We analytically investigate the switching rate in a two-photon driven Kerr oscillator with finite detuning and two-photon dissipation. This system exhibits quantum bistability and supports a logical manifold for a bosonic qubit. Using Kramer's theory together with the $P$-representation, we derive an analytical expression for the bit-flip error rate within the potential-barrier approximation. The agreement is demonstrated between analytical calculations and numerical simulations obtained by diagonalization of the Liouvillian superoperator. In the purely dissipative limit, the switching rate increases monotonically with detuning, as the two metastable states approach each other in phase space. However, the exponential contribution to the bit-flip rate exhibits a nontrivial dependence on system parameters, extending beyond the naive scaling with the average photon number. In the presence of large Kerr nonlinearity, the switching rate becomes a nonmonotonic function of the detuning and reaches a minimum at a finite detuning. This effect arises because detuning lowers the activation barrier for weak nonlinearity but increases it for large ones, ensuring a minimum of the switching-rate at nonzero detuning. These results establish key conditions for optimizing the performance of critical cat qubits and are directly relevant for the design of scalable superconducting bosonic quantum architectures.
We analytically investigate the switching rate in a two-photon driven Kerr oscillator with finite detuning and two-photon dissipation. This system exhibits quantum bistability and supports a logical manifold for a bosonic qubit. Using Kramer's theory together with the P-representation, we derive an analytical expression for the bit-flip error rate within the potential-barrier approximation. The agreement is demonstrated between analytical calculations and numerical simulations obtained by diagonalization of the Liouvillian superoperator. In the purely dissipative limit, the switching rate increases monotonically with detuning, as the two metastable states approach each other in phase space. However, the exponential contribution to the bit-flip rate exhibits a nontrivial dependence on system parameters, extending beyond the naive scaling with the average photon number. In the presence of large Kerr nonlinearity, the switching rate becomes a nonmonotonic function of the detuning and reaches a minimum at a finite detuning. This effect arises because detuning lowers the activation barrier for weak nonlinearity but increases it for large ones, ensuring a minimum of the switching-rate at nonzero detuning. These results establish key conditions for optimizing the performance of critical cat qubits and are directly relevant for the design of scalable superconducting bosonic quantum architectures.
Encoding quantum information onto bosonic systems is a promising route to quantum error correction. In a cat code, this encoding relies on the confinement of the system's dynamics onto the two-dimensional manifold spanned by Schrödinger cats of opposite parity. In dissipative cat qubits, an engineered dissipation scheme combining two-photon drive and loss has been used to autonomously stabilize this manifold, ensuring passive protection against bit-flip errors, regardless of their origin. In Kerr cat qubits, where highly-performing gates can be engineered, two-photon drive and Kerr nonlinearity cooperate to confine the system to a two-fold degenerate ground state manifold spanned by cats of opposite parity. Dissipative, Hamiltonian, and hybrid confinements have been investigated at resonance. Here, we propose a critical cat code, where both two-photon loss and Kerr nonlinearity are present, and the two-photon drive is allowed to be out of resonance. The performance of this code is assessed via the spectral theory of Liouvillians in all configurations, from the purely dissipative to the Kerr limit. We show that large detunings and small, but non-negligible, two-photon loss rates are fundamental to achieve optimal performance. We demonstrate that the competition between nonlinearity and detuning results in a first-order dissipative phase transition, leading to a squeezed vacuum steady state. To achieve the maximal suppression of the logical bit-flip rate requires initializing the system in the metastable state emerging from the first-order transition, and we detail a protocol to do so. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts characterizing multiple-qubit operations, opening venues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures.
We study the quantum dynamics of a single bosonic cavity subject to two-photon driving and two-photon dissipation in the presence of finite detuning. Exploiting a hidden time-reversal symmetry, the Wigner representation and the WKB method, we introduce an effective phase-space potential for description of the steady state. It reveals two attracting points, which are metastable due to quantum fluctuations. By employing the Keldysh real-time path integral formalism, we compute the instanton trajectory governing the quantum activation process between these attractors and establish a fundamental connection with the Wigner representation. This relation unifies the steady-state phase-space description with dynamical quantum activation processes. We also derive an analytical expression for the decoherence rate of the system. Our work provides a coherent theoretical framework for analyzing quantum bistability, metastability, and decoherence in driven-dissipative nonlinear resonators, with direct implications for the design of bosonic qubits and quantum information processing.
Kerr nonlinear oscillators driven by a two-photon process are promising systems to encode quantum information and to ensure a hardware-efficient scaling towards fault-tolerant quantum computation. In this paper, we show that an extra control parameter, the detuning of the two-photon drive with respect to the oscillator resonance, plays a crucial role in the properties of the defined qubit. At specific values of this detuning, we benefit from strong symmetries in the system, leading to multiple degeneracies in the spectrum of the effective confinement Hamiltonian. Overall, these degeneracies lead to a stronger suppression of bit-flip errors. We also study the combination of such Hamiltonian confinement with colored dissipation to suppress leakage outside of the bosonic code space. We show that the additional degeneracies allow us to perform fast and high-fidelity gates while preserving a strong suppression of bit-flip errors.
Bosonic cat qubits promise hardware-efficient quantum error correction because their logical bit-flip rate is exponentially suppressed with the photon number of the cat state. However, several experiments report a saturation of this suppression at large photon numbers, thus limiting the achievable protection. Combining quantum-trajectory simulations, semiclassical analysis, and Liouvillian spectral methods, we investigate the properties of bit flips in realistic dissipative cat qubits, where a memory mode hosting quantum information interacts with a dissipative buffer cavity. We show that bit flips are dynamical processes inherently involving both the memory and buffer, and therefore cannot be captured by single-mode approximate descriptions. We identify a reflection symmetry, resulting in a phase-locking condition at the semiclassical level and for quantum trajectories, as the main requirement for regular bit-flip dynamics. Its breakdown is the origin of the saturation, and we find that it occurs when two conditions are met. First, the adiabatic approximation, where the state of the buffer instantaneously follows that of the memory, must not be valid, which typically happens at large photon numbers. Second, key parameters such as the cross-Kerr interaction and dephasing must be present, leading to irregular dynamics in which memory fluctuations are amplified by the buffer during bit flips. In this regime, we find that bit flips manifest as chaotic bursts within otherwise regular dynamics, as evidenced by both changes in the topology of quantum trajectories and in the Liouvillian spectrum and its associated eigenmodes involved in these switching events. Finally, we verify our predictions against experimental data, highlighting the detrimental role of dissipative chaotic behavior in bosonic error-correcting codes.
Kerr nonlinear oscillators driven by a two-photon process are promising systems to encode quantum information and to ensure a hardware-efficient scaling towards fault-tolerant quantum computation. In this paper, we show that an extra control parameter, the detuning of the two-photon drive with respect to the oscillator resonance, plays a crucial role in the properties of the defined qubit. At specific values of this detuning, we benefit from strong symmetries in the system, leading to multiple degeneracies in the spectrum of the effective confinement Hamiltonian. Overall, these degeneracies lead to a stronger suppression of bit-flip errors. We also study the combination of such Hamiltonian confinement with colored dissipation to suppress leakage outside of the bosonic code space. We show that the additional degeneracies allow us to perform fast and high-fidelity gates while preserving a strong suppression of bit-flip errors.
Quantum computing crucially relies on maintaining quantum coherence for the duration of a calculation. Bosonic quantum error correction protects this coherence by encoding qubits into superpositions of noise-resilient oscillator states. In the case of the Kerr-cat qubit (KCQ), these states derive their stability from being the quasi-degenerate ground states of an engineered Hamiltonian in a driven nonlinear oscillator. KCQs are experimentally compatible with on-chip architectures and high-fidelity operations, making them promising candidates for a scalable bosonic quantum processor. However, their bit-flip time must increase further to fully leverage these advantages. Here, we present direct evidence that the bit-flip time in a KCQ is limited by leakage out of the qubit manifold and experimentally mitigate this process. We coherently control the leakage population and measure it to be > 9%, twelve times higher than in the undriven system. We then cool this population back into the KCQ manifold with engineered dissipation, identify conditions under which this suppresses bit-flips, and demonstrate increased bit-flip times up to 3.6 milliseconds. By employing both Hamiltonian confinement and engineered dissipation, our experiment combines two paradigms for Schrödinger-cat qubit stabilization. Our results elucidate the interplay between these stabilization processes and indicate a path towards fully realizing the potential of these qubits for quantum error correction.