE OPERATION: Leakage population is fit as a function of code rounds to extract parameters for a rate equation.The leakage growth is measured during bit-flip-code operation.
- III. RATE EQUATIONS FOR LEAKAGE DURING CODE OPERATION: Applying reset to measure qubits increases the effective leakage decay rate during stabilizer rounds.The reset condition changes the established leakage-growth behavior.
- III. RATE EQUATIONS FOR LEAKAGE DURING CODE OPERATION: The reset-condition estimate of γ↓ assumes γ↑ equals the no-reset value and uses the average measure-qubit error across all rounds for p∞.These assumptions are noted in the rate analysis and Table S1.
IV. THE CHECKERBOARD PATTERN IN THE pij-MATRIX
The p_ij correlation matrix shows a checkerboard pattern: correlations are larger at odd round separations and smaller, sometimes negative, at even separations. This pattern arises from repeated X gates alternating a data qubit’s state, making subsequent energy relaxation events preferentially occur at odd separations.
- Odd-separation correlations are larger, while even-separation correlations are smaller and can be negative.
- Repeated X gates alternate the data qubit between |0⟩ and |1⟩, producing positive odd-separation and negative even-separation relaxation correlations.
- The correlations gradually decay as the separation between relaxation events increases.
- The checkerboard pattern is more pronounced for a single measure qubit affected by both neighboring data qubits.
V. STATISTICS AND POSTSELECTION IN THE BIT-FLIP CODE
The bit-flip-code benchmarks average many randomized realizations, while postselection removes atypical high-logical-error events. The retained dataset excludes approximately 0.8% of measurements.
- 40 random data-qubit bitstrings and 1000 repetitions per bitstring produced 40,000 total realizations for Figs. 4, 5, and 6.
- The leakage-population measurements used 20 random initial data-qubit bitstrings and 5000 repetitions for each bitstring.
- A moving-average logical-error threshold of 25% identified event starts, removing around 1000 realizations for each event.
- Postselection removed approximately 0.8% of the data; the total average logical error was below 3%, but the event-period moving average could reach 50%.