Abstract
Quantum error correction is based on a feedback loop. This feedback generally corresponds to a classical controller with classical signals from quantum measurements as input, and classical signals controlling the quantum evolution of the physical system encoding a logic qubit as output. Quantum error correction can also exploit the dissipation associated with the phenomenon of decoherence. Called autonomous correction by physicists, it then uses feedback where the controller is a dissipative quantum auxiliary system. This talk focuses on the development of such quantum controllers for stabilizing gate states in a harmonic oscillator, states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computing and corresponding to GKP qubits. In two preprints(arXiv:2304.01425 and arXiv:2304.03806), the design and convergence analysis of such controllers is based on the master equations giving the time evolution of the density operator, perturbation theory (averaging and singular perturbations), Lyapunov functions and the spectrum associated with certain Witten Laplacians. Such quantum controllers should be able to be realized experimentally with superconducting quantum circuits and technologies close to those used for cat qubits (" cat-qubits ") where " bit-flip " errors are exponentially suppressed(arXiv:2204.09128). For GKP qubits, such controllers could then exponentially suppress both " bit-flip " and " phase-flip " errors. They could therefore lead to a logical qubit whose physical support would be a harmonic oscillator equipped with a quantum controller.
