Spin Kerr-cat qubits

Abstract: The use of noise-robust qubit encodings provides a way of extending the lifetime of quantum information at the hardware level. In this work, we introduce the spin Kerr-cat encoding, which leverages a clock transition in the spectrum of quadrupolar nuclei (having spin length $I\geq 1$) to achieve a first-order suppression of noise leading to qubit dephasing. The basis states of the spin Kerr-cat qubit are given by the two lowest levels of a $\mathbb{Z}_2$-symmetric nuclear-spin Hamiltonian and are well approximated by spin cat states. We compute the dephasing time of the spin Kerr-cat qubit under a model of $1/f$ noise, as well as relaxation of the qubit due to breaking of the $\mathbb{Z}_2$ symmetry by charge-noise-induced fluctuations of the quadrupolar tensor. Using measured parameters for antimony (${}^{123}\mathrm{Sb}$) donors in silicon, we estimate that a coherence time of $T_2^*=100$ s could be achieved with this encoding. We propose a two-qubit gate mediated by hopping electrons and estimate that with an enhancement of measured quadrupolar splittings by a factor of $\approx 4$, a gate fidelity of $99\%$ could be achieved for spin Kerr-cat qubits encoded in ${}^{123}\mathrm{Sb}$ nuclear spins, neglecting errors that impact the electron while it is being shuttled and read out.

• The paper introduces a novel qubit encoding that leverages clock transitions in quadrupolar nuclear spins to achieve first-order insensitivity to environmental fluctuations.
• It employs a Z₂-symmetric nuclear-spin Hamiltonian to approximate the computational basis with high-fidelity spin cat states, exceeding 99% accuracy under certain conditions.
• Simulations reveal that passive dephasing suppression and effective electron-mediated two-qubit gates can yield coherence times up to 100 seconds and gate fidelities nearing 99%.

Spin Kerr-cat Qubits: Noise-Resilient Encoding in Quadrupolar Nuclear Spins

Overview and Motivation

The paper "Spin Kerr-cat qubits" (2604.19687) introduces a new qubit encoding for nuclear spins with spin quantum number $I \geq 1$, specifically targeting quadrupolar nuclei in silicon. This encoding leverages a clock transition in the nuclear spin spectrum originating from the quadrupolar interaction between the nucleus and the electric-field gradient (EFG) at its site. Clock transitions lead to first-order insensitivity to environmental fluctuations. The basis states of the spin Kerr-cat qubit are the two lowest eigenstates of a $\mathbb{Z}_2$-symmetric nuclear-spin Hamiltonian, well-approximated by spin cat states.

This approach exploits the inherent noise bias in nuclear spins, where $T_1$ typically far exceeds $T_2^*$. The methodology is passive, providing a hardware-level suppression of dephasing without any active quantum error correction or dynamical decoupling.

Quadrupole Hamiltonian and Clock Transitions

Quadrupolar nuclei possess an electric quadrupole moment, coupling to EFGs and generating a nuclear Hamiltonian with nonlinearity:

$$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$

where $Q$ quantifies the quadrupole strength, and $\eta$ is the asymmetry parameter. When the external magnetic field aligns with the principal axis of the EFG tensor (PAS), the Hamiltonian acquires a $\mathbb{Z}_2$ symmetry; its eigenstates become parity eigenstates under $\hat{\Pi} = e^{i\pi \hat{I}_z}$.

The interplay between Zeeman and quadrupolar terms results in avoided crossings for same-parity states and true crossings for opposite parity. Clock transitions occur at field strengths where the qubit splitting $\Delta$ between the two lowest states is locally maximal and first-order insensitive to magnetic-field fluctuations.

Figure 1: (a) Nuclear spectrum for $\mathbb{Z}_2$0, $\mathbb{Z}_2$1 with even/odd parity labeling. Avoided crossings occur only within parity sectors; clock transition is the red dot-dash line. (b) Minimal splitting between lowest parity doublet—a natural qubit operating point.

Spin Kerr-cat Basis States: Cat-State Approximation

The two computational basis states, $\mathbb{Z}_2$2 and $\mathbb{Z}_2$3, are the lowest eigenstates of $\mathbb{Z}_2$4, both parity eigenstates, and are well approximated by spin cat states:

$\mathbb{Z}_2$5

For $\mathbb{Z}_2$6 and $\mathbb{Z}_2$7, the fidelity between the true eigenstates and cat-state ansatz exceeds $\mathbb{Z}_2$8. This is shown numerically and visualized with spin Wigner functions.

Figure 2: (a) Fidelity vs $\mathbb{Z}_2$9 for cat-state approximation. (b) Polar angle $T_1$0 as a function of $T_1$1 defining cat states. (c) Magnetic field $T_1$2 (in units of $T_1$3) for clock transition.

The analogy with bosonic Kerr-cat qubits is technically rigorous: under Holstein-Primakoff, the quadrupole Hamiltonian maps to a squeezed Kerr oscillator for $T_1$4.

Decoherence Suppression Analysis

Dephasing

Operating at the clock transition ensures the qubit splitting is first-order insensitive to external field and EFG fluctuations. For $T_1$5 and quasistatic noise, the coherence decay is governed by second-order derivatives, resulting in a quadratic enhancement of $T_1$6:

$T_1$7

where $T_1$8 is a dimensionless coefficient extracted from numerics.

Figure 3: Coefficient $T_1$9 governing curvature of qubit splitting at the clock transition.

For experimentally realistic values ($T_2^*$0 kHz, $T_2^*$1 ms), the suppressed dephasing time can reach $T_2^*$2 s.

Relaxation from Charge Fluctuators

Quadrupolar coupling introduces a new channel: fluctuations in the EFG tensor arising from nearby charge fluctuators (TLFs), breaking the Hamiltonian's symmetry and causing bit-flip ($T_2^*$3) and leakage errors.

The perturbation scales as $T_2^*$4 with distance $T_2^*$5 between TLF and nucleus. For $T_2^*$6 nm, the bit-flip time exceeds $T_2^*$7 s for $T_2^*$8 kHz, making charge-noise-induced relaxation highly subdominant unless fluctuators are extremely close.

Control, Readout, and Two-Qubit Gates

Single-qubit Gates

Standard NMR techniques suffice for $T_2^*$9 and $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$0 rotations, with selective AC magnetic fields. Rotating-wave approximation restricts transitions to the Kerr-cat subspace, yielding efficient qubit operations.

Electron-spin Mediated Two-qubit Gates

Entangling operations are implemented via the hyperfine interaction with a mobile electron. Loading an ancillary electron onto the donor generates an Ising-like coupling and enables a controlled-phase gate logic, up to single-spin rotations. The electron can mediate gates between spatially separated Kerr-cat qubits via shuttling.

Figure 4: Electron tunneling schematic. Electron-nuclear hyperfine interaction generates a CZ gate between spin Kerr-cat qubits.

Gate Fidelity and Requirements

Numerical simulations show CZ gate fidelities reaching $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$1 when $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$2 is enhanced by a factor of $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$3 over currently measured values, and $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$4.

Figure 5: CZ gate fidelity contour vs $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$5 and $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$6; the $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$7 boundary indicated by a dashed line.

Initialization and Readout

Initial state preparation involves measurement in the $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$8 basis, rotation to a spin coherent state, and parity-projection using the electron-mediated entangling operation. Readout similarly leverages electron-spin measurement, mapping nuclear parity onto electron basis.

Practical Implications and Future Prospects

The encoding presented is highly promising for solid-state quantum computation where nuclear-spin coherence is critical. Passive suppression of dephasing at the hardware level, with no dynamical decoupling or QEC overhead, aligns with the architectural strengths of silicon-based donors.

For scalability, high uniformity in strain fields and deterministic donor placement will be required. The ability to tune $$H_q = \frac{Q}{2} [3\hat{I}_z^2 + \frac{\eta}{2} (\hat{I}_+^2 + \hat{I}_-^2) - \hat{\mathbf{I}}^2]$$9 and $Q$0 is necessary and could be facilitated via strain engineering or nanometer-scale actuators. Given the requirements on magnetic field alignment and EFG engineering, the method is best suited for systems where site-specific environmental control is feasible.

Future directions include deploying Kerr-cat qubits as quantum memories for electron-spin qubits, combining the advantages of long $Q$1 and fast manipulation. Further studies on combinatorial error channels and robust error correction tailored for this bias are warranted.

Conclusion

The spin Kerr-cat encoding for quadrupolar nuclear spins in silicon achieves first-order insensitivity to environmental fluctuations via clock transitions and parity symmetry. This approach enables dramatic suppression of dephasing, robust two-qubit gates, and highly efficient parity-based readout. Practical deployment will require precise engineering of EFGs and control of local environments, offering a scalable, noise-resilient architecture for solid-state quantum computing.