Quantum error correction [1, 2, 3, 4] provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. Here, we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: a distance-7 code, and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of
Λ=2.14±0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit’s lifetime by a factor of
2.4±0.3 . Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 μs at distance-5 up to a million cycles, with a cycle time of 1.1 μs. We also run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or
3 × 109 cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.
Large error rates of quantum bits (qubits) are one of the main difficulties in the development of quantum computing. Performing quantum error correction (QEC) with surface codes is considered the most promising approach to reduce the error rates of qubits effectively. To perform error correction, we need an error-decoding unit, which estimates errors in the noisy physical qubits repetitively, to create a robust logical qubit. While complicated graph-matching problems must be solved within a strict time restriction for the error decoding, several hardware implementations that satisfy the restriction at a large code distance have been proposed.
In this article, we investigate the role of local information in the decoding of the repetition and surface error correction codes for the protection of quantum states. Our key result is an improvement in resource efficiency when local information is taken into account during the decoding process: the code distance associated with a given logical error rate is reduced with a magnitude depending on the proximity of the physical error rate to the accuracy threshold of the code. We also briefly discuss an averaged approach with local information for table lookup and localized decoding schemes, an expected breakdown of these effects for large-scale systems, and the importance of this resource reduction in the near term.
We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. These results fit well known scaling laws for studied physical error rates and code distances, but break down in some other regimes. Increasing the code distance still reduces logical error rates, but returns diminish quickly beyond a certain point. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error to provably good accuracy quickly. We include an analysis of measurement errors to allow for a more complete comparison of different surface code implementations. We extend this counting method to a particular locally-correlated error model and another globally-correlated error model.
Practical quantum computing requires error rates much lower than those achievable with physical qubits. Surface codes provide a reliable method to implement logical qubits using physical qubits. However, when these physical qubits are transmitted within quantum networks, their error rates can vary due to differences in the routing paths. Our experimental results indicate that the geometric arrangement of these physical qubits within a surface code significantly impacts its logical error rate. In this paper, we address this issue by proposing disparity surface codes, which are a group of geometric configurations for surface codes that adapt them for transmission within quantum networks. For a distance-
ddisparity surface code, only
dphysical qubits need to be transmitted with low error rates, and we identify multiple equivalent choices for selecting these
dqubits. Additionally, we propose Di-MWPM, a new decoder designed to exploit physical error rate disparity for enhanced error correction. Results demonstrate that disparity surface codes maintain favorable performance in different network scenarios, even under high noise conditions. Our Di-MWPM decoder achieves improved Pauli-error thresholds.
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.