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arXiv:2509.07693

Simulation of one and two qubit superconducting quantum gates under the non-Markovian $1/f$ noise

Yinjia Chen, Shuocang Zhang, Qiang Shi

Published Sep 9, 2025• 0 citations
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Abstract

Non-Markovian $1/f$ noise consists a dominant source of decoherence in superconducting qubits, yet its slow 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. By using the approach, it is first shown that perturbative quantum master equations may fail to reproduce the correct dephasing dynamics of a qubit coupled to slow baths. We then analyze dynamical decoupling sequences by including effects of finite pulse duration. It is found that different pulse sequences results in different behavior in error accumulation: all X-CPMG sequences exhibit linear scaling with parity effects, Y-CPMG follows quadratic growth, and alternating XY-type sequences can suppress the error accumulation significantly. Finally, we extend the framework to two-qubit cross-resonance (CR) gates, reconstructing the full Choi matrix and Pauli Transfer Matrix (PTM) to identify the incoherent error induced by $1/f$ noise. Together, these results establish HEOM as a robust methodology for simulating the environmental noise in superconducting circuits and provide new insights into error mechanisms in both single- and two-qubit gates.

DOI
10.48550/arxiv.2509.07693

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Simulation of one and two qubit superconducting quantum gates under the non-Markovian 1/f1/f1/f noise
Non-Markovian 1/f1/f1/f noise consists a dominant source of decoherence in superconducting qubits, yet its slow 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/f1/f1/f noise. By using the approach, it is first shown that perturbative quantum master equations may fail to reproduce the correct dephasing dynamics of a qubit coupled to slow baths. We then analyze dynamical decoupling sequences by including effects of finite pulse duration. It is found that different pulse sequences results in different behavior in error accumulation: all X-CPMG sequences exhibit linear scaling with parity effects, Y-CPMG follows quadratic growth, and alternating XY-type sequences can suppress the error accumulation significantly. Finally, we extend the framework to two-qubit cross-resonance (CR) gates, reconstructing the full Choi matrix and Pauli Transfer Matrix (PTM) to identify the incoherent error induced by 1/f1/f1/f noise. Together, these results establish HEOM as a robust methodology for simulating the environmental noise in superconducting circuits and provide new insights into error mechanisms in both single- and two-qubit gates.
Abstract Non-Markovian noise can be a significant source of errors in superconducting qubits. We develop gate sequences utilising mirrored pseudoidentities that allow us to characterise and model the effects of non-Markovian noise on both idle and driven qubits. We compare three approaches to modelling the observed noise: (i) a Markovian noise model, (ii) a model including interactions with a two-level system (TLS), (iii) a model utilising the post Markovian master equation, which we show to be equivalent to the qubit-TLS model in certain regimes. When running our noise characterisation circuits on a superconducting qubit device we find that purely Markovian noise models cannot reproduce the experimental data. Our model based on a qubit-TLS interaction, on the other hand, is able to closely capture the observed experimental behaviour for both idle and driven qubits. We investigate the stability of the noise properties of the hardware over time, and find that the parameter governing the qubit-TLS interaction strength fluctuates significantly even over short time-scales of a few minutes. Finally, we evaluate the changes in the noise parameters when increasing the qubit drive pulse amplitude. We find that although the hardware noise parameters fluctuate significantly over different days, their drive pulse induced relative variation is rather well defined within computed uncertainties: both the phase error and the qubit-TLS interaction strength change significantly with the pulse strength, with the phase error changing quadratically with the amplitude of the applied pulse. Since our noise model can closely describe the behaviour of idle and driven qubits, it is ideally suited to be used in the development of quantum error mitigation and correction methods.
Non-Markovian noise can be a significant source of errors in superconducting qubits. We develop gate sequences utilising mirrored pseudoidentities that allow us to characterise and model the effects of non-Markovian noise on both idle and driven qubits. We compare three approaches to modelling the observed noise: (i) a Markovian noise model, (ii) a model including interactions with a two-level system (TLS), (iii) a model utilising the post Markovian master equation (PMME), which we show to be equivalent to the qubit-TLS model in certain regimes. When running our noise characterisation circuits on a superconducting qubit device we find that purely Markovian noise models cannot reproduce the experimental data. Our model based on a qubit-TLS interaction, on the other hand, is able to closely capture the observed experimental behaviour for both idle and driven qubits. We investigate the stability of the noise properties of the hardware over time, and find that the parameter governing the qubit-TLS interaction strength fluctuates significantly even over short time-scales of a few minutes. Finally, we evaluate the changes in the noise parameters when increasing the qubit drive pulse amplitude. We find that although the hardware noise parameters fluctuate significantly over different days, their drive pulse induced relative variation is rather well defined within computed uncertainties: both the phase error and the qubit-TLS interaction strength change significantly with the pulse strength, with the phase error changing quadratically with the amplitude of the applied pulse. Since our noise model can closely describe the behaviour of idle and driven qubits, it is ideally suited to be used in the development of quantum error mitigation and correction methods.
We investigate the possibility of achieving high-fidelity universal two-qubit gates by supplementing optimal tuning of individual qubits with dynamical decoupling (DD) of local 1/f1/f1/f noise. We consider simultaneous local pulse sequences applied during the gate operation and compare the efficiencies of periodic, Carr-Purcell, and Uhrig DD with hard $\ensuremath{\pi}$ pulses along two directions (${\ensuremath{\pi}}_{z/y}$ pulses). We present analytical perturbative results (Magnus expansion) in the quasistatic noise approximation combined with numerical simulations for realistic 1/f1/f1/f noise spectra. The gate efficiency is studied as a function of the gate duration, of the number nnn of pulses, and of the high-frequency roll-off. We find that the gate error is nonmonotonic in nnn, decreasing as ${n}^{\ensuremath{-}\ensuremath{\alpha}}$ in the asymptotic limit, $\ensuremath{\alpha}\ensuremath{\ge}2$, depending on the DD sequence. In this limit ${\ensuremath{\pi}}_{z}$-Urhig is the most efficient scheme for quasistatic 1/f1/f1/f noise, but it is highly sensitive to the soft UV cutoff. For small number of pulses, ${\ensuremath{\pi}}_{z}$ control yields anti-Zeno behavior, whereas ${\ensuremath{\pi}}_{y}$ pulses minimize the error for a finite nnn. For the current noise figures in superconducting qubits, two-qubit gate errors $\ensuremath{\sim}{10}^{\ensuremath{-}6}$, meeting the requirements for fault-tolerant quantum computation, can be achieved. The Carr-Purcell-Meiboom-Gill sequence is the most efficient procedure, stable for 1/f1/f1/f noise with UV cutoff up to gigahertz.
Abstract Decoherence in superconducting qubits arises from both multilevel dynamics and structured environmental noise, yet perturbative models cannot capture all resulting signatures. Here, EmuPlat couples instruction-set-architecture-level waveform generation to the hierarchical equations of motion under 1 / f non-Markovian pure dephasing. In the resulting non-perturbative regime—where filter-function predictions become quantitatively uninformative—Carr–Purcell–Meiboom–Gill (CPMG) scaling of a three-level superconducting transmon yields one calibration result, two physical findings, and one structural null. Y-CPMG exhibits axis-dependent scaling-law breakdown—non-monotonic decoherence, partial coherence revival, and pronounced X – Y population asymmetry ( 0.204 vs < 0.01 )—driven by third-level anharmonicity amplified by bath memory; X-CPMG maintains well-behaved power-law scaling with a finite- n transient excess consistent n-Markovian bath-memory effects. This null result is equally informative: waveform-level differences—Standard versus virtual pulse processing unit (VPPU) realizations—remain undetectable across all coupling strengths. This shows that rotating-frame pure-dephasing coupling renders control-layer detail invisible to scaling observables. These findings define testable predictions, the most experimentally accessible requiring only qualitative verification.
Decoherence in superconducting qubits arises from both multilevel dynamics and structured environmental noise, yet perturbative models cannot capture all resulting signatures. Here, EmuPlat couples instruction-set-architecture-level waveform generation to the hierarchical equations of motion HEOM under 1/f1/f1/f non-Markovian pure dephasing. In the resulting non-perturbative regime -- where filter-function predictions become quantitatively uninformative -- CPMG scaling of a three-level superconducting transmon yields one calibration result, two physical findings, and one structural null. Y-CPMG exhibits axis-dependent scaling-law breakdown -- non-monotonic decoherence, partial coherence revival, and pronounced X--Y population asymmetry (0.2040.2040.204 vs < 0.01{<}\,0.01<0.01) -- driven by third-level anharmonicity amplified by bath memory; X-CPMG maintains well-behaved power-law scaling with a finite-nnn transient excess consistent with non-Markovian bath-memory effects. This null result is equally informative: waveform-level differences -- Standard versus VPPU realisations -- remain undetectable across all coupling strengths. This shows that rotating-frame pure-dephasing coupling renders control-layer detail invisible to scaling observables. These findings define testable predictions, the most experimentally accessible requiring only qualitative verification.
Accurate noise characterization is essential for scaling quantum processors toward fault-tolerant operation. Although reduced qubit dynamics are often modeled with Markovian master equations, present-day devices can exhibit memory effects generated by residual qubit-qubit couplings, structured environments, and finite bath correlation times. Here we develop a channel-resolved, Post-Markovian Master Equation model for non-Markovian noise and test it in superconducting qubits. Using idle-evolution tomography on IBM Quantum processors, we identify complementary operational signatures of non-Markovianity, including violations of CP-divisibility and revivals of distinguishability-based information-backflow measures. We further derive a closed-form spectator-ZZZZZZ model with local dissipation and show that it captures the observed transverse Bloch-vector revivals while leaving the longitudinal relaxation mode Markovian within the model. The fitted closed-form dynamics enable an analytical reconstruction of the transverse memory kernel, whose damped oscillatory structure captures the non-Markovian correction beyond the fitted Markovian baseline. Two-qubit tomography shows buildup and revivals of quantum mutual information on comparable timescales, supporting spectator-induced crosstalk as an important contributor to the observed memory effects. Our results connect operational non-Markovianity diagnostics, microscopic crosstalk modeling, and reduced memory-kernel reconstruction in a single experimental framework for superconducting quantum hardware.