Kerr-Cat Qubit

Figure

Description

The Kerr-cat qubit is a superconducting bosonic qubit that encodes quantum information in two coherent-state superpositions $|\pm \alpha ⟩$ of a Kerr nonlinear oscillator stabilized by a resonant two-photon drive. The two computational states are the even and odd Schrödinger cat states $|{\mathcal{C}}_{\alpha }^{\pm }⟩\propto |\alpha ⟩\pm |-\alpha ⟩$, which are confined to a two-dimensional manifold within the oscillator’s infinite-dimensional Hilbert space by the interplay of the Kerr nonlinearity and the parametric drive.

The key feature of the Kerr-cat qubit is exponential noise bias: bit-flip errors (transitions between $|\alpha ⟩$ and $|-\alpha ⟩$) require tunneling through a phase-space barrier that grows with $|\alpha {|}^2$, and are therefore exponentially suppressed with increasing cat size. Phase-flip errors (relative phase between $|\alpha ⟩$ and $|-\alpha ⟩$) grow only linearly with $|\alpha {|}^2$. This extreme noise asymmetry can be exploited by tailored quantum error correction codes (e.g., repetition codes for phase flips only), dramatically reducing the overhead for fault-tolerant quantum computing.

Demonstrated by Grimm et al. (2020) at Yale using a SNAIL-based parametric oscillator, and further developed by the Amazon AWS Center for Quantum Computing in collaboration with Yale.

Hamiltonian

The Kerr-cat Hamiltonian in the rotating frame of the oscillator is:

$H=-K{\hat{a}}^{†2}{\hat{a}}^2+{\epsilon }_2({\hat{a}}^{†2}+{\hat{a}}^2)$

where $K$ is the Kerr nonlinearity (self-Kerr), ${\hat{a}}^{(†)}$ are the ladder operators, and ${\epsilon }_2$ is the two-photon drive amplitude.

The two classical stable states are coherent states $|\pm \alpha ⟩$ with amplitude $\alpha =\sqrt{{\epsilon }_2/K}$. The quantum ground states are the even/odd cat states:

$|{\mathcal{C}}_{\alpha }^{\pm }⟩=\frac{1}{\sqrt{2(1\pm e^{-2|\alpha {|}^2})}}(|\alpha ⟩\pm |-\alpha ⟩)$

The energy gap protecting against bit-flips scales as:

$\Delta E_{\text{gap}}\propto |\alpha {|}^{2}K$

and the bit-flip rate is suppressed as ${\Gamma }_{\text{bf}}\propto e^{-2|\alpha {|}^2}$.

Motivation

Standard bosonic codes (GKP, binomial) require complex syndrome extraction. The Kerr-cat qubit achieves autonomous error bias through its Hamiltonian structure alone: the two-photon drive continuously stabilizes the cat-state manifold without active feedback. By concentrating errors into a single channel (phase flips), the Kerr-cat enables a path to fault-tolerant quantum computing using simpler, lower-overhead error correction — potentially a repetition code rather than a full surface code. This could reduce the number of physical qubits needed for a logical qubit by an order of magnitude.

Experimental Status

First demonstration — Grimm et al. (2020):

• Generated and stabilized Schrödinger cat states using a SNAIL-based Kerr nonlinear resonator with single-mode squeezing.
• Demonstrated quantum non-demolition readout of the cat qubit state via dispersive coupling to an ancilla transmon.
• Showed >10× improvement in transverse relaxation time over Fock-state encoding.

Exponential bit-flip suppression — Lescanne et al. (2020):

• Bit-flip times exceeding $1\text{ms}$ demonstrated with noise bias ratios $>10^4$.

Bias-preserving gates — Puri et al. (2020):

• Demonstrated bias-preserving CNOT and Toffoli gates that maintain the noise asymmetry during gate operations.

Repetition code combination — Guillaud and Mirrahimi (2019):

• Showed theoretically that combining cat qubits with a repetition code achieves exponential suppression of both error types simultaneously.

Autoparametric stabilization — Réglade et al. (2024):

• Eliminated the need for an external two-photon pump via autoparametric design.

Key Metrics

Metric	Value	Notes	Fidelity reference
Bit-flip time	>1 ms	Exponentially suppressed with $\bar{n}$	Lescanne et al. 2020
Phase-flip time	1–10 μs	Grows linearly with $\bar{n}$; dominant error	Grimm et al. 2020
Noise bias ratio	$10^3$–$10^4$	${\Gamma }_{\text{phase}}/{\Gamma }_{\text{bit-flip}}$	Lescanne et al. 2020
Cat size $\bar{n}$	2–8 photons	Larger $\bar{n}$ increases bias but also phase-flip rate	Grimm et al. 2020
Kerr nonlinearity $K/2\pi$	1–10 MHz	Set by SNAIL or transmon nonlinearity	Grimm et al. 2020
1Q gate fidelity (Z-rotation)	99%+	Bias-preserving via detuning of drive	Puri et al. 2020
2Q gate fidelity (ZZ-CNOT)	98–99%	Bias-preserving CNOT	Puri et al. 2020
Operating temperature	10–20 mK	Dilution refrigerator	—

References

Original demonstration

• A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit,” Nature 584, 205 (2020) — arXiv:1907.12131

Bit-flip suppression

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

Bias-preserving gates

• S. Puri et al., “Bias-preserving gates with stabilized cat qubits,” Sci. Adv. 6, eaay5901 (2020)

Related theory

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

Linked Papers

• grimm-2020-kerr-cat
• puri-2020-bias-preserving-gates
• lescanne-2020-bit-flip-suppression

Related Entries

• cat-codes — general cat code framework underlying the Kerr-cat encoding
• transmon — provides the Kerr nonlinearity in some implementations
• gkp-codes — alternative bosonic code with different error correction strategy
• erasure-qubit — complementary approach to reducing QEC overhead via error conversion
• quarton-coupler — purely quartic coupler engineering strong nonlinear interactions