Dynamical Decoupling and Zero-Noise Extrapolation Under Colored Noise

Zero-noise extrapolation (ZNE) is an established quantum error mitigation technique that infers noise-free expectation values by scaling circuit noise and extrapolating to the zero-noise limit. Standard ZNE formulations implicitly assume memoryless Markovian gate noise, where error rates scale linearly with circuit duration. In physical solid-state quantum processors dominated by colored $1/f^\alpha$ environmental dephasing, this fundamental assumption fails: non-Markovian noise induces superlinear coherence decay $\chi(t) \propto t^{1+\alpha}$, causing standard polynomial extrapolation to diverge. In this work, we demonstrate how inserting dynamical decoupling (DD) sequences into idle circuit intervals spectrally filters low-frequency noise components, linearizing the effective decay profile into a Markovian regime $\chi_{\text{eff}}(t) \propto t$. We benchmark this combined DD–ZNE framework by reconstructing the order parameter across the quantum phase transition of a 1D Transverse-Field Ising Model (TFIM). Our results demonstrate that while standalone ZNE fails under non-Markovian dephasing, the combined DD–ZNE approach eliminates extrapolation divergence and reliably recovers ideal quantum observables. Key Highlights: Theoretical Breakdown: Analytical derivation showing why polynomial ZNE diverges under $1/f^\alpha$ dephasing environments due to superlinear temporal scaling. Filter Function Engineering: Demonstration of spectator-qubit XY-4 pulse insertion as a high-pass spectral filter that suppresses low-frequency noise and restores linear decay dynamics. Phase Transition Benchmark: Validation on a 4-qubit 1D TFIM order parameter ($\langle \sigma_z^{(0)} \sigma_z^{(1)} \rangle$) across $g \in [0.1, 2.0]$, demonstrating a $>69\%$ error reduction over unmitigated circuits and preventing non-physical polynomial divergence.