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.
Miguel Casanova Kentaro Ohki Francesco Ticozzi