A ``spectator state" approach to mitigate non-Markovian and coherent errors in quantum control of qudits*

Detecting mitigating and correcting errors in quantum control is among the most pertinent contemporary problems in quantum technologies. Here we develop a new technique of in-situ error characterization and mitigation which can be implemented in quantum systems with a large Hilbert space such as qudits and multi-qubit systems. We construct two 2- dimensional subspaces --- a code space C where the logical qubit is encoded and a ``spectator'' space P, which is orthogonal to the code space, but they are both within the qudit Hilbert space. While the qubit is a part of a quantum circuit and can evolve in time according to the gates in the circuit, the spectator remains idle. Note that we can make simultaneous measurements on the two subspaces. Errors of physical origin acting on the qudit Hilbert space would affect both the logical qubit and the spectator and therefore, it can be detected and even partially characterized using measurements on the spectator qubit. We consider the projections of the error operator on C and P represented as Pauli transfer T and T' respectively. Note that we can directly measure T', but not T. Under reasonable assumptions regarding the physical origin of the errors, T can be inferred from T' acting on the spectator qubit and the latter can be measured without affecting the qubit. We show numerically that for typical and physically motivated errors on the qudit, T and T' are strongly correlated and share a large mutual information, implying that one can be inferred from the other. We use the numerical data to learn an affine map Φ which maps T' to T, i.e., T ≈Φ(T'). We also show that the inversion of a suitable spectator space's logical Pauli transfer matrix can effectively mitigate the noise on the two modes bosonic system or two qudits system.