Cat Qubit (Cat Codes)

Figure

Description

Cat qubits encode quantum information in superpositions of coherent states of a harmonic oscillator (typically a superconducting microwave cavity). The logical states are:

$|0_L⟩={\mathcal{N}}_{+}(|\alpha ⟩+|-\alpha ⟩),|1_L⟩={\mathcal{N}}_{-}(|\alpha ⟩-|-\alpha ⟩)$

where $|\alpha ⟩$ is a coherent state with mean photon number $\bar{n}=|\alpha {|}^2$, and ${\mathcal{N}}_{\pm }$ are normalization constants.

The key insight is that these states have exponentially different parity: $|0_L⟩$ lives in the even-photon Fock subspace and $|1_L⟩$ in the odd-photon subspace. Single-photon loss (the dominant error) maps $|0_L⟩\leftrightarrow |1_L⟩$ — a detectable bit flip. But phase-flip errors (rotations in the coherent-state basis) are exponentially suppressed with increasing $|\alpha {|}^2$, because the overlap $⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$ vanishes exponentially.

Modern implementations use two-photon driven-dissipative stabilization: a nonlinear element (Josephson junction or SNAIL) drives two-photon exchange between the cavity and a lossy buffer, confining the cavity state to the manifold spanned by $|\alpha ⟩$ and $|-\alpha ⟩$. This makes the cat qubit autonomous — it continuously corrects itself against single-photon loss while maintaining exponential bit-flip protection.

Alice & Bob (Paris) is developing cat qubits as their primary architecture, demonstrating bit-flip times exceeding 10 seconds with $|\alpha {|}^2\approx 16$.

Hamiltonian

The driven-dissipative cat qubit is stabilized by the effective two-photon process:

$H_{\text{drive}}=g_2(a^{†2}-{\alpha }^{\ast 2})(a^2-{\alpha }^2)$

with engineered two-photon dissipation:

$\mathcal{L}[\rho ]={\kappa }_2\mathcal{D}[a^2-{\alpha }^2]\rho$

where ${\kappa }_2$ is the two-photon loss rate. The steady states of this dissipator are exactly $|\pm \alpha ⟩$.

The cat qubit error rates scale as:

• Bit-flip rate: ${\Gamma }_{\text{bf}}\propto {\kappa }_1\bar{n}$ (linear in photon loss rate)
• Phase-flip rate: ${\Gamma }_{\text{pf}}\propto e^{-2\bar{n}}$ (exponentially suppressed)

Motivation

Standard bosonic codes (like GKP) require complex active error correction. The cat qubit achieves exponential suppression of one error type (phase flips) autonomously through hardware design, reducing the problem to correcting only bit flips. Combined with a repetition code for the remaining bit-flip errors, this creates an asymmetric error correction scheme with favorable resource scaling.

Experimental Status

First demonstration of exponential bit-flip suppression — Lescanne et al. (2020):

• Demonstrated exponential suppression of phase-flip errors with increasing $\bar{n}$
• Used two-photon driven-dissipative stabilization in a superconducting cavity
• Achieved bit-flip times scaling with ${\kappa }_2/{\kappa }_1$ ratio

Bias-preserving CNOT gate — Guillaud and Mirrahimi (2019):

• Proposed and analyzed CNOT gate between cat qubits that maintains the exponential error bias
• Gate fidelity ~99% while preserving the noise asymmetry
• Compatible with repetition code for full fault tolerance

Alice & Bob scaling results (2024):

• Bit-flip times exceeding 10 s achieved with $|\alpha {|}^2\approx 16$
• Demonstrated the practical viability of the cat qubit architecture for error-biased quantum computing

Key Metrics

Metric	Value	Notes	Fidelity reference
Bit-flip time	0.1–10+ s	Scales with $\bar{n}$ and ${\kappa }_2/{\kappa }_1$	Lescanne et al. 2020
Phase-flip time	1–100 μs	Scales as $e^{-2\bar{n}}/{\kappa }_{\varphi }$	Lescanne et al. 2020
Mean photon number $\bar{n}$	4–16	Typical operating regime	Lescanne et al. 2020
${\kappa }_2/{\kappa }_1$	10–100	Two-photon vs single-photon loss ratio	Lescanne et al. 2020
CNOT gate fidelity	~99%	Bias-preserving gate	Guillaud and Mirrahimi 2019
Operating temperature	10–20 mK	Dilution refrigerator	—

References

Original proposal

• M. Mirrahimi et al., “Dynamically protected cat-qubits: a new paradigm for universal quantum computation,” New J. Phys. 16, 045014 (2014) — arXiv:1312.2017

Experimental demonstrations

• R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator,” Nat. Phys. 16, 509 (2020) — arXiv:1907.11729

Related theory

• J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation,” Phys. Rev. X 9, 041053 (2019) — arXiv:1904.09474

Linked Papers

• mirrahimi-2014-cat-codes

Related Entries

• gkp-codes — alternative bosonic code with grid-state encoding
• kerr-cat-qubit — related cat qubit variant using Kerr nonlinearity
• circuit-qed — underlying hardware platform
• transmon — ancilla qubit used in cat qubit control
• binomial-codes — another bosonic code family
• bosonic-qubit — parent category entry for bosonic encodings