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arXiv:2108.12225 · Physical Review Research

All-optical cat-code quantum error correction

Jacob Hastrup, Ulrik L. Andersen

Published Oct 31, 2022• 37 citations
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Abstract

The cat code is a promising encoding scheme for bosonic quantum error correction as it allows for correction against losses---the dominant error mechanism in most bosonic systems. However, it has remained unclear how the required syndrome measurement and recovery can be implemented in the optical regime. Here, we introduce a teleportation-based error-correction scheme for the cat code, using elements suitable for an optical setting. The scheme detects and corrects single-photon losses while restoring the amplitude of the cat states, thereby greatly suppressing the accumulation of errors in lossy channels.

DOI
10.1103/physrevresearch.4.043065

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The cat code is a promising encoding scheme for bosonic quantum error correction as it allows for correction against losses---the dominant error mechanism in most bosonic systems. However, it has remained unclear how the required syndrome measurement and recovery can be implemented in the optical regime. Here, we introduce a teleportation-based error-correction scheme for the cat code, using elements suitable for an optical setting. The scheme detects and corrects single-photon losses while restoring the amplitude of the cat states, thereby greatly suppressing the accumulation of errors in lossy channels.
We propose a teleportation-based scheme to implement a universal set of quantum gates with a four-component cat code, assisted by appropriate entangled resource states and photon-number-resolving detection. The four-component cat code features the ability to recover from single-photon loss. Here we propose a concrete procedure to correct the single-photon loss, including detecting the single-photon-loss event and recovering the initial states. By concatenating with standard qubit error-correcting codes, we estimate the loss threshold for fault-tolerant quantum computation and obtain a significant improvement over the two-component cat code.
We analyze a generalized quantum error-correction code against photon loss where a logical qubit is encoded into a subspace of a single oscillator mode that is spanned by distinct multicomponent cat states (coherent-state superpositions). We present a systematic code construction that includes the extension of an existing one-photon-loss code to higher numbers of losses. When subject to a photon loss (amplitude damping) channel, the encoded qubits are shown to exhibit a cyclic behavior where the code and error spaces each correspond to certain multiples of losses, half of which can be corrected. As another generalization we also discuss how to protect logical qudits against photon losses, and as an application we consider a one-way quantum communication scheme in which the encoded qubits are periodically recovered while the coherent-state amplitudes are restored as well at regular intervals.
We present a systematic study of Schrödinger cat codes constructed from Kerr-type coherent states, including displaced Kerr coherent states and Barut--Girardello Kerr coherent states, each admitting two distinct families determined by the sign of the Kerr nonlinearity. By tuning the Kerr parameter and coherent-state amplitude, these states interpolate between su(2)\mathfrak{su}(2)su(2), su(1,1)\mathfrak{su}(1,1)su(1,1) coherent states, providing a unified and versatile foundation for this type of bosonic quantum error correction. Unlike standard two-component Schrödinger cat codes, where a single photon-loss event induces an uncorrectable bit-flip, the nonlinear phase-space structure of Kerr cat states enables simultaneous detection and correction of both photon-loss and dephasing errors within a unified recovery framework, with optimal recovery operations determined via convex optimization. We demonstrate that Kerr cat encodings significantly outperform conventional cat codes under combined loss and dephasing noise, and that judicious parameter optimization can suppress both error channels to a level that reduces the overhead of additional error correction layers. We further show that Kerr-deformed coherent-state manifolds under engineered two-photon driving emerge as effective steady states of driven-dissipative dynamics, with single-photon decoherence strongly suppressed and leakage outside the protected manifold appearing only as higher-order corrections in the deformation strength. Our extended formalism identifies generalized Kerr Schrödinger cat codes as promising candidates for fault-tolerant bosonic quantum computation in experimental platforms such as nonlinear photonics.
It is known that multiphoton states can be protected from decoherence due to a passive loss channel by applying noiseless attenuation before and noiseless amplification after the channel. In this work, we propose the combined use of multiphoton subtraction on four-component cat codes and teleamplification to effectively suppress errors under detection and environmental losses. The back-action from multiphoton subtraction modifies the encoded qubit encoded on cat states by suppressing the higher photon numbers, while simultaneously ensuring that the original qubit can be recovered effectively through teleamplification followed by error correction, thus preserving its quantum information. With realistic photon subtraction and teleamplification-based scheme followed by optimal error-correcting maps, one can achieve a worst-case fidelity (over all encoded pure states) of over 93.5% (82% with only noisy teleamplification) at a minimum success probability of about 3.42%, under a 10% environmental-loss rate, 95% detector efficiency and sufficiently large cat states with the coherent-state amplitudes of 2. This sets a promising standard for combating large passive losses in quantum-information tasks in the noisy intermediate-scale quantum (NISQ) era, such as direct quantum communication or the storage of encoded qubits on the photonic platform.
Bosonic codes have seen a resurgence in interest for applications as varied as fault-tolerant quantum architectures, quantum enhanced sensing, and entanglement distribution. Cat codes have been proposed as low-level elements in larger architectures, and the theory of rotationally symmetric codes more generally has been significantly expanded in the recent literature. The fault-tolerant preparation and maintenance of cat-code states as a standalone quantum error correction scheme remains, however, limited by the need for robust state preparation and strong intermode interactions. In this work, we consider the teleportation-based correction circuit for cat-code quantum error correction. We show that the class of acceptable ancillary states is broader than is typically acknowledged, and exploit this to propose the use of many-component bridge states, which, though not themselves in the cat-code space, are nonetheless capable of syndrome extraction in the regime where nonlinear interactions are a limiting factor.
We have generalised an all-optical telecorrection protocol for the higher orders of the cat code, and show that with these higher orders we can achieve target performance at substantially reduced iteration counts at the cost of a higher mean photon-number. We also introduce a probabilistic scheme for correcting deformation of the state, which highlights two interesting abilities of telecorrection: to encode new sets of transformations, and to change the basis of the code. We find that for a target channel fidelity of 99.9%99.9\%99.9% over a channel with 1 dB1\text{ dB}1 dB of loss, a third-order cat code requires 707070 times fewer telecorrection iterations than a first-order one, at a cost of a 3.63.63.6-fold increase in mean photon-number.