A critical Schrödinger cat qubit
The main road to quantum error correction consists of encoding a logical qubit onto several physical qubits. This redundancy makes it possible to detect and correct errors without affecting the stored quantum information. An alternative way to quantum error correction is bosonic quantum codes, exploiting the redundancy of the harmonic ladder of a bosonic oscillator. A pivotal example of a bosonic code is the cat code, whose encoding lies in the two-dimensional manifold spanned by Schrödinger cats of opposite parity. This code space is autonomously stabilized by the balance between two-photon drive and two-photon loss, the latter preventing leakage outside of the code manifold: a dissipative stabilization. A different protocol to confine the cat states uses two-photon drive and Kerr nonlinearity. Dissipative, Hamiltonian, and hybrid confinement mechanisms have been investigated at resonance, i.e., for driving frequencies matching that of the cavity. 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. We show that large detunings and small, but non-negligible, two-photon loss rates are fundamental to achieve optimal performance. The competition between nonlinearity and detuned two-photon drive triggers a first-order dissipative phase transition, and we show that achieving the maximal suppression of the logical bit-flip rate requires the initialization of the system in the metastable state emerging from the first-order transition. 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.
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