Rotation-Symmetric Quantum Codes

• Rotation-symmetric codes are quantum error-correcting codes defined in bosonic modes that use discrete phase-space rotation symmetry to detect and correct photon loss and dephasing.
• They incorporate explicit constructions like cat and binomial codes, employing symmetrization in the Fock-space to optimize error correction with defined trade-offs in error-model parameters.
• Recent advances extend these codes to multimode and random ensembles, integrating symmetry expansion, fault-tolerant protocols, and efficient state preparation for scalable quantum communication.

Rotation-symmetric codes are a class of quantum error-correcting codes defined in the infinite-dimensional Hilbert space of a bosonic mode, whose structure is determined by invariance under discrete phase-space rotation. The underlying symmetry enables robust error detection and correction against dominant noise channels, particularly photon loss and dephasing, in both circuit QED and all-optical platforms. Notable representatives include cat codes and binomial codes, while recent developments extend rotation symmetry to multi-mode encodings and random code ensembles. The defining mathematical, operational, and error-correction properties of these codes have been systematically elucidated, revealing precise trade-offs, optimal error correction, and powerful error mitigation strategies based on symmetry expansion.

1. Formal Structure of Rotation-Symmetric Bosonic Codes

Let $\hat{a}$ and $\hat{a}^\dag$ denote bosonic annihilation and creation operators, and $\hat{n} = \hat{a}^\dag \hat{a}$ the number operator. The $N$-fold discrete phase rotation is generated by

$R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$

A state $|\psi\rangle$ is $N$-fold rotation-symmetric if $R_N |\psi\rangle = |\psi\rangle$. The codespace $\mathcal{C}_N$ is defined as the $+1$ eigenspace of $\hat{a}^\dag$0, isolated by the projector

$\hat{a}^\dag$1

which projects onto Fock states $\hat{a}^\dag$2 with $\hat{a}^\dag$3.

Logical basis states are constructed by selecting a primitive state $\hat{a}^\dag$4 and applying symmetrization: \begin{align*} |0_N, \Phi\rangle &\propto \Pi_N |\Phi\rangle, \ |1_N, \Phi\rangle &\propto R\left(\frac{\pi}{N}\right) \Pi_N |\Phi\rangle. \end{align*} In Fock basis, the support of $\hat{a}^\dag$5 is on $\hat{a}^\dag$6; $\hat{a}^\dag$7 is on $\hat{a}^\dag$8.

Prominent specializations include:

• Cat codes: $\hat{a}^\dag$9 (coherent state); codewords are superpositions of $\hat{n} = \hat{a}^\dag \hat{a}$0.
• Binomial codes: $\hat{n} = \hat{a}^\dag \hat{a}$1 a finite Fock-space superposition, with codewords \begin{align*} |0_N\rangle &= \sum_{m\;\mathrm{even}} \sqrt{C_m} |m N\rangle, \ |1_N\rangle &= \sum_{m\;\mathrm{odd}} \sqrt{C_m} |m N\rangle. \end{align*} Such codewords exactly inhabit the rotation-symmetric subspace (Endo et al., 2022, Marinoff et al., 2023, Grimsmo et al., 2019, Totey et al., 2023).

2. Error-Correction Properties and Code Distance

The essential error basis for rotation-symmetric codes consists of shift–rotation operators: $\hat{n} = \hat{a}^\dag \hat{a}$2 where $\hat{n} = \hat{a}^\dag \hat{a}$3 ($\hat{n} = \hat{a}^\dag \hat{a}$4-step generalization as $\hat{n} = \hat{a}^\dag \hat{a}$5).

The Knill–Laflamme conditions for these codes yield the code distance vector

$\hat{n} = \hat{a}^\dag \hat{a}$6

where $\hat{n} = \hat{a}^\dag \hat{a}$7 is the maximum number of detectable photon-loss events, and $\hat{n} = \hat{a}^\dag \hat{a}$8 quantifies the minimal distinguishable phase (rotation) error (Marinoff et al., 2023). This sets a strict trade-off: increasing $\hat{n} = \hat{a}^\dag \hat{a}$9 enhances loss protection but reduces phase error resilience.

Explicit error correction proceeds by measuring the stabilizers ($N$0 for number-shifts, $N$1 for modular rotation), extracting the syndromes, and applying a recovery operator constructed from the error basis. Teleportation-based protocols using dual-basis ancillae and controlled-rotations provide software-trackable recovery, optimal for continuous-variable architectures (Grimsmo et al., 2019).

3. Error Models, Performance, and Noise Robustness

Rotation-symmetric codes exhibit distinct responses to photon loss, Gaussian and non-Gaussian dephasing, as well as non-Markovian environments:

• Photon loss: Loss of $N$2 photons maps codewords to mutually orthogonal error spaces, so single-photon loss can be detected and corrected. In the presence of higher-order loss, logical errors occur when $N$3 (Grimsmo et al., 2019, Marinoff et al., 2023).
• Dephasing: Symmetry ensures phase errors of magnitude less than $N$4 can be detected.
• Random telegraph noise (RTN): Non-Markovian dephasing arising from bistable fluctuators leads to oscillatory fidelity dynamics. The Breuer–Laine–Piilo measure of non-Markovianity grows linearly with $N$5. Higher $N$6 induces faster dephasing-revival oscillations, presenting both opportunities and timing challenges for optimal QEC (Udupa et al., 13 May 2025).

Numerical benchmarks across loss and dephasing rates confirm that for moderate-to-large loss with low dephasing, rotation-symmetric codes (including random code variants) surpass cat and binomial codes, while structured codes dominate at high dephasing (Totey et al., 2023).

The interplay is summarized by $N$7, enforcing an intrinsic trade-off and guiding code parameter selection (Marinoff et al., 2023, Totey et al., 2023).

4. Code Construction: Cat, Binomial, Random, and Multimode Generalizations

Cat and binomial codes are concrete families extensively analyzed in circuit QED and optics. Their explicit Fock-space constructions ensure number-parity and phase-symmetry protection (Endo et al., 2022, Nehra et al., 2021).

Random rotation codes extend this landscape by sampling Fock-space amplitudes according to Haar measure. "One-primitive" and "two-primitive" ensembles allow systematic exploration of the code space bounded by a number cutoff. The best two-primitive random codes outperform fixed cat and binomial codes in high-loss, low-dephasing regimes, owing to effective Fock-weight variational optimization (Totey et al., 2023).

Multimode extensions employ group-theoretic frameworks to encode qudits in $N$8 bosonic modes, implementing Pauli operators via linear optics. Two-mode binomial codes constructed in this formalism exhibit simultaneous protection against loss and correlated dephasing, without the trade-offs present in single-mode codes. This facilitates exact correction of dephasing-induced errors and allows qudit encoding (Ahmed et al., 28 Aug 2025).

5. Practical State Preparation, Error Mitigation, and Fault-Tolerant Protocols

State preparation: All-optical engineering protocols generate rotation-symmetric code states by coherent photon subtraction from two-mode squeezed vacua, followed by photon-number-resolving (PNR) detection. Heralding rates at $N$9–$R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$0 per shot with established squeezing levels ($R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$1–$R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$2 dB) confirm near-term feasibility. Detector inefficiency and loss can be managed so that state-preparation infidelity remains $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$3, compatible with fault-tolerance (Nehra et al., 2021).

Error mitigation: Symmetry expansion (SE) leverages intrinsic code-group symmetry to realize virtual projections onto the codespace during both preparation and measurement. The process, implementable with an ancilla and two controlled-rotation gates, suppresses leading-order photon-loss errors exactly up to order $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$4. Sampling cost scales as $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$5 for photon loss $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$6, remaining modest for near-term devices (Endo et al., 2022). SE is compatible with dispersive, circuit-QED, and trapped ion platforms.

Fault tolerance: Advanced protocols concatenate rotation-symmetric codes with repetition and Bacon-Shor subsystem codes. Fault-tolerance thresholds of $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$7–$R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$8 per mode are observed, with syndrome measurements delegated to CROT gates and minimal hardware overhead. Destructive measurements and non-Gaussian ancillae are integral in this architecture (Grimsmo et al., 2019).

6. Applications and Performance in Communication and Computation

Rotation-symmetric codes underpin high-performance quantum communication, with particular utility in quantum repeaters for long-distance key distribution. Comparative studies reveal:

• Binomial codes outclass cat codes and squeezed-cat codes in secret-key rate, success probability, and resource overhead.
• GKP-like rotation codes achieve further improvement by combining translation and rotation symmetry, at cost of increased complexity.
• Resource scaling: Binomial codes enable up to $R_N = \exp\left(i\frac{2\pi}{N}\hat{n}\right).$9 km spacing between repeater stations for target SKRs, with optimized mean photon number $|\psi\rangle$0, and clear advantages in cost per kilometer in realistic models (Li et al., 2023).

Design guidelines emphasize matching rotation order $|\psi\rangle$1 to the dominant error (larger $|\psi\rangle$2 for loss, smaller $|\psi\rangle$3 for dephasing), optimizing $|\psi\rangle$4, and using moderate squeezing. Random and multimode codes provide additional flexibility for hardware and noise constraints (Totey et al., 2023, Ahmed et al., 28 Aug 2025).

7. Theoretical Developments and Future Directions

Recent advances include explicit error propagation analyses using rotation-adapted error bases, analytical demonstration of code distances $|\psi\rangle$5, and the construction of group-covariant multimode codes with arbitrary-dimension logical encoding (Marinoff et al., 2023, Ahmed et al., 28 Aug 2025). Research continues into resource-efficient state preparation, advanced error mitigation strategies beyond symmetry expansion, and integration into measurement-based computation and photonic network architectures (Endo et al., 2022, Nehra et al., 2021).

The ubiquity and hardware efficiency of rotation-symmetric codes highlight their promise for fault-tolerant, scalable quantum processing and communication under realistic noise and operational constraints.