Circuit-level fault tolerance of cat codes

Long D. H. My; Shushen Qin; Hui Khoon Ng

doi:10.22331/q-2025-07-23-1810

Circuit-level fault tolerance of cat codes

Long D. H. My^1,2, Shushen Qin², and Hui Khoon Ng^1,2

¹Yale-NUS College, Singapore
²Centre for Quantum Technologies, National University of Singapore, Singapore

Published:	2025-07-23, volume 9, page 1810
Editor:	Pei Zeng
Eprint:	arXiv:2406.04157v3
Doi:	https://doi.org/10.22331/q-2025-07-23-1810
Citation:	Quantum 9, 1810 (2025).

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Updated after initial publication: This publication was updated to version v3 after the initial publication.

Abstract

Bosonic codes encode quantum information into a single infinite-dimensional physical system endowed with error correction capabilities. This reduces the need for complex management of many physical constituents compared with standard approaches employing multiple physical qubits. Recent discussions of bosonic codes centre around correcting only boson-loss errors, with phase errors either actively suppressed or deferred to subsequent layers of encoding with standard qubit codes. Rotationally symmetric bosonic (RSB) codes, which include the well-known cat and binomial codes, are capable of simultaneous correction of loss and phase errors, offering an alternate route that deals with arbitrary errors already at the base layer. Here, we investigate the robustness of such codes, moving away from the more idealistic past studies towards a circuit-level noise analysis closer to the practical situation where every physical component in the device is potentially faulty. We extend the concept of fault tolerance to the case of RSB codes, and then examine the performance of two known error correction circuits under circuit-level noise. Our analysis reveals a significantly more stringent noise threshold for fault-tolerant operation than found in past works; nevertheless, we show how, through waiting-time optimization and the use of squeezing, we can restore the noise requirements to a regime achievable with near-term quantum hardware. While our focus here is on cat codes for concreteness, a similar analysis applies for general RSB codes.

Featured image: Left figure: Error correction circuit for cat codes, originally proposed in Grimsmo et al. [PRX 10, 011058 (2020)], now with circuit-level noise, i.e., potential faults at every circuit location (dashed boxes). Right figure: Performance of cat codes of different code orders ($N=2, 3, 4$) in the presence of circuit-level noise, with phase and loss error probabilities $\gamma_\mathrm{ph}$ and $\gamma_\mathrm{loss}$, respectively. We show curves for fixed time between error-correction cycles ("fixed wait”), optimised wait time (“varying wait”), and for squeezed cat codes with optimised wait time. The region to the left and below each curve is where error correction outperforms the no-error-correction situation. We find significantly more stringent noise requirements in this circuit-level noise situation than reported in past studies; squeezing and wait-time optimization, however, bring those requirements back to levels feasible for near-term experiments.

Popular summary

There is currently intense interest in building large-scale quantum computers to solve problems intractable on traditional computers. Quantum computers are extremely sensitive to errors, and accurate, useful computation can be completed only if the operations are carried in a fault-tolerant manner, to neutralize the effects of errors due to imperfections in the physical implementation.

Fault-tolerant quantum computing schemes are built upon error-correcting codes. An attractive class of codes, well-suited for experimental implementation, is the family of rotationally symmetric bosonic (RSB) codes, which includes cat codes and binomial codes as well-studied instances. These codes store computational information in bosonic degrees of freedom, within a specially chosen subspace of the infinite-dimensional bosonic state space. The infinite-dimensional state space permits simultaneous correction of loss and phase errors—common in bosonic systems—within a single physical system. This is to be contrasted with qubit schemes where the information is stored across multiple physical qubits, requiring then complicated management of multi-component systems, with worsening complexity as one scales up the noise protection by using more and more qubits.

Of course, just having a code that can correct errors is not enough. The error correction procedure that carries out the error removal has itself to be tolerant against faults, to realise the correction capabilities. Past analyses of RSB codes took only limited account of imperfections in the error correction; a realistic assessment of performance, which we provide here, has to consider errors that can happen anywhere in the physical device.

Here, we extend the notion of quantum fault tolerance to the case of RSB codes, giving a set of conditions on the allowed errors and circuit properties. We then use these conditions to analyze the performance of error-correction circuits for cat codes. Not surprisingly, we find significantly more stringent noise threshold requirements than reported in past studies. However, we offer solutions, through wait-time optimization and squeezing, that bring the noise requirements back to levels feasible for near-term experiments.

► BibTeX data

@article{My2025circuitlevelfault,
  doi = {10.22331/q-2025-07-23-1810},
  url = {https://doi.org/10.22331/q-2025-07-23-1810},
  title = {Circuit-level fault tolerance of cat codes},
  author = {My, Long D. H. and Qin, Shushen and Ng, Hui Khoon},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1810},
  month = jul,
  year = {2025}
}

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Cited by

[1] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

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