Kerr-Cat Qubit

Figure

Description

The Kerr-cat qubit is a superconducting bosonic qubit formed in a Kerr-nonlinear oscillator driven near twice its resonance frequency. In the usual logical convention, the computational states are the two opposite coherent states

$|0_L⟩\simeq |\alpha ⟩,|1_L⟩\simeq |-\alpha ⟩.$

The exactly orthogonal even- and odd-parity cat states

$|{\mathcal{C}}_{\alpha }^{\pm }⟩\propto |\alpha ⟩\pm |-\alpha ⟩$

are the complementary $X_L$-basis states. Keeping these two bases distinct is essential: tunnelling between the two phase-space wells produces logical bit flips, whereas single-photon loss acts approximately as $\hat{a}|\pm \alpha ⟩=\pm \alpha |\pm \alpha ⟩$ and therefore produces the dominant logical phase flips.

The phase-space separation of the wells grows with the mean photon number $\bar{n}=|\alpha {|}^2$. Their overlap is $⟨-\alpha |\alpha ⟩=e^{-2|\alpha {|}^2}$, which underlies the exponential suppression of bit flips in the ideal large-cat regime. Phase-flip errors from photon loss instead grow roughly linearly with $\bar{n}$, producing a strongly biased but not error-free qubit.

Hamiltonian

A representative rotating-frame Hamiltonian is

$\frac{H}{\hslash }=\Delta {\hat{a}}^{†}\hat{a}-K{\hat{a}}^{†2}{\hat{a}}^2+{\epsilon }_2{\hat{a}}^{†2}+{\epsilon }_2^{\ast }{\hat{a}}^2,$

where $K>0$ is the self-Kerr strength, ${\epsilon }_2$ is the complex two-photon-drive amplitude, and $\Delta$ is the detuning between the oscillator and half the pump frequency. For the resonant case $\Delta =0$, choose the phase convention in which ${\epsilon }_2>0$ and define $\alpha =\sqrt{{\epsilon }_2/K}$. The Hamiltonian can then be factorized as

$\frac{H}{\hslash }=-K({\hat{a}}^{†2}-{\alpha }^2)({\hat{a}}^2-{\alpha }^2)+K|\alpha {|}^4.$

Thus $|\pm \alpha ⟩$, and equivalently their even/odd superpositions $|{\mathcal{C}}_{\alpha }^{\pm }⟩$, span an exactly degenerate eigenspace of the ideal model. They should not be described as ordinary ground states of an unbounded rotating-frame quartic Hamiltonian. In the large-cat limit, the separation to the first excited doublet is approximately

${\Delta }_{\mathrm{gap}}/\hslash \simeq 4K|\alpha {|}^2.$

Finite detuning is not merely a nuisance: the detuned Kerr-cat demonstrated in 2026 increases the coherent-state lifetime and enables a directly benchmarked control toolbox.

Motivation

Hamiltonian confinement gives the Kerr-cat a native error asymmetry without repeated syndrome extraction at the physical-qubit level. It does not autonomously correct photon loss: the dominant phase errors still require an outer code. If control and readout preserve the bias, repetition, XZZX, or other asymmetric-noise codes can exploit the rare bit-flip channel and lower fault-tolerance overhead.

The engineering challenge is to retain that bias during gates. Strong pumps can heat auxiliary circuit modes, leakage can leave the protected manifold, and an idling-lifetime ratio can substantially overestimate the bias of an actual gate. Modern experiments therefore benchmark operations directly rather than inferring gate quality from $T_{\mathrm{bit}}/T_{\mathrm{phase}}$ alone.

Experimental Status

Hamiltonian proposal — Puri, Boutin, and Blais (2017):

• Proposed preparing and controlling cat states in a two-photon-driven Kerr-nonlinear resonator and identified the degenerate encoded manifold.

First complete Kerr-cat demonstration — Grimm et al. (2020):

• Used a SNAIL-based nonlinear resonator to demonstrate stabilization, universal single-qubit control, and single-shot cat-quadrature readout.
• Reported process fidelities of $85.5\%$ for identity, $85.7\%$ for $X(\pi /2)$, and $81.1\%$ for $Z(\pi /2)$; these are historical benchmarks, not current records.

Scalable planar architecture — Hajr et al. (2024):

• Added an on-chip band-block filter in a two-dimensional circuit, obtained $99.6\%$ QND readout fidelity at $|\alpha {|}^2=8$, and measured coherent-state lifetimes near $0.95\mathrm{ms}$ at that cat size.

Oscillator control — Ding et al. (2025):

• Demonstrated a parametrically driven conditional-displacement interaction between a Kerr-cat and a high-Q cavity, enabling universal cavity control while frequency-selective dissipation suppressed heating-induced cavity dephasing.

Direct operation benchmarking — Qing et al. (2026):

• Demonstrated a detuned Kerr-cat in a planar circuit with initialization, universal one-qubit control, and QND readout.
• Measured a $1.2\mathrm{ms}$ bit-flip time and $4\mu \mathrm{s}$ phase-flip time, plus SPAM-free fidelities of $99.18(7)\%$ for the bias-preserving $Z(\pi /2)$ gate and $92.5(2)\%$ for $X(\pi /2)$.
• Direct dihedral randomized benchmarking found operational noise bias above $100$ and up to about $250$, showing why lifetime ratios alone are insufficient.

Bias-preserving Kerr-cat–Kerr-cat entangling gates remain an active development target; the widely cited CNOT and Toffoli constructions in Puri et al. (2020) are proposals, not experimental demonstrations.

Key Metrics

Metric	Value	Notes	Fidelity reference
QND readout fidelity	$99.6\%$	Planar resonant Kerr-cat, $|\alpha {|}^2=8$	Hajr et al. 2024
Coherent-state (bit-flip) lifetime	$\sim 0.95\mathrm{ms}$	Resonant planar device near $|\alpha {|}^2=8$	Hajr et al. 2024
Detuned coherent-state lifetime	$1.2\mathrm{ms}$	About $3\times$ the resonant value in the same 2026 study	Qing et al. 2026
Detuned phase-flip time	$4\mu \mathrm{s}$	Dominant error channel	Qing et al. 2026
$Z(\pi /2)$ process fidelity	$99.18(7)\%$	SPAM-free gate-set tomography; bias-preserving operation	Qing et al. 2026
$X(\pi /2)$ process fidelity	$92.5(2)\%$	SPAM-free gate-set tomography; tunnelling operation	Qing et al. 2026
Directly benchmarked operation bias	$>100$, up to $\sim 250$	Dihedral randomized benchmarking, not a lifetime-ratio estimate	Qing et al. 2026
Kerr strength $K/2\pi$	$0.93(3)\mathrm{MHz}$	Kerr-cat ancilla used for conditional cavity control	Ding et al. 2025

References

Hamiltonian proposal

• S. Puri, S. Boutin, and A. Blais, “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving,” npj Quantum Information 3, 18 (2017) — arXiv:1605.09408

First demonstration

• A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit,” Nature 584, 205–209 (2020) — arXiv:1907.12131

Bias-preserving gate theory

• S. Puri et al., “Bias-preserving gates with stabilized cat qubits,” Science Advances 6, eaay5901 (2020) — arXiv:1905.00450
• J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation,” Physical Review X 9, 041053 (2019) — arXiv:1904.09474

Modern experiments

• A. Hajr et al., “High-Coherence Kerr-Cat Qubit in 2D Architecture,” Physical Review X 14, 041049 (2024) — arXiv:2404.16697
• A. Z. Ding et al., “Quantum control of an oscillator with a Kerr-cat qubit,” Nature Communications 16, 5279 (2025) — arXiv:2407.10940
• B. Qing et al., “Quantum benchmarking of high-fidelity noise-biased operations on a detuned Kerr-cat qubit,” PNAS 123, e2520479123 (2026) — arXiv:2411.04442

Linked Papers

• puri-2017-engineering-states-light
• grimm-2020-kerr-cat
• puri-2020-bias-preserving-gates
• guillaud-2019-repetition-cat-qubits
• hajr-2024-high-coherence-kerr-cat
• ding-2025-kerr-cat-oscillator-control
• qing-2026-detuned-kerr-cat-benchmarking

Evergreen context

• bosonic-code-hierarchy — where the Kerr-cat sits among cat, GKP, and binomial bosonic encodings
• josephson-junction-as-nonlinear-element — how the SNAIL / junction nonlinearity implements the driven Kerr oscillator
• noise-bias-and-asymmetric-error-channels — why exponentially rare bit flips change the error-correction strategy
• erasure-error-vs-pauli-error — contrast with architectures that lower overhead by flagging loss instead of biasing Pauli noise

Related Entries

• cat-codes — umbrella entry spanning Hamiltonian and dissipative cat stabilization
• bosonic-qubit — parent oscillator-encoding family
• transmon — common ancilla and circuit ingredient, but a distinct encoding
• gkp-codes — alternative bosonic code with shift-error correction
• erasure-qubit — complementary route to lower-overhead fault tolerance