Kerr-Cat Qubit: Noise-Biased Bosonic Encoding

• Kerr-cat qubit is a bosonic qubit encoded in the low-energy manifold of a Kerr-nonlinear oscillator using a two-photon drive, forming even/odd cat states.
• It leverages a double-well phase-space structure with coherent states at ±α to exponentially suppress bit-flip errors while dephasing scales linearly.
• Engineered dissipation and precise pulse shaping enable robust initialization, gate operations, and high-fidelity readout, making it promising for scalable quantum computing.

A Kerr-cat qubit is a bosonic qubit encoded in the low-energy manifold of a Kerr-nonlinear oscillator subject to a two-photon drive. In phase space, the drive and nonlinearity generate a double-well structure with coherent-state lobes near $\pm \alpha$; in the quantum description, the corresponding even and odd Schrödinger-cat states form a protected or strongly noise-biased qubit manifold. The platform has developed from early demonstrations of stabilization, gates, and readout in superconducting resonators to a broader family of Hamiltonian, dissipative, and hybrid encodings, together with increasingly detailed analyses of leakage, nonadiabatic transport, and open-system critical phenomena (Grimm et al., 2019, Hajr et al., 2024, Wiggins, 29 Jun 2026).

1. Hamiltonian structure and logical encoding

A representative rotating-frame Hamiltonian for the Kerr-cat qubit is

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$

while detuned formulations include

$$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$

or

$$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$

In each case, the essential ingredients are the Kerr nonlinearity and the coherent two-photon drive, which together create a phase-space double well whose minima are centered at coherent states $|\pm \alpha\rangle$ with $\alpha^2=\epsilon_2/K$ or $\alpha_0=\sqrt{p_0/K}$, depending on convention (Grimm et al., 2019, Wiggins, 29 Jun 2026, Adinolfi et al., 2 Nov 2025).

The corresponding cat states are

$$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$

Some works identify the logical basis directly with the parity eigenstates $\ket{\mathcal C_\alpha^\pm}$, while others define logical $Z$ states as

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$0

This distinction is not merely notational: it determines whether the dominant physical processes are interpreted as parity flips inside the cat manifold or as inter-well transitions between coherent-state lobes (Adinolfi et al., 2 Nov 2025).

Several papers emphasize that the qubit manifold is separated from higher manifolds by a gap that grows with cat size. In the 2D superconducting realization, the double-well energy landscape is described with a gap

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$1

whereas shifted-Fock analyses of the ideal Kerr-cat Hamiltonian identify the first excited manifold at detuning $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$2 from the cat manifold. These formulations are technically different but converge on the same physical picture: the encoded qubit occupies a low-dimensional manifold stabilized by a combination of nonlinearity and parametric driving (Hajr et al., 2024, Putterman et al., 2021).

2. Noise bias, excited manifolds, and logical failure modes

The Kerr-cat qubit is valued for its strongly biased noise. In the 2D architecture, the leading-order error scalings are stated as

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$3

so enlarging $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$4 exponentially suppresses bit flips at the cost of only a linear increase in dephasing. In the coherent-state basis $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$5, bit-flip errors correspond to transitions between wells; in the parity basis $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$6, single-photon loss mixes parity and acts within or near the logical manifold (Hajr et al., 2024).

A central qualification is that the logical manifold is not perfectly closed. In the experimental study of controlled dissipation, single-photon loss at rate $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$7 induces “quantum heating” into higher manifolds $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$8, creating a leakage population $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$9. Once population reaches $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$0, the reduced barrier in that manifold permits fast inter-well tunneling at rate $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$1, so leakage and logical bit flips become directly linked. Experimentally, the undriven oscillator has leakage $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$2, while the driven KCQ at $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$3, $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$4 reaches steady $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$5, twelve times higher than $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$6 (Adinolfi et al., 2 Nov 2025).

Measured coherence metrics reflect this interplay between bias and leakage. In the 2D SNAILmon implementation, the bit-flip time rises from $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$7 at $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$8 to peaks $$H(t)=\omega_0\,a^\dagger a + K\,a^{\dagger2}a^2 + [\epsilon(t)\,a^{\dagger2}+\epsilon^*(t)\,a^2]$$9 at $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$0 and $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$1 at $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$2, exceeding $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$3 by $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$4. Over the same range, the phase-flip time decreases from $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$5 at $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$6 to $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$7 at $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$8, consistent with the stated linear dependence on $$H_{KCQ}/\hbar = \Delta\,\hat a^\dagger\hat a -K\,\hat a^{\dagger2}\hat a^2 +2\bigl(\hat a^2+\hat a^{\dagger2}\bigr).$$9 (Hajr et al., 2024).

Leakage-limited bias has also been quantified directly in dissipation-assisted experiments. Without cooling, $|\pm \alpha\rangle$0 at $|\pm \alpha\rangle$1, $|\pm \alpha\rangle$2; with cooling on resonance $|\pm \alpha\rangle$3, $|\pm \alpha\rangle$4 extends to $|\pm \alpha\rangle$5; and at the optimal working point $|\pm \alpha\rangle$6, $|\pm \alpha\rangle$7 reaches $|\pm \alpha\rangle$8, while $|\pm \alpha\rangle$9 remain unchanged, yielding a noise bias $\alpha^2=\epsilon_2/K$0 (Adinolfi et al., 2 Nov 2025).

3. Initialization, gate synthesis, and readout

Initialization is typically performed by adiabatically ramping on the two-photon pump so that a bare oscillator state is mapped into the cat manifold. In the 2019 experiment, the stabilization pulse sequence ramps on the two-photon pump $\alpha^2=\epsilon_2/K$1 over $\alpha^2=\epsilon_2/K$2 with a tanh-shaped envelope to map a Fock qubit into the cat manifold. A later experiment showed that this process is limited by the squeezing pump-induced frequency shift, modeled as $\alpha^2=\epsilon_2/K$3, and that dynamic compensation using

$\alpha^2=\epsilon_2/K$4

improves the initialization fidelity from $\alpha^2=\epsilon_2/K$5 to $\alpha^2=\epsilon_2/K$6, with a projected fidelity of $\alpha^2=\epsilon_2/K$7 after excluding state preparation and measurement errors (Grimm et al., 2019, Xu et al., 2024).

Single-qubit control has been realized in several ways. In the 3D cavity experiment, a continuous $\alpha^2=\epsilon_2/K$8 rotation is driven by an additional one-photon term $\alpha^2=\epsilon_2/K$9 at $\alpha_0=\sqrt{p_0/K}$0, with demonstrated $\alpha_0=\sqrt{p_0/K}$1 duration $\alpha_0=\sqrt{p_0/K}$2 and process fidelity $\alpha_0=\sqrt{p_0/K}$3. A discrete $\alpha_0=\sqrt{p_0/K}$4 gate is obtained by turning off the two-photon pump for $\alpha_0=\sqrt{p_0/K}$5, giving fidelity $\alpha_0=\sqrt{p_0/K}$6 (Grimm et al., 2019).

In the 2D architecture, the gate set was extended using fast Rabi oscillations and phase modulation of the stabilization drive. There, $\alpha_0=\sqrt{p_0/K}$7 is performed in $\alpha_0=\sqrt{p_0/K}$8 for $\alpha_0=\sqrt{p_0/K}$9, and a two-segment phase-modulation waveform produces $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$0 in $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$1 at $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$2. Process tomography at $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$3 gives $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$4 for $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$5 and $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$6 for $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$7. The same platform reports quantum non-demolition readout fidelity $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$8 for a cat with $$\ket{\mathcal C_\alpha^{\pm}} =\mathcal N_\pm(\ket{+\alpha}\pm\ket{-\alpha}), \qquad \mathcal N_\pm=\bigl[2(1\pm e^{-2|\alpha|^2})\bigr]^{-1/2}.$$9 photons and $\ket{\mathcal C_\alpha^\pm}$0 at $\ket{\mathcal C_\alpha^\pm}$1 using a $\ket{\mathcal C_\alpha^\pm}$2 readout (Hajr et al., 2024).

Theoretical work on universal control under frequency uncertainty has emphasized that gates about nontrivial axes require more elaborate synthesis than $\ket{\mathcal C_\alpha^\pm}$3-type operations. For the effective Kerr oscillator model, truncated-Gaussian $\ket{\mathcal C_\alpha^\pm}$4-gates achieve average infidelity below $\ket{\mathcal C_\alpha^\pm}$5 for $\ket{\mathcal C_\alpha^\pm}$6, DRAG-assisted $\ket{\mathcal C_\alpha^\pm}$7 gates achieve $\ket{\mathcal C_\alpha^\pm}$8 for $\ket{\mathcal C_\alpha^\pm}$9 and $Z$0–$Z$1, and adiabatic robust-line $Z$2 gates achieve average infidelities $Z$3 to $Z$4 in $Z$5–$Z$6 (Seifert et al., 12 Mar 2025).

Readout has progressed from proof-of-principle to high-fidelity operation. The original cat-quadrature readout reported total single-shot fidelity $Z$7 and repeatability $Z$8 under continuous stabilization, whereas later 2D implementations achieved state-of-the-art fidelities $Z$9 for bosonic qubits (Grimm et al., 2019, Hajr et al., 2024).

4. Engineered dissipation and manifold stabilization

A recurring result across the literature is that Hamiltonian confinement by itself does not eliminate leakage. One approach is colored dissipation: an engineered single-photon loss spectrum resonant only with code-space-to-excited-space transitions while strongly suppressing loss channels internal to the cat manifold. In the shifted-Fock description this yields an effective dissipator

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$00

with

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$01

Numerically, a single filter with $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$02 suppresses the idling bit-flip rate by $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$03 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$04, and the added phase-flip penalty becomes negligible for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$05 (Putterman et al., 2021).

An experimentally demonstrated variant is frequency-selective single-photon dissipation that cools leakage from $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$06 back to $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$07. The effective cooling channel

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$08

improves performance only when $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$09 exceeds the tunneling rate $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$10. Under those conditions, the leakage population is cooled from $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$11 to $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$12, and the bit-flip time reaches $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$13 (Adinolfi et al., 2 Nov 2025).

Another stabilization route uses a quantum circuit refrigerator based on photon-assisted electron tunneling. There, QCR-induced deexcitation rates can be changed by more than four orders of magnitude by tuning the bias voltage, the QCR-induced bit-flip rate scales as

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$14

and a representative parameter set yields $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$15 and $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$16 while preserving the biased-noise property of the Kerr-cat qubit (Masuda et al., 2024).

The same general principle appears in ancilla-based architectures. In the experiment on quantum control of a storage cavity with a Kerr-cat ancilla, thermal population of the first excited cat manifold caused excess dephasing of the storage mode. By applying frequency-selective dissipation at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$17, with $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$18 at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$19, the storage $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$20 fully recovers to the bare limit $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$21 within experimental uncertainty for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$22–$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$23 (Ding et al., 2024).

5. Nonautonomous transport, dissipative criticality, and Liouvillian structure

The nonautonomous character of the Kerr-cat qubit has become a major topic because state preparation and gate execution are performed with explicitly time-dependent pulses rather than frozen-time Hamiltonians. In the driven-qubit transport study, a logic gate is modeled as a transient deformation of the phase-space double well. In the conservative limit, the two wells are separated by a homoclinic separatrix; a time-dependent pulse splits the stable and unstable manifolds, creating turnstile lobes that carry trajectories across the dividing surface between wells. On the quantum side, the same pulse can corrupt the encoded bit (Wiggins, 29 Jun 2026).

The central result of that analysis is that corruption depends on the full temporal protocol, not on pulse strength alone. For pulse-amplitude sweeps at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$24, $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$25, the classical leak fraction rises from $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$26 below threshold $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$27 to $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$28 by $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$29, while the logical trace distance collapses from $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$30 at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$31 to $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$32 at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$33. For fixed $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$34, short pulses $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$35 produce large classical leak and minimal trace distance, whereas wide pulses $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$36 suppress transport and partially recover the trace distance. A Loschmidt echo evaluated shortly after the pulse end predicts the much later quantum outcome with correlation $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$37 (Wiggins, 29 Jun 2026).

Related semiclassical work derives a nonautonomous quintic normal form for ramped state preparation and a Melnikov threshold for fast gate pulses. In the resonant case $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$38, reduction to a local invariant graph produces

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$39

with moving branches $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$40 that organize nonautonomous state formation. For gate pulses, the leading-order signed splitting of the separatrix is measured by

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$41

and the threshold curve $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$42 provides a geometric indicator for the onset of lobe-mediated transport (Wiggins, 27 Apr 2026).

Open-system criticality has also been analyzed spectrally. The “critical cat code” combines Kerr nonlinearity, two-photon loss, and nonzero detuning; large detunings and small, but non-negligible, two-photon loss rates are identified as fundamental to optimal performance. The model exhibits a first-order dissipative phase transition at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$43, where the steady-state photon number jumps from a cat-like state to a squeezed vacuum, while above the transition a long-lived metastable manifold supports logical encoding for times $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$44. In the hybrid regime, $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$45 reaches as low as $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$46 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$47, $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$48, and the architecture is reported to be particularly resistant to random frequency shifts characterizing multiple-qubit operations (Gravina et al., 2022).

Non-Hermitian structure has been studied in the cat subspace through Liouvillian exceptional points. For the driven-dissipative Kerr-cat qubit, an order-2 LEP occurs at

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$49

marking a change from overdamped relaxation to underdamped oscillation, with Wigner-function negativity revivals in the underdamped regime. A later extension identifies third-order Liouvillian exceptional points arising from the competition between bidirectional cat-state jumps and a weak single-photon drive, together with a winding-number topological characterization (Han et al., 2 Feb 2026, Han et al., 23 Nov 2025).

6. Implementations, hybrid couplings, and generalizations

Superconducting implementations presently span 3D cavity devices, 2D planar circuits, and alternate nonlinear elements. The 2D SNAILmon architecture uses an on-chip band-block filter that provides $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$50 of isolation at the qubit frequency with $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$51 loss over a $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$52 bandwidth, thereby solving the drive-decay trade-off that had previously complicated strong two-photon pumping (Hajr et al., 2024). A different proposal based on Symmetrically Threaded SQUIDs derives the static effective Hamiltonian

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$53

and predicts $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$54–$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$55 at $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$56 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$57, $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$58, and $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$59, while emphasizing reduced sensitivity to higher-order photon dissipation relative to SNAIL circuits (Bhandari et al., 2024).

A major caveat is that the ideal single-mode picture can fail in multimode circuits. In the Floquet-Markov treatment of a driven SNAIL mode coupled to a buffer, multiphoton resonances generated by the buffer can sharply degrade coherence above a critical drive amplitude. That analysis attributes the sudden reduction of tunneling time to quasidegenerate hybridization between Floquet replicas and buffer excitations, and recommends choosing the buffer-drive detuning $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$60 negative to avoid such resonances (Benhayoune-Khadraoui et al., 8 Jul 2025). This directly contradicts the simplified view that increasing the pump amplitude alone monotonically improves protection.

The Kerr-cat qubit has also been developed as a biased ancilla for hybrid architectures. A beam-splitter interaction between a KCQ and a transmon realizes an effective

$$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$61

Experimentally, the interaction rate scales as expected with $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$62 and $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$63, with $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$64, $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$65 up to $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$66 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$67 and $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$68 photons, transmon coherence under interaction $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$69, and KCQ bit-flip lifetime $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$70 from earlier work (Cochran et al., 26 Nov 2025).

Generalizations extend the concept beyond microwave oscillators. The spin Kerr-cat encoding uses quadrupolar nuclei with spin length $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$71 and a $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$72-symmetric nuclear-spin Hamiltonian. For $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$73 donors in silicon, the estimated coherence time is $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$74, and with an enhancement of measured quadrupolar splittings by a factor of $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$75, a two-qubit gate fidelity of $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$76 is estimated, neglecting errors that impact the electron while it is being shuttled and read out (McIntyre et al., 21 Apr 2026).

Recent proposals also move the Kerr-cat qubit into networked and generalized-code settings. Time-dependent two-photon drives can generate flying cat-qubit states from vacuum, with generation fidelity $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$77 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$78 and logical $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$79 rotations retaining fidelities above $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$80 for $$H_{\rm cat} =\hbar\Bigl[-K\,\hat a^{\dagger2}\hat a^2 +\epsilon_{2}\,\hat a^{\dagger2} +\epsilon_{2}^*\,\hat a^2\Bigr],$$81 (Erneman et al., 29 Jun 2026). More abstractly, generalized Kerr-cat codes built from displaced Kerr coherent states and Barut--Girardello Kerr coherent states are reported to outperform conventional cat codes under combined loss and dephasing noise, with optimal recovery operations determined via convex optimization (Viladomat et al., 12 Jun 2026).

These developments indicate that the Kerr-cat qubit is no longer a single device concept but a family of noise-biased bosonic encodings whose practical performance depends on the joint design of Hamiltonian confinement, dissipation engineering, pulse shaping, and electromagnetic environment.