Quantum error correction promises a viable path to fault-tolerant computations, enabling exponential error suppression when the device's error rates remain below the protocol's threshold. However, this threshold strongly depends on the classical method used to decode the measurements of the syndrome. These classical algorithms traditionally only interpret binary data, ignoring valuable information contained in complete analog measurement data. We present the first large-scale experimental verification of soft decoding by implementing repetition codes up to distance 51 on superconducting hardware. Unlike previous demonstrations limited to small code distances, this extensive range allows us to confirm a fundamental improvement in the exponential scaling of error suppression rather than a mere constant-factor reduction in error rates. We observe a 25% increase in the threshold, which yields logical error rates up to 30 times lower than standard hard decoding. Analyzing the trade-off between information volume and decoding performance, we show that a single byte of information per measurement suffices to reach optimal decoding. This underscores the effectiveness and practicality of soft decoding on hardware, including in time-sensitive contexts such as real-time decoding.
Quantum error correction promises a viable path to fault-tolerant computations, enabling exponential error suppression when the device's error rates remain below the protocol's threshold. However, this threshold strongly depends on the classical method used to decode the measurements of the syndrome. These classical algorithms traditionally only interpret binary data, ignoring valuable information contained in complete analog measurement data. We present the first large-scale experimental verification of soft decoding by implementing repetition codes up to distance 51 on superconducting hardware. Unlike previous demonstrations limited to small code distances, this extensive range allows us to confirm a fundamental improvement in the exponential scaling of error suppression rather than a mere constant-factor reduction in error rates. We observe a 25% increase in the threshold, which yields logical error rates up to 30 times lower than standard hard decoding. Analyzing the trade-off between information volume and decoding performance, we show that a single byte of information per measurement suffices to reach optimal decoding. This underscores the effectiveness and practicality of soft decoding on hardware, including in time-sensitive contexts such as real-time decoding.
Abstract Imperfect measurements are a prevalent source of error across quantum computing platforms, significantly degrading the logical error rates achievable on current hardware. To mitigate this issue, rich measurement data referred to as soft information has been proposed to efficiently identify and correct measurement errors as they occur. In this work, we model soft information decoding across a variety of physical qubit platforms and decoders and showcase how soft information can make error correction viable at lower code distances and higher physical error rates than is otherwise possible. We simulate the effects of soft information decoding on quantum memories for surface codes and bivariate bicycle codes, and evaluate the error suppression performance of soft decoders against traditional decoders. Our simulations show that soft information decoding on near-term devices can provide up to 11% higher error suppression on superconducting qubits and up to 20% stronger error suppression on neutral atom qubits. These accuracy gains correspond to 13% and 33% reductions in the physical qubit footprint of superconducting and neutral atom devices respectively when operating at a logical error rate of 10 −6 , showcasing that soft information is a powerful tool for reducing the cost and complexity of large-scale fault-tolerant quantum computers.
Quantum error correction allows for quantum information to be preserved using logical qubits, which are subject to lower error rates than their constituent physical qubits. The degree of error suppression depends on the choice of error correcting code and distance, the underlying physical error rate, and the accuracy of the decoder. While traditional decoders utilise a binary (hard) syndrome, recent work shows that additional (soft) information captured during qubit readout can be effectively utilised to improve decoding accuracy. In this work, we present experimental results from a distance-three surface code implemented on transmon qubits, where we perform Z-stabiliser measurements to protect the state of the logical qubit against bit-flip errors. We initialise the logical qubit in one of 16 possible computational states representing the logical zero state, and perform repeated stabiliser checks over a variable number of rounds to preserve the state over time. We compare the decoding performance for a hard minimum-weight perfect matching decoder against a soft variant where rich measurement information is incorporated, and demonstrate an improved logical fidelity. Additionally, we employ a recurrent neural network decoder with both soft and hard variants and observe improved performance when soft information is used. The general nature of soft information makes it widely applicable to different physical qubit platforms, where it can be leveraged to shorten measurement times and improve the logical fidelity in quantum error correction experiments. Pre-print available at arXiv:2403.00706.
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.
Quantum error correction enables the preservation of logical qubits with a lower logical error rate than the physical error rate, with performance depending on the decoding method. Traditional error decoding approaches, relying on the binarization (`hardening') of readout data, often ignore valuable information embedded in the analog (`soft') readout signal. We present experimental results showcasing the advantages of incorporating soft information into the decoding process of a distance-three (d=3) bit-flip surface code with transmons. To this end, we use the 3×3 data-qubit array to encode each of the 16 computational states that make up the logical state ∣0L⟩, and protect them against bit-flip errors by performing repeated Z-basis stabilizer measurements. To infer the logical fidelity for the ∣0L⟩ state, we average across the 16 computational states and employ two decoding strategies: minimum weight perfect matching and a recurrent neural network. Our results show a reduction of up to 6.8% in the extracted logical error rate with the use of soft information. Decoding with soft information is widely applicable, independent of the physical qubit platform, and could reduce the readout duration, further minimizing logical error rates.
Quantum error correction can suppress logical errors exponentially with code distance when physical noise is below threshold. Realizing this on hardware requires tight coordination among quantum gates, mid-circuit measurements, and qubit coherence. Device variation, noise asymmetry, and correlations across space and time complicate the translation of these capabilities into logical error suppression. Here, we study rotated surface-code memory on an IBM Nighthawk r2 processor. Performance varies across tested processor placements, which we attribute to differences in qubit quality. We optimize the code by combining short readout, fused measurement and reset, and an XX echo on data qubits during ancilla readout. Compared with standard readout without reset or echo, the optimized syndrome extraction circuit reduces the median per-round logical error rate by factors of 3.8 in the X basis and 1.4 in the Z basis. In memory experiments of up to 150 rounds, the distance advantage weakens, alongside long-lived correlations that may reflect a persistent error component. Under our optimization recipe, increasing the protected distance from 3 to 6 while holding the other distance fixed reduces the per-round logical error rate from 0.53% to 0.24% in Z memory and from 0.52% to 0.22% in X memory. The corresponding error-suppression factors under maximum-likelihood decoding are 1.68 and 1.85. These results connect circuit compilation and optimization to surface-code memory performance and limitations, and motivate QPU and QEC circuit codesign for fault-tolerant quantum computing primitives on current cloud hardware.
Quantum error mitigation (QEM) is typically viewed as a suite of practical techniques for today's noisy intermediate-scale quantum devices, with limited relevance once fault-tolerant quantum computers become available. In this work, we challenge this conventional wisdom by showing that QEM can continue to provide substantial benefits in the era of quantum error correction (QEC), and in an even more efficient manner than it does on current devices. We introduce a framework for logical-level QEM that leverages soft information naturally produced by QEC decoders, requiring no additional data, hardware modifications, or runtime overhead beyond what QEC protocols already provide. Within this framework, we develop and analyze three logical-level QEM techniques: post-selection and runtime abort policies, probabilistic error cancellation, and zero-noise extrapolation. Our techniques reduce logical error rates by more than 100x while discarding fewer than 0.1% of shots; they also provide in situ characterization of logical channels for QEM protocols. As a proof of principle, we benchmark our approach using a surface-code architecture and two state-of-the-art decoders based on tensor-network contraction and minimum-weight perfect matching. We evaluate logical-level QEM on random Clifford circuits and molecular simulation algorithms and find that, compared to previous approaches relying on QEC only or QEC combined with QEM, we can achieve up to 87.4% spacetime overhead savings. Our results demonstrate that logical-level QEM with QEC decoder soft information can reliably improve logical performance, underscoring the efficiency and usefulness of QEM techniques for fault-tolerant quantum computers.