Non-Markovian $1/f$ noise is a major source of decoherence in superconducting qubits, yet its low-frequency nature poses a significant challenge for accurate simulation. Here, we develop a hierarchical equations of motion (HEOM) framework that enables efficient and reliable modeling of qubit dynamics and gate operations under $1/f$ noise. Using this approach, we first show that perturbative quantum master equations fail to reproduce the correct dephasing dynamics of a qubit coupled to $1/f$ noise. We then analyze dynamical decoupling using the Carr-Purcell-Meiboom-Gill (CPMG) protocol with finite-duration pulses. The results reveal that errors accumulate linearly with parity effects in $X$-CPMG, quadratically in $Y$-CPMG, and are strongly suppressed in alternating $\mathit{XY}$ sequences. Finally, we extend the framework to simulate a two qubit cross-resonance gate and reconstruct the Pauli transfer matrix to characterize the error channels induced by $1/f$ noise. Together, these results establish HEOM as a robust and accurate method for simulating the slow non-Markovian $1/f$ noise in superconducting qubits and provide insights into error mechanisms in both single and two qubit gates.

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• Noise
• Open quantum systems & decoherence
• Quantum gates
• Superconducting qubits
• Hierarchical equations of motion
• Non-Markovian processes