Repetition Cat Codes for Quantum Error Correction

• Repetition cat codes are quantum error-correcting codes that encode logical qubits using superpositions of coherent states arranged in phase space, balancing excitation loss and dephasing errors.
• They optimize code parameters like the coherent amplitude and logical subspace index to mitigate bit-flip and dephasing errors, achieving robust error suppression in lossy bosonic channels.
• Their design underpins scalable quantum memories and efficient quantum repeater protocols, demonstrating hardware-friendly performance in superconducting and photonic platforms.

Repetition cat codes are quantum error-correcting codes based on superpositions of coherent states (“cats”) in bosonic modes, optimized for suppressing both excitation loss and dephasing errors in bosonic systems. The concept was introduced in the context of lossy bosonic channels, with particular emphasis on balancing two major classes of logical errors—bit-flip errors from excessive excitation loss and dephasing errors from environmental back-action—by appropriate choice of code parameters. These codes form the foundation for efficient, hardware-friendly quantum memories and high-rate quantum repeater protocols.

1. Construction of Repetition Cat Codes: Logical Subspace and Amplitude Optimization

Repetition cat codes encode a logical qubit in a single bosonic mode using superpositions of $2d$ coherent states, symmetrically located on a circle in phase space. The codespace is constructed via orthonormal “cat states”: $$|C_\alpha^{n}\rangle = \frac{1}{\sqrt{2d\,\mathcal{N}_n(\alpha)}} \sum_{k=0}^{2d-1} \omega^{-kn} |\omega^k \alpha\rangle, \quad \omega = e^{i\pi/d},$$

$$\mathcal{N}_n(\alpha) = \sum_{k=0}^{2d - 1} \omega^{-k n} e^{(\omega^k - 1)\alpha^2}.$$

The logical subspace is chosen as a pair of $s$-labeled states: $$|0\rangle_L^s = |C_\alpha^s\rangle, \qquad |1\rangle_L^s = |C_\alpha^{s+d}\rangle,$$ with $s = 0, 1, \ldots, d-1$.

A central optimization is the selection of the coherent amplitude $\alpha$ and subspace index $s$ such that the mean photon number is equalized between logical states: $$\langle C_\alpha^s| a^\dagger a | C_\alpha^s\rangle = \langle C_\alpha^{s+d}| a^\dagger a | C_\alpha^{s+d}\rangle.$$ Matching not only suppresses environment-accessible “which-state” information but also cancels leading order contributions to dephasing. This parameterization typically reduces dephasing error terms in the effective diamond norm to $$\mathcal{O}\left[\left(\Delta \frac{\pi^2}{d^2}\right)^2\right]$$, an order lower in $\Delta = \gamma \alpha^2$ (excitation loss).

2. Logical Error Mechanisms and Their Trade-off

Two major types of logical errors are addressed:

• Bit-flip error ($X$ error): Occurs when the number of lost excitations exceeds the code’s correction capacity ($k \ge d$). Explicitly, for loss indices $k$ satisfying $(2m+1)d \le k \le 2(m+1)d-1$, a logical $X$ operation is induced. The bit-flip error probability is:

$\epsilon_f = \sum_{k=0}^{d-1} T_{k+d}$

where $T_{k+d}$ is the probability to lose enough excitations to cross the code’s correctable threshold.

• Dephasing error ($Z$ error): Generated by differences in higher Fock-space moments of the logical states and quantum back-action, producing decoherence. Leading-order dephasing error is

$$\epsilon_d \approx \frac{1}{2} e^{-4\alpha^2 \sin^2 \frac{\pi}{2d} (\cdots)}$$

with terms sensitive to $\Delta,\,s,\,\alpha$.

The crucial trade-off is that a larger $\alpha$ (more separated coherent states) exponentially suppresses dephasing but increases the probability of uncorrectable excitation loss. The global minimization of logical error (total decoherence rate)

$\Gamma \approx \epsilon_f + \epsilon_d$

is achieved by tuning $\alpha$ and $s$ together. For $d > 2$ and in the small $\Delta$ regime, the optimal value of $\alpha^2$ satisfies: $$\alpha_o^2 \approx \frac{W\left[ \frac{4 \sin^2(\pi/2d) - \gamma}{(d-2)\gamma} \left(\frac{d!}{2}\pi^4 d^4\right)^{\frac{1}{d-2}} \right]}{\frac{4 \sin^2(\pi/2d) - \gamma}{d-2}}$$ with $W$ the Lambert W function.

3. Decoherence Suppression and Recovery Operations

Losses on the bosonic channel are represented using Kraus operators

$$\mathcal{W}_k = \frac{1}{\sqrt{k!}} \gamma^{k/2} (1-\gamma)^{a^\dagger a/2} a^k,$$

where each jump $k$ both damps amplitude and introduces dephasing. The designed recovery operation $\mathcal{R}$ includes:

• A $\mathbb{Z}_d$ measurement (detecting in which coset modulo $d$ the loss landed),
• A conditional photon-addition unitary $U_k$,
• Amplitude restoration $S_{\alpha'}$ (restoring $\alpha$ after loss).

By orchestrating these steps and matching the mean photon number between logical states, the code suppresses back-action-induced decoherence from $\mathcal{O}(\Delta\,\pi^2/d^2)$ to $\mathcal{O}[(\Delta\,\pi^2/d^2)^2]$ in the effective Pauli channel. This approach has experimental support from circuit QED demonstrations surpassing the “break-even” threshold for bosonic QEC.

4. Application in Quantum Repeaters and Communication Rate Enhancement

Repetition cat codes offer high mode efficiency in one-way quantum rewiring architectures, as each bosonic mode supports a complete logical qubit. In quantum repeaters, the total channel is modeled as

$\mathcal{E}^N = (\mathcal{R} \circ \mathcal{L})^N,$

with $N$ repeater stations, and error correction performed after each segment. The secure key rate per mode (SKRPM) in quantum key distribution scenarios is given by

$\mathrm{Rate} = 1 - H(Q_Z) - H(Q_X),$

where $H(p)$ is the binary entropy function and $Q_Z/Q_X$ are logical $Z$ and $X$ error rates after all correction. For high coupling efficiency $\eta \approx 99.5\%$, the secure key rate per mode with cat codes is shown to be significantly higher than with conventional discrete-variable schemes, including quantum parity and polynomial codes—a consequence of mode efficiency and optimized code design.

5. Broader Connections and Applications

The architecture and optimization methods for repetition cat codes naturally generalize to other continuous-variable codes, such as Gottesman–Kitaev–Preskill (GKP) and binomial codes. Optimization of the logical subspace and amplitude is a general principle for CV code design aiming to reduce extractable syndrome information and limit phase decoherence. This positions repetition cat codes as a leading candidate for:

• High-density, long-lifetime quantum memories in superconducting cavities,
• Efficient quantum repeaters and long-distance QKD,
• Ancillary modules in universal CV quantum computation and distributed CV QEC networks.

6. Summary and Significance

Repetition cat codes achieve optimal suppression of both excitation loss and dephasing by tuning manifold parameters to balance logical $X$ and $Z$ error rates. The code structure, syndrome measurement, and recovery sequence are engineered to minimize decoherence in lossy bosonic channels. This tailored approach allows for high secure communication rates per mode and hardware-efficient error correction, forming a robust foundation for scalable quantum communication and memory using continuous-variable encodings.

The integrated strategies outlined—viewing cat codes as repetition codes with optimized subspace and recovery—enable a practical avenue toward high-coherence quantum networks and quantum memories in contemporary superconducting and photonic platforms (Li et al., 2016).