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arXiv:2409.13025 · Nature

Hardware-efficient quantum error correction via concatenated bosonic qubits

Harald Putterman, Kyungjoo Noh, Connor T. Hann, Gregory S. MacCabe, Shahriar Aghaeimeibodi, Rishi N. Patel, Menyoung Lee, William M. Jones, Hesam Moradinejad, Roberto Rodríguez, Neha Mahuli, Jefferson Rose, John Clai Owens, Harry Levine, Emma Rosenfeld, Philip Reinhold, Lorenzo Moncelsi, Joshua Ari Alcid, Nasser Alidoust, Patricio Arrangoiz-Arriola, James W. Barnett, Przemysław Bienias, Hugh A. Carson, Cliff Chen, Li Chen, Harutiun Chinkezian, Eric M. Chisholm, Ming-Han Chou, Aashish A. Clerk, Andrew Clifford, R. Cosmic, Ana Valdés-Curiel, Erik Davis, Laura DeLorenzo, John M. D'Ewart, Art Diky, Nathan D’Souza, Philipp T. Dumitrescu, Shmuel Eisenmann, E. S. Elkhouly, Glen Evenbly, Michael T. Fang, Yawen Fang, Matthew J. Fling, Warren Fon, Gabriel Garcia, Alexey Vyacheslavovich Gorshkov, Julia A. Grant, Mason Gray, Sebastian Grimberg, Arne L. Grimsmo, Arbel Haim, Justin Hand, He Yuan, Mike Hernandez, David J. Hover, Jimmy S. C. Hung, Matthew Hunt, Joseph K. Iverson, Ignace Jarrige, Jean-Christophe Jaskula, Liang Jiang, Mahmoud Kalaee, Rassul B. Karabalin, Peter J. Karalekas, Andrew J. Keller, Amirhossein Khalajhedayati, Aleksander Kubica, Hanho Lee, Catherine Leroux, Simon Lieu, Victor Ly, Keven Villegas Madrigal, Guillaume Marcaud, Gavin McCabe, Cody Miles, Ashley Milsted, Joaquín Minguzzi, Anurag Mishra, Biswaroop Mukherjee, Mahdi Naghiloo, Eric Oblepias, Gerson Ortuno, Jason Pagdilao, Nicola Pancotti, Ashley Panduro, Jean-Philip Paquette, Minje Park, G. A. Peairs, David Perello, Eric C. Peterson, Sophia Ponte, John P. Preskill, Johnson Qiao, Gil Refael, Rachel Resnick, Alex Retzker, Omar A. Reyna, Marcus C. Runyan, Colm A. Ryan

Published Feb 26, 2025• 108 citations
arXivHTMLPDF

Abstract

To solve problems of practical importance1,2, quantum computers probably need to incorporate quantum error correction, in which a logical qubit is redundantly encoded in many noisy physical qubits3–5. The large physical-qubit overhead associated with error correction motivates the search for more hardware-efficient approaches6–18. Here, using a superconducting quantum circuit19, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance d = 5 (ref. 10). A stabilizing circuit passively protects cat qubits against bit flips20–24. The repetition code, using ancilla transmons for syndrome measurement, corrects cat qubit phase flips. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below the threshold. The logical bit-flip error is suppressed with increasing cat qubit mean photon number, enabled by our realization of a cat-transmon noise-biased CX gate. The minimum measured logical error per cycle is on average 1.75(2)% for the distance-3 code sections, and 1.65(3)% for the distance-5 code. Despite the increased number of fault locations of the distance-5 code, the high degree of noise bias preserved during error correction enables comparable performance. These results, where the intrinsic error suppression of the bosonic encodings enables us to use a hardware-efficient outer error-correcting code, indicate that concatenated bosonic codes can be a compelling model for reaching fault-tolerant quantum computation. Bosonic qubits can be engineered to feature intrinsic protection against certain kinds of errors, which makes quantum error correction across many bosonic qubits possible with less overhead.

DOI
10.1038/s41586-025-08642-7

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To solve problems of practical importance1,2, quantum computers probably need to incorporate quantum error correction, in which a logical qubit is redundantly encoded in many noisy physical qubits3–5. The large physical-qubit overhead associated with error correction motivates the search for more hardware-efficient approaches6–18. Here, using a superconducting quantum circuit19, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance d = 5 (ref. 10). A stabilizing circuit passively protects cat qubits against bit flips20–24. The repetition code, using ancilla transmons for syndrome measurement, corrects cat qubit phase flips. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below the threshold. The logical bit-flip error is suppressed with increasing cat qubit mean photon number, enabled by our realization of a cat-transmon noise-biased CX gate. The minimum measured logical error per cycle is on average 1.75(2)% for the distance-3 code sections, and 1.65(3)% for the distance-5 code. Despite the increased number of fault locations of the distance-5 code, the high degree of noise bias preserved during error correction enables comparable performance. These results, where the intrinsic error suppression of the bosonic encodings enables us to use a hardware-efficient outer error-correcting code, indicate that concatenated bosonic codes can be a compelling model for reaching fault-tolerant quantum computation. Bosonic qubits can be engineered to feature intrinsic protection against certain kinds of errors, which makes quantum error correction across many bosonic qubits possible with less overhead.
Dissipative cat qubits are a promising physical platform for quantum computing, since their large noise bias can enable more hardware-efficient quantum error correction. In this work we theoretically study the long-term prospects of a hybrid cat-transmon quantum computing architecture where dissipative cat qubits play the role of data qubits, and error syndromes are measured using ancillary transmon qubits. The cat qubits’ noise bias enables more hardware-efficient quantum error correction, and the use of transmons allows for practical, high-fidelity syndrome measurement. While correction of the dominant cat Z errors with a repetition code has recently been demonstrated in experiment, here we show how the architecture can be scaled beyond a repetition code. In particular, we propose a cat-transmon entangling gate that enables the correction of residual cat X errors in a thin rectangular surface code, so that logical error can be arbitrarily suppressed by increasing code distance. We numerically estimate logical memory performance, finding significant overhead reductions in comparison to architectures without biased noise. For example, with current state-of-the-art coherence, physical error rates of 10 − 3 and noise biases in the range 10 3 – 10 4 are achievable. With this level of performance, the qubit overhead required to reach algorithmically relevant logical error rates with the cat-transmon architecture matches that of an unbiased-noise architecture with physical error rates in the range 10 − 5 – 10 − 4 .
Dissipative cat qubits exponentially suppress one Pauli error channel with the mean photon number, leaving the conjugate bit-flip error as the dominant failure mode. This strong noise bias makes the full machinery of general quantum error correction unnecessary: a code need only protect against a single error type, and any classical linear code can be promoted to a Clifford stabilizer code that does exactly this. We use this observation to build a bit-flip-only quantum Reed-Solomon (RS) code. Starting from the maximum-distance-separable RS code [7, 3, 5] over GF(232^323), we binary-expand it to the linear code [21, 9, 6] over GF(2) and realize it as a [[21, 9, dX=6,dZ=1d_X = 6, d_Z = 1dX​=6,dZ​=1]] bit-flip code whose stabilizers are products of ZZZ operators. Because no phase-flip correction is attempted, the construction discards the redundancy that standard quantum RS codes spend on correcting ZZZ errors -- which a strongly biased cat qubit renders unnecessary -- and yields a shallow Clifford circuit that samples directly in Stim. Errors are decoded by an optimal bounded-distance syndrome-lookup table. We then introduce a Tornado architecture: a two-layer concatenation that wraps every position of the outer RS code in an inner distance-three repetition code, yielding a [[63, 9, 18]] code decoded by a two-stage inner majority vote and outer lookup decoder. Monte-Carlo simulation shows that at a physical bit-flip rate p=0.1p = 0.1p=0.1 the Tornado code reaches a logical error rate pL≈5.3×10−3p_L \approx 5.3 \times 10^{-3}pL​≈5.3×10−3, below both parent codes, and that its logical error rate scales as pL∝p6p_L \propto p^6pL​∝p6 at low ppp, in contrast to p2p^2p2 for the repetition code and p3p^3p3 for the standalone RS code. We give the exact construction, the error and circuit model, an asymptotic scaling analysis, and an honest account of the overhead cost and single-shot assumptions.
Scalable fault-tolerant quantum computation requires quantum error-correcting codes that simultaneously support universal logical operations, suppress hardware-specific noise, and enable efficient handling of photon-loss errors. Bosonic encodings such as the dual-rail and cat codes each offer attractive features but also exhibit important limitations when used in isolation. The dual-rail code enables efficient single-photon-loss detection by converting leakage out of the computational subspace induced by photon-loss errors into an erasure error. In contrast, the cat code provides a resource-efficient, bias-tailored error-correction scheme with bias-preserving logical gate operations. Here, we introduce the dual-rail cat code (DRCC), a concatenated bosonic encoding that combines an inner cat code with an outer dual-rail structure, thereby inheriting and enhancing the advantages of both constituent codes. We analyse the error-correction properties of the DRCC and propose a deterministic single-photon-loss correction protocol by concatenating it with an outer repetition code. Exploiting the code's intrinsic noise bias, we construct a universal set of logical gates using only beam-splitter interactions and demonstrate that all logical operations preserve the erasure-biased noise structure. The DRCC offers several distinctive advantages, including the absence of relative geometric phases during gate operations, deterministic erasure detection and correction, and simultaneous syndrome extraction without interrupting stabilisation. These features make the DRCC a promising bosonic code for hardware-efficient, bias-preserving, and erasure-resilient fault-tolerant quantum computation.
Stabilized cat qubits that possess biased noise channel with bit-flip errors exponentially smaller than phase-flip errors. Together with a set of bias-preserving (BP) gates, cat qubits are a promising candidate for realizing hardware efficient quantum error correction and fault-tolerant quantum computing. Compared to dissipatively stabilized cat qubits, the Kerr cat qubits can in principle support faster gate operations with higher gate fidelity, benefiting from the large energy gap that protects the code space. However, the leakage of the Kerr cats can increase the minor type of errors and compromise the noise bias. Both the fast implementation of gates and the interaction with environment can lead to such detrimental leakage if no sophisticated controls are applied. In this work, we introduce new fine-control techniques to overcome the above obstacles for Kerr cat qubits. To suppress the gate leakage, we use the derivative-based transition suppression technique to design derivative-based controls for the Kerr BP gates. We show that the fine-controlled gates can simultaneously have high gate fidelity and high noise bias and when applied to concatenated quantum error correction, can not only improve the logical error rate but also reduce resource overhead. To suppress the environment-induced leakage, we introduce colored single-photon dissipation, which can continuously cool the Kerr cats and suppress the minor errors while not enhancing the major errors.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Biased-noise qubits, in which one type of error (e.g. XXX- and YYY-type errors) is significantly suppressed relative to the other (e.g. ZZZ-type errors), can significantly reduce the overhead of quantum error correction. Codes such as the rectangular surface code or XZZX code substantially reduce the qubit overhead under biased noise, but they still face challenges. The rectangular surface code suffers from a relatively low threshold, while the XZZX code requires twice as many physical qubits to maintain the same code distance as the surface code. In this work, we introduce a 2D local code construction that outperforms these codes for noise biases η≥7×104η\ge 7\times10^{4}η≥7×104, reducing the qubit overhead by over 50% at pZ=10−3p_Z=10^{-3}pZ​=10−3 and η=2×106η= 2 \times 10^6η=2×106 to achieve a logical error rate of 10−1210^{-12}10−12. Our construction relies on the concatenation of two classical codes. The inner codes are repetition phase-flip codes while the outer codes are high-rate bit-flip codes enabled by their implementation at the logical level, which circumvents device connectivity constraints. These results indicate that under sufficiently biased noise, it is advantageous to address phase-flip and bit-flip errors at different layers of the coding scheme. The inner code should prioritize a high threshold for phase-flip errors, while the bit-flip outer code should optimize for encoding rate efficiency. In the strong biased-noise regime, high-rate outer codes keep the overhead for correcting residual bit-flip errors comparable to that of the repetition code itself, meaningfully lower than that required by earlier approaches.