Limitations to Dynamical Error Suppression and Gate-Error Virtualization from Temporally Correlated Nonclassical Noise

Realistic multi-qubit noise processes often result in error mechanisms that are not captured by the probabilistic, Markovian error models commonly employed in circuit-level analyses of quantum fault-tolerance. By working within an open-quantum system Hamiltonian formulation, we revisit the validity of the notion of a constant gate error in the presence of noise that is both {\em temporally correlated and nonclassical}, and whose impact is mitigated through perfect instantaneous dynamical decoupling subject to finite timing constraints. We study a minimal exactly solvable single-qubit model under Gaussian quantum dephasing noise, showing that the fidelity of a dynamically protected idling gate can depend strongly on its {\em location in the circuit and the history of applied control}, even when the system-side error propagation is fully removed through perfect reset operations. For digital periodic control, we prove that, under mild conditions on the low-frequency behavior of the nonclassical noise spectrum, the gate fidelity saturates at a value that is strictly smaller than the one attainable in the absence of control history; the presence of high-frequency noise peaks is also found as especially harmful, due to the possible onset of control-induced resonance effects. We explicitly relate these features to the evolution of the bath statistics during the computation, which has not been fully accounted for in existing treatments. We find that only if decoupling can keep the qubit highly pure over a timescale larger than the correlation time of the noise, the bath approximately converges to its original statistics and a stable-in-time control performance is recovered. Implications of this costly re-equilibration of the quantum bath statistics for layered quantum fault-tolerant architectures are discussed.