Finding Quantum Codes via Riemannian Optimization

We propose a novel optimization scheme designed to find optimally correctable subspace codes for a known quantum noise channel. To each candidate subspace code we first associate a universal recovery map, as if the code was perfectly correctable, and aim to maximize a performance functional that combines a modified channel fidelity with a tuneable regularization term that promotes simpler codes. With this choice optimization is performed only over the set of codes, and not over the set of recovery operators. The set of codes of fixed dimension is parametrized as a complex-valued Stiefel manifold: the resulting non-convex optimization problem is then solved by gradient-based local algorithms. When perfectly correctable codes cannot be found, a second optimization routine is run on the recovery Kraus map, also parametrized in a suitable Stiefel manifold via Stinespring representation. To test the approach, correctable codes are sought in different scenarios and compared to existing ones: three qubits subjected to bit-flip errors (single and correlated), four qubits undergoing local amplitude damping and five qubits subjected to local depolarizing channels. Approximate codes are found and tested for the previous examples as well pure non-Markovian dephasing noise acting on a $7/2$ spin, induced by a $1/2$ spin bath, and the noise of the first three qubits of IBM's \texttt{ibm\_kyoto} quantum computer. The fidelity results are competitive with existing iterative optimization algorithms, with respect to which we maintain a strong computational advantage, while obtaining simpler codes.