Quantum Error Correction via Riemannian Optimisation

A quantum error correction protocol consists of (1) a set of quantum states faithfully representing some logical information (the code), and (2) a recovery map that is able to correct the effect of environmental noise on such states. The aim of the work is to develop an optimisation scheme that is able to find optimally correctable subspace codes for a known quantum noise channel, leveraging Petz recovery maps and nonlinear optimization methods based on the Stiefel manifold. In fact, in the proposed approach, given a candidate subspace to be associated to a code, we fix the recovery map as if the code was perfectly correctable. Therefore, optimisation must only be considered over the set of valid codes, not over the set of recovery operators. Optimisation over the codes is equivalent to optimisation over the complex-valued Stiefel manifold of the corresponding dimensions. In this thesis, which includes a brief mathematical introduction to the relevant elements of quantum mechanics, quantum error correction and Riemannian optimisation, the theoretical basis for the problem is developed and gradient-based local optimisation algorithms are discussed and tested, as well as two different kinds of global optimisers on the Stiefel manifold. The global optimisation algorithms are based on simulated annealing with intermittent diffusion, and a consensus based algorithm. Using these algorithms, correctable codes are found and compared to existing ones for three qubits subjected to bit-flip errors (single and correlated), four qubits undergoing local amplitude damping and five qubits subjected to local depolarising channels, and general single qubit errors.