Limitations to Gate-Error Virtualization from Temporally Correlated Nonclassical Noise

Realistic multi-qubit noise processes often show significant temporal correlations, resulting in error mechanisms not captured by the ''local, Markovian (Pauli)'' error models commonly employed in circuit-level analyses of quantum fault-tolerance (FT). Within a Hamiltonian formulation, we revisit the validity of the notion of a constant gate error in the presence of temporally correlated noise that is nonclassical, and perfect instantaneous dynamical decoupling (DD) with finite resources. We study a minimal exactly-solvable single-qubit model under Gaussian quantum dephasing noise, showing that the fidelity of a DD-protected idling gate can depend strongly on the applied control history, even when the system-side error propagation is removed through perfect reset operations. We attribute this history-dependence of the gate fidelity to the evolution of the bath statistics during the computation, which has not been fully accounted for in existing treatments. Only if DD can keep the qubit highly pure over a timescale larger than the correlation time of the nonclassical noise, the bath approximately converges to its original statistics. Implications of this costly 're-equilibration' of the quantum bath statistics for FT control are discussed.