Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2311.13670 · Physical Review A

Explicit error-correction scheme and code distance for bosonic codes with rotational symmetry

Benjamin Marinoff, Miles Bush, Joshua Combes

Published Mar 27, 2024• 5 citations
arXivHTMLPDF

Abstract

Bosonic codes with rotational symmetry are currently one of the best-performing quantum error-correcting codes. Little is known about error propagation and code distance for these rotation codes in contrast with qubit codes and bosonic codes with translation symmetry. We use a general purpose error basis that is naturally suited to codes with rotation symmetry to compute how errors propagate through gates. This error basis allows us to give an explicit error detection, decoding, and correction scheme for any code with rotation symmetry. We also prove that codes with an $N$-fold rotation symmetry have a distance of $({d}_{n},{d}_{\ensuremath{\theta}})=(N,\ensuremath{\pi}/N)$ with respect to number and rotation errors.

DOI
10.1103/physreva.109.032436

Related papers

Quantum Computing With Rotation-Symmetric Bosonic Codes

Arne L. Grimsmo, Joshua Combes, Ben Q. Baragiola

Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error-correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on the concatenation of number-phase codes and Bacon-Shor subsystem codes.

•Nov 2, 2024•120 citations

Bosonic Cyclic Codes: Trading Stabilizers for Gaussian Non-Clifford Phase Gates

Owen C. Wetherbee, Yijia Xu, Victor V. Albert +2 more

Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger $SU(2)$ symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.

quant-pharXiv:2606.11010•Jun 10, 2026•0 citations

Quantum error mitigation for rotation-symmetric bosonic codes with symmetry expansion

Suguru Endo, Yasunari Suzuki, Kento Tsubouchi +4 more

The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat and binomial codes achieve the breakeven point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, the state preparation fidelity needs to be improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and postprocessing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. Then, we consider two types of stabilization: photon number and phase stabilization. Photon number stabilization can be performed by leveraging the rotation operators. On the other hand, the phase errors can be suppressed via the extension towards phase direction by leveraging the recently proposed projective squeezing method. To implement our method, we use an ancilla qubit and a randomly generated controlled gate between the bosonic code states and the ancilla qubit. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing.

arXiv:2211.06164•Physical Review A•Jun 2, 2025•7 citations

The performance of random bosonic rotation codes

Saurabh Totey, Akira Kyle, Liu, Steven +3 more

Bosonic error correcting codes utilize the infinite dimensional Hilbert space of a harmonic oscillator to encode a qubit. Bosonic rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes.We define two different notions of random bosonic rotation codes and numerically explore their performance against loss and dephasing. We find that the best random rotation codes can outperform cat and binomial codes in a certain parameter regime where loss is large and dephasing errors are small.

arXiv:2311.16089•Nov 27, 2023•0 citations

Generalized number-phase lattice encoding of a bosonic mode for quantum error correction

Dong-Long Hu, Weizhou Cai, Chang‐Ling Zou +1 more

Bosonic systems offer unique advantages for quantum error correction, as a single bosonic mode provides a large Hilbert space to redundantly encode quantum information. However, previous studies have been limited to exploiting symmetries in the quadrature phase space. Here we introduce a unified framework for encoding a qubit utilizing the symmetries in the phase space of number and phase variables of a bosonic mode. The logical codewords form lattice structures in the number-phase space, resulting in rectangular, oblique, and diamond-shaped lattice codes. Notably, oblique and diamond codes exhibit a number-phase vortex effect, where number-shift errors induce discrete phase rotations as syndromes, enabling efficient correction via phase measurements. These codes show significant performance advantages over conventional quadrature codes against dephasing noise in the potential one-way quantum communication applications. Our generalized number-phase codes open up new possibilities for fault-tolerant quantum computation and extending the quantum communication range with bosonic systems. Current bosonic quantum error correction codes exploit displacement or rotational symmetries in the quadrature phase space. Here, the authors generalise the concept by looking at potential symmetries and encoding in a broader number-phase space, where the known cat and binomial codes would correspond to rectangular lattices and be completed by other lattice codes like oblique and diamond.

arXiv:2508.12354•Nature Communications•Aug 16, 2025•4 citations

Quantum Error Correction beyond S U ( 2 ) : Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry

Arda Aydin, Victor V. Albert, Alexander Barg

We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group S U ( q ) , we prove that many codes and their gates in S U ( q ) can be interconverted between the three state spaces. We construct code instances for all three spaces using classical ℓ 1 codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length N by constructing ℓ 1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

arXiv:2509.20545•PRX Quantum•Jan 14, 2026•1 citation
Bosonic codes with rotational symmetry are currently one of the best-performing quantum error-correcting codes. Little is known about error propagation and code distance for these rotation codes in contrast with qubit codes and bosonic codes with translation symmetry. We use a general purpose error basis that is naturally suited to codes with rotation symmetry to compute how errors propagate through gates. This error basis allows us to give an explicit error detection, decoding, and correction scheme for any code with rotation symmetry. We also prove that codes with an NNN-fold rotation symmetry have a distance of (dn,d\ensuremathθ)=(N,\ensuremathπ/N)({d}_{n},{d}_{\ensuremath{\theta}})=(N,\ensuremath{\pi}/N)(dn​,d\ensuremathθ​)=(N,\ensuremathπ/N) with respect to number and rotation errors.
Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error-correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on the concatenation of number-phase codes and Bacon-Shor subsystem codes.
Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger SU(2)SU(2)SU(2) symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.
The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat and binomial codes achieve the breakeven point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, the state preparation fidelity needs to be improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and postprocessing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. Then, we consider two types of stabilization: photon number and phase stabilization. Photon number stabilization can be performed by leveraging the rotation operators. On the other hand, the phase errors can be suppressed via the extension towards phase direction by leveraging the recently proposed projective squeezing method. To implement our method, we use an ancilla qubit and a randomly generated controlled gate between the bosonic code states and the ancilla qubit. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing.
Bosonic error correcting codes utilize the infinite dimensional Hilbert space of a harmonic oscillator to encode a qubit. Bosonic rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes.We define two different notions of random bosonic rotation codes and numerically explore their performance against loss and dephasing. We find that the best random rotation codes can outperform cat and binomial codes in a certain parameter regime where loss is large and dephasing errors are small.
Bosonic systems offer unique advantages for quantum error correction, as a single bosonic mode provides a large Hilbert space to redundantly encode quantum information. However, previous studies have been limited to exploiting symmetries in the quadrature phase space. Here we introduce a unified framework for encoding a qubit utilizing the symmetries in the phase space of number and phase variables of a bosonic mode. The logical codewords form lattice structures in the number-phase space, resulting in rectangular, oblique, and diamond-shaped lattice codes. Notably, oblique and diamond codes exhibit a number-phase vortex effect, where number-shift errors induce discrete phase rotations as syndromes, enabling efficient correction via phase measurements. These codes show significant performance advantages over conventional quadrature codes against dephasing noise in the potential one-way quantum communication applications. Our generalized number-phase codes open up new possibilities for fault-tolerant quantum computation and extending the quantum communication range with bosonic systems. Current bosonic quantum error correction codes exploit displacement or rotational symmetries in the quadrature phase space. Here, the authors generalise the concept by looking at potential symmetries and encoding in a broader number-phase space, where the known cat and binomial codes would correspond to rectangular lattices and be completed by other lattice codes like oblique and diamond.
We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group S U ( q ) , we prove that many codes and their gates in S U ( q ) can be interconverted between the three state spaces. We construct code instances for all three spaces using classical ℓ 1 codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length N by constructing ℓ 1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.