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.
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.
Cat qubits, a type of bosonic qubit encoded in a harmonic oscillator, can exhibit an exponential noise bias against bit-flip errors with increasing mean photon number. Here, we focus on cat qubits stabilized by two-photon dissipation, where pairs of photons are added and removed from a harmonic oscillator by an auxiliary, lossy buffer mode. This process requires a large loss rate and strong nonlinearities of the buffer mode that must not degrade the coherence and linearity of the oscillator. In this work, we show how to overcome this challenge by coloring the loss environment of the buffer mode with a multipole filter and optimizing the circuit to take into account additional inductances in the buffer mode. Using these techniques, we achieve near-ideal enhancement of cat-qubit bit-flip times with increasing photon number, reaching over 0.1 s with a mean photon number of only 4. Concurrently, our cat qubit remains highly phase coherent, with phase-flip times corresponding to an effective lifetime of T 1 , eff ≃ 70 μ s , comparable with the bare oscillator lifetime. We achieve this performance even in the presence of an ancilla transmon, used for reading out the cat-qubit states, by engineering a tunable oscillator-ancilla dispersive coupling. Furthermore, the low nonlinearity of the harmonic oscillator mode allows us to perform pulsed cat-qubit stabilization, an important control primitive, where the stabilization can remain off for a significant fraction (e.g., two-thirds) of a 3 μ s cycle without degrading bit-flip times. These advances are important for the realization of scalable error correction with cat qubits, where large noise bias and low phase-flip error rate enable the use of hardware-efficient outer error-correcting codes.
Autonomous quantum error correction (QEC) stabilizes a logical manifold through dissipative events that emit into output channels, which are typically accessible to measurement. These signals are often discarded, and whether they contain useful information about logical failures remains generally unclear. Using quantum trajectories, we show that in dissipatively stabilized cat qubits bit flips are not silent logical errors: each flip is accompanied by a strong, time-localized photon burst from the dissipative buffer. Photon counting and homodyne monitoring can therefore herald the loss of logical information without interrupting the autonomous stabilization: bit flips in dissipative cat qubits are erasures. More broadly, our results show that the emitted signals of engineered reservoirs can act as built-in failure monitors for autonomous QEC, turning rare logical faults into erasures available to a decoder and reducing fault-tolerance overhead. To this end, we develop a general framework, based on past quantum states and number-resolved master equations, to quantify the detectability of such logical failures in autonomous QEC from the emitted signal.
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.
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.
Bistable systems are central to information processing, from classical bits to quantum qubits. Their robustness relies on mitigating switching errors, or bit flips. While a classical potential barrier suppresses thermal bit flips, quantum systems also suffer from phase-flip errors that degrade superpositions. "Cat" qubits leverage the principle of a potential barrier to exponentially suppress bit flips, but at the expense of a marginal increase in phase flips. This unique error bias simplifies quantum error correction, making cat qubits a promising platform for fault-tolerant quantum computation.This thesis first investigates the scaling of bit-flip errors in cat qubits as a function of their size. Using path integral techniques, we predict the switching rate in a single-mode bistable open quantum system. Drawing inspiration from systems satisfying detailed balance, we identify a set of non-idealities that nonetheless preserve the optimal scaling of ideal cat qubits.We then turn to the analysis of a superconducting circuit composed of Josephson junctions that implements a dissipative cat qubit. The nonlinear dissipation that is key to stabilizing a cat qubit originates from a parametrically activated coupling between the memory and a lossy mode. Our perturbative analysis uncovers parasitic drive-activated nonlinearities that amplify phase-flip errors as the cat-state stabilization strength increases, which ultimately alters the performance of a repetition code.Finally, we present the integration of the methods developed in this thesis into a complete simulation stack for superconducting circuits. We exemplify this integration with an ab initio study of a recently developed cascaded single-microwave-photon detector.
Cat qubits, for which logical |0⟩ and |1⟩ are coherent states |±α⟩ of a harmonic mode, offer a promising route towards quantum error correction. Using dissipation to our advantage so that photon pairs of the harmonic mode are exchanged with single photons of its environment, it is possible to stabilize the logical states and exponentially increase the bit-flip time of the cat qubit with the photon number |α|2 . A large two-photon dissipation rate κ2 ensures fast qubit manipulation and short error-correction cycles, which are instrumental to correct the remaining phase-flip errors in a repetition code of cat qubits. Here, we introduce and operate an autoparametric superconducting circuit that couples a mode containing the cat qubit to a lossy mode whose frequency is set at twice that of the cat mode. This passive coupling does not require a parametric pump, and it reaches a rate κ2/2π≈2 MHz . With such a strong two-photon dissipation, bit-flip errors of the autoparametric cat qubit are prevented for a characteristic time up to 0.3 s with only a mild impact on phase-flip errors. In addition, we illustrate how the phase of a quantum superposition between |α⟩ and |−α⟩ can be arbitrarily changed by driving the harmonic mode while keeping the engineered dissipation active. Published by the American Physical Society 2024