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Journal of High School Science

Noise-dependent limits of dynamical decoupling: fidelity and variance dynamics in single-qubit systems

Eli Kim

Published Apr 27, 2026• 0 citations
Publisher

Abstract

Decoherence is a primary barrier to practical quantum computation, causing quantum states to degrade through phase randomization and irreversible energy relaxation. This study presents a simulation-based evaluation of echo-based dynamical decoupling (DD) pulse sequences as a software-level mitigation strategy in single-qubit systems. Using the Qiskit Aer density matrix simulator, four strategies—baseline, Hahn Echo, Carr-Purcell-Meiboom-Gill (CPMG), and Mixed Synchronization—were evaluated across low-frequency phase noise, high-frequency phase noise, and combined T₁ + T₂ noise regimes. Hahn Echo restored mean fidelity to 100.00% under low-frequency noise (baseline: 57.90%), while CPMG achieved 97.62% under high-frequency noise (baseline: 2.45%). Under combined noise, all strategies showed limited improvement, with Mixed-Sync reaching 65.34% versus 60.95%. Coefficient of variation (CV) analysis across 500 Monte Carlo trials showed that CPMG reduced purity-decay variance threefold under high-frequency noise, with bootstrap confidence intervals confirming statistical significance. Time-resolved analysis showed that CPMG suppressed variance growth throughout evolution, reducing variance growth from +267% (baseline) to +89%, while baseline and Hahn Echo exhibited progressive amplification of variance. These results show that dynamical decoupling influences the temporal structure of noise, not just mean fidelity. In T₂-dominated regimes, pulse sequences constrain variability; in mixed regimes, they partially shape it; and in T₁-dominated regimes, both mean performance and variability converge to hardware-limited behavior. DD effectively corrects reversible phase errors but cannot overcome irreversible T₁ loss, with implications for NISQ circuit design.

DOI
10.64336/001c.161307

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Decoherence is a primary barrier to practical quantum computation, causing quantum states to degrade through phase randomization and irreversible energy relaxation. This study presents a simulation-based evaluation of echo-based dynamical decoupling (DD) pulse sequences as a software-level mitigation strategy in single-qubit systems. Using the Qiskit Aer density matrix simulator, four strategies—baseline, Hahn Echo, Carr-Purcell-Meiboom-Gill (CPMG), and Mixed Synchronization—were evaluated across low-frequency phase noise, high-frequency phase noise, and combined T₁ + T₂ noise regimes. Hahn Echo restored mean fidelity to 100.00% under low-frequency noise (baseline: 57.90%), while CPMG achieved 97.62% under high-frequency noise (baseline: 2.45%). Under combined noise, all strategies showed limited improvement, with Mixed-Sync reaching 65.34% versus 60.95%. Coefficient of variation (CV) analysis across 500 Monte Carlo trials showed that CPMG reduced purity-decay variance threefold under high-frequency noise, with bootstrap confidence intervals confirming statistical significance. Time-resolved analysis showed that CPMG suppressed variance growth throughout evolution, reducing variance growth from +267% (baseline) to +89%, while baseline and Hahn Echo exhibited progressive amplification of variance. These results show that dynamical decoupling influences the temporal structure of noise, not just mean fidelity. In T₂-dominated regimes, pulse sequences constrain variability; in mixed regimes, they partially shape it; and in T₁-dominated regimes, both mean performance and variability converge to hardware-limited behavior. DD effectively corrects reversible phase errors but cannot overcome irreversible T₁ loss, with implications for NISQ circuit design.
Abstract Dynamical decoupling (DD) is a quantum error-mitigation technique based on applying sequences of control pulses to suppress decoherence. Here we perform a numerical study to compare different DD protocols, namely Carr-Purcell-Meiboom-Gill (CPMG), Periodic Dynamical Decoupling (PDD), Concatenated Dynamical Decoupling (CDD) and Uhrig Dynamical Decoupling (UDD) applied to three commonly used noise power spectral densities: (i) white, (ii) 1/f, (iii) 1/f 2 and their hybrid combinations. By ignoring qubit relaxation, we simulate these protocols focusing purely on dephasing noise. We then consider independent multi-axial noise with uniaxial control pulses. The effective coherence times T2,eff are extracted for each case and compared with the baseline free induction decay T ∗ 2 [6]. In all these simulations, the time-domain decoherence profiles follow a stretched exponential form ∼ exp −(t/T2,eff) β with the effective coherence times dependent on the noise spectrum considered, the strength of the noise, and the DD sequence employed. An exception is observed for the UDD under specific noise conditions where it cannot be described by a simple stretched exponential decay form.
One of the simplest and least resource-intensive methods to suppress decoherence for qubit operations, namely, dynamical decoupling (DD), is investigated for a broad range of realistic noise sources with time-retarded feedback. By way of example, the Carr-Purcell-Meiboom-Gill sequence is analyzed in a numerically rigorous manner accounting also for correlations between qubits and environments. Since experimentally noise sources are characterized through spectral densities, we adopt the spin-boson model as a suitable framework to describe the qubit dynamics under DD for a given spectral density J ( ω ) ∝ ω s . Motivated by the situation for superconducting qubits, the spectral exponent s is varied from s = 1 (Ohmic bath) to a substantially small value 0 < s ≪ 1 (deep sub-Ohmic bath), in order to investigate the impact of time-nonlocal back action on DD performances for enhanced coherence times. As reference to the DD schemes, dynamics of a single qubit subject to Ramsey sequences without any pules and Hahn echo sequences are also investigated.
We demonstrate the use of dynamic decoupling techniques to extend the coherence time of a single memory qubit by nearly two orders of magnitude. By extending the Hahn spin-echo technique to correct for unknown, arbitrary polynomial variations in the qubit precession frequency, we show analytically that the required sequence of π-pulses is identical to the Uhrig dynamic decoupling (UDD) sequence. We compare UDD and Carr–Purcell–Meiboom–Gill (CPMG) sequences applied to a single 43 Ca + trapped-ion qubit and find that they afford comparable protection in our ambient noise environment.
Recent studies have shown that applying a sequence of Hahn spin-echo pulses to a qubit system at judiciously chosen intervals can, in certain noise environments, greatly improve the suppression of phase errors compared to traditional dynamical decoupling approaches. By enforcing a simple analytical condition, we obtain sets of dynamical decoupling sequences that are designed for optimized noise filtration, but are independent of the noise spectrum up to a single scaling factor set by the coherence time of the system. These sequences are tested in a model qubit system, ;{9}Be;{+} ions in a Penning trap. Our combined theoretical and experimental studies show that in high-frequency-dominated noise environments with sharp high-frequency cutoffs this approach may suppress phase errors orders of magnitude more efficiently than comparable techniques can.
In quantum information processing, it is vital to protect the coherence of qubits in noisy environments. Dynamical decoupling (DD), which applies a sequence of flips on qubits and averages the qubit-environment coupling to zero, is a promising strategy compatible with other desired functionalities, such as quantum gates. Here, we review the recent progresses in theories of dynamical decoupling and experimental demonstrations. We give both semiclassical and quantum descriptions of the qubit decoherence due to coupling to noisy environments. Based on the quantum picture, a geometrical interpretation of DD is presented. The periodic Carr-Purcell-Meiboom-Gill DD and the concatenated DD are reviewed, followed by a detailed exploration of the recently developed Uhrig DD, which employs the least number of pulses in an unequally spaced sequence to suppress the qubit-environment coupling to a given order of the evolution time. Some new developments and perspectives are also discussed.
Currently available quantum computers are prone to errors. Circuit optimization and error mitigation methods are needed to design quantum circuits to achieve better fidelity when executed on NISQ hardware. Dynamical decoupling (DD) is generally used to suppress the decoherence error and different DD strategies have been proposed. Moreover, the circuit fidelity can be improved by pulselevel optimization, such as creating hardware-native pulse-efficient gates. This paper implements all the popular DD sequences and evaluates their performances on IBM quantum chips with different characteristics for various well-known quantum applications. Also, we investigate combining DD with pulse-level optimization method and apply them to QAOA to solve Max-Cut problem. Based on the experimental results, we found that DD can be a benefit for only certain types of quantum algorithms, while the combination of DD and pulse-level optimization methods always has a positive impact. Finally, we provide several guidelines for users to learn how to use these pulse-level noise mitigation methods to build circuits for quantum applications with high fidelity.