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Optimal Clock Speed of Single-Qubit Operations on Open Quantum Systems

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Fidelity of a single-qubit Hadamard gate is maximized at drive amplitude sqrt((1/T1 + 1/T2)/τc), the optimal clock speed set by the competition between relaxation and drive-induced decoherence.

arxiv 1908.00443 v4 pith:YBFGMUJA submitted 2019-08-01 quant-ph

Nilanjana Chanda , Rangeet Bhattacharyya This is my paper · ORCID
classification quant-ph
keywords quantumdrive-amplitudefidelityopenoperationssingle-qubitsystemsclock
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

Imagine a quantum bit, or qubit, controlled by a microwave pulse. The stronger the pulse, the faster the qubit flips, so a gate operation finishes more quickly. In a perfect, isolated world, you would always use the strongest pulse possible.

But real qubits sit in a noisy environment. Two things go wrong. First, the environment relaxes the qubit, corrupting its state; the longer the gate takes, the more damage relaxation does. Second, the drive pulse itself, by shaking the qubit, creates additional decoherence; this drive-induced decoherence grows with the pulse strength. So there is a tradeoff: a weak pulse is slow, letting relaxation act for a long time; a strong pulse is fast but adds extra noise from the drive.

The authors use a specific master equation, previously developed by their group, which includes both effects. For a Hadamard gate applied to a spin qubit, they compute the fidelity, how close the final state is to the ideal result. The fidelity is non-monotonic: it rises as the pulse strengthens, then falls as drive-induced decoherence takes over. The maximum occurs at a drive amplitude sqrt((1/T1 + 1/T2)/τc), where T1 and T2 are the relaxation times and τc is the correlation time of environmental fluctuations. This optimum is the 'optimal clock speed' of the gate.

Extended reading notes

Core claim

The fidelity of single-qubit gate operations on open quantum systems has a maximum value corresponding to an optimum drive-amplitude, ω1^opt = sqrt(R_eff/τc), with R_eff = 1/T1 + 1/T2. If correct, single-qubit operations on open quantum systems have an optimal clock speed.

Load-bearing premise

The result is contingent on the fluctuation-regulated quantum master equation (Eq. 1) from Ref. [22], specifically its second-order drive-induced decoherence term proportional to ω1^2 τc. It also assumes the gate pulses are fast compared with relaxation (ω1 >> 1/T1, 1/T2) so that the decay factor is exponential with rate R_eff + ω1^2 τc. If the DiD term had a different scaling, or if higher-order terms matter near the optimum, the formula for ω_opt would change.

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Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the frQME and its DiD term, with tau_c and R_eff as input parameters. No new entities are postulated. The optimization itself introduces no free parameters.

free parameters (2)
  • tau_c
    Correlation time of the environmental fluctuations in the frQME; appears in the DiD term and determines ω_opt. Not fitted in this paper, but taken as input from the model.
  • R_eff = 1/T1 + 1/T2
    Combined relaxation rate from qubit-environment coupling; an environment parameter. Appears in the denominator of ω_opt.
assumptions (4)
  • domain assumption The fluctuation-regulated quantum master equation (Eq. 1) from Ref. [22] is valid.
    The entire analysis builds on this master equation, which includes drive-induced decoherence terms ω1^2 τc. Not re-derived in this paper.
  • domain assumption Pulse duration is short compared with relaxation times (ω1 >> 1/T1, 1/T2), allowing the loss factor to be exponential with rate R_eff + ω1^2 τc.
    Used in the 3-pulse analysis and carried over to the Hadamard gate. This justifies the simple exponential decay factor.
  • domain assumption The environment is in equilibrium and the system-environment coupling is isotropic, so no first-order coupling and no cross terms survive the ensemble average.
    Simplifies Eq. (2) to Eq. (3).
  • standard math The fidelity measure is the Uhlmann-Jozsa fidelity, and the initial state is pseudopure with polarization m.
    Standard definition in quantum information.

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Pith. "Pith review of Optimal Clock Speed of Single-Qubit Operations on Open Quantum Systems." pith.science (2026). https://pith.science/paper/YBFGMUJA

@misc{pith2026190800443,
  author       = {Pith},
  title        = {Pith review of: Optimal Clock Speed of Single-Qubit Operations on Open Quantum Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBFGMUJA}},
  note         = {Machine review of arXiv:1908.00443}
}
read the original abstract

Efficient implementation of quantum algorithms requires single- or multi-qubit gates with high fidelity. In this report, we report that the fidelity of single-qubit gate operations on open quantum systems has a maximum value corresponding to an optimum value of the drive-amplitude in the presence of drive-induced decoherence. To show this, we use a previously reported fluctuation-regulated quantum master equation [Phys. Rev. A 97, 063837 (2018)]. The fidelity is found to be a function of the drive-induced dissipative terms as well as the relaxation terms arising from the qubit-environment coupling; as a result, it behaves non-monotonically with the drive-amplitude. The existence of an optimum drive-amplitude implies that the single-qubit operations on open quantum systems would have an optimal clock speed.

Figures

Figures reproduced from arXiv: 1908.00443 by the authors.

Figure 1
Figure 1. FIG. 1: The filled contours show the fidelity as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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