Chiral cat code: Enhanced error correction induced by higher-order nonlinearities

TL;DR

The paper addresses the susceptibility of Kerr-cat qubits to dephasing by introducing a chiral cat qubit that leverages higher-order nonlinearities to create a chiral phase-space topology with a code space $|\pm \alpha\rangle$ and an error space $|\pm \alpha_H\rangle$. By engineering bistability via detuning and higher-order terms, bit-flip errors are funneled into a high-photon trap where they can be detected and corrected without degrading stored quantum information, even under large dephasing. The authors develop a concrete recovery protocol based on optimal-control of system parameters, achieving fidelities around 99.3% and enabling simultaneous correction of bit- and phase-flip errors in a concatenated repetition code; they also quantify the performance through Liouvillian eigenmodes, showing exponential scaling of the error rate with the code size. This work demonstrates how the phase-space topology of driven-dissipative bosonic systems can substantially enhance bosonic codes, potentially reducing hardware overhead for fault-tolerant quantum computing and motivating further exploration of chiral, detuned, and higher-order–driven bosonic codes.

Abstract

We introduce a Schrödinger chiral cat qubit, a novel bosonic quantum code generalizing Kerr cat qubits that exploits higher-order nonlinearities. Compared to a standard Kerr cat, the chiral cat qubit allows additional correction of bit-flip errors within the Hilbert space of a single bosonic oscillator. Indeed, this code displays optical bistability, i.e., the simultaneous presence of multiple long-lived states. Two of them define the code space and two define an error space. Thanks to the chiral structure of the phase space of this system, the error space can be engineered to ``capture'' bit flip events in the code space (a bit-flip trap), without affecting the quantum information stored in the system. Therefore, it is possible to perform detection and correction of errors. We demonstrate how this topological effect can be particularly efficient in the presence of large dephasing. We provide concrete examples of the performance of the code and show the possibility of applying quantum operations rapidly and efficiently. Beyond the interest in this single technological application, our work demonstrates how the topology of phase space can enhance the performance of bosonic codes.

Figures (13)

• Figure 1: Bloch sphere representation of the code and error spaces of a chiral cat, spanned by the low- ($\ket{\pm \alpha}$) and high-amplitude ($\ket{\pm \alpha_H}$) lobes, respectively. Upon an appropriate choice of the system parameters, dephasing and photon loss induce a passage from the low- to the high-amplitude lobes (a bit-flip trap that captures the state without degrading quantum information), making it possible to perform error detection and correction.
• Figure 2: Schematics of various types of cat states. The panels show the mean photon number as a function of the two-photon drive amplitude. Solid lines represent the steady-state photon number, while the dashed line represents the prediction of the semiclassical approximation. The insets show a sketch of the Wigner function, with the solid arrows indicating main the error processes and dashed ones depicting suppressed errors. (a) For a pure dissipative leghtas2015 Kerr grimm2020 cat, increasing the two-photon drive linearly increases the photon number. For a given drive, the system spans a manifold characterized by two states of opposite phase $\rho_{\pm \alpha} \simeq \ketbra{\pm \alpha}$. Bit flip errors than take the form of a passage between these two states. (b) For a critical cat gravina2023criticalcat (a hybrid cat is simultaneously stabilized by Kerr and two-photon loss and operated in the presence of detuning), the system displays optical bistability, i.e., the simultaneous presence of multiple solution according to the semiclassical approximation. The optimal regime of operation is one where a cat state will eventually decay into the vacuum, and quantum information is encoded in the metastable manifold spanned by squeezed states of opposite phases. The dominant source of errors is a leakage, corresponding to a jump to the vacuum ($\rho_{\pm\alpha} \to \ketbra{0}$). (c) The chiral cat is a hybrid cat with additional higher-order nonlinearity. The manifold $\rho_{\pm \alpha} \simeq \ketbra{\pm \alpha}$ coexists with a larger-photon number one at $\rho_{\pm \alpha_H} \simeq \ketbra{\pm \alpha_H}$, both capable of hosting cat-like states. The high manifold acts as a bit-flip trap where the jump $\rho_{\alpha} \leftrightarrow \rho_{-\alpha}$ is suppressed in favour of $\rho_{\pm \alpha} \to \rho_{\pm \alpha_H}$. Finally, as the photon number between the two manifolds is different, the error can be detected and corrected. Parameters $(\mathrm{MHz})$: $\kappa_1/2\pi = 0.005$, $K_2/2\pi = -5$, $\kappa_\phi = 0$ and incrementally we set (a) $\Delta = K_3 = 0$, (b) $\Delta/2\pi = -25$ and $\kappa_2 / \abs{K_2} = 0.01$, and (c) $K_3/2\pi = 0.15$.
• Figure 3: The Liouvillian gap describing the bit-flip error rate as a function of the mean photon number ($\abs{\alpha}^2$) for the dissipative (orange) and Kerr (blue) cats in a regime with negligible dephasing. Parameters as in \ref{['fig:main-sketch']} except $\epsilon_s$ which is chosen to match the mean photon number.
• Figure 4: As a function of the mean-photon number, the Liouvillian gap describing the bit-flip error rate in (a) dissipative, (b) Kerr, (c) hybrid and (d) critical cat. We show three different regimes: single-photon dominated at $\kappa_\phi = 0.1 \kappa_1$ (solid lines), equal error rate $\kappa_\phi = \kappa_1$ (dashed lines) and dephasing dominated $\kappa_\phi = 10 \kappa_1$ (dotted lines). Parameters as in \ref{['fig:main-sketch']}. Detuning is zero for dissipative, Kerr and hybrid cats, but an optimal detuning is chosen for the critical cat.
• Figure 5: Phase-space representation of the manifold for (a) dissipative, (b) Kerr or hybrid, (c) critical, and (d) chiral cat states, and corresponding vector field obtained by the semiclassical analysis. While for a standard dissipative, Kerr, or hybrid cat errors can bring the manifold out of the bottom its potential, in the chiral case the system is attracted to the high-photon manifold. Parameters ($\mathrm{MHz}$) as in \ref{['fig:main-sketch']} with $\kappa_\phi = 10^{-4}\kappa_1$. Detuning and two-photon driving are chosen as to keep the code space with eight photons: (a) $\epsilon_s / 2\pi = 20$, (b) $\epsilon_s / 2\pi = 20$, (c) $\Delta/2\pi = -32$ and $\epsilon_s / 2\pi = 3.6$, and (d) $\Delta/2\pi = -4$ and $\epsilon_s / 2\pi = 13.2$.