From virtual Z gates to virtual Z pulses

Christopher K. Long and Crispin H. W. Barnes

Preprint published: 16 September 2025 18:37:59 UTC

DOI: 10.48550/arXiv.2509.13453

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Abstract

Virtual $Z$ gates have become integral for implementing fast, high-fidelity single-qubit operations. However, virtual $Z$ gates require that the system’s two-qubit gates are microwave-activated or normalise the single-qubit $Z$ rotations—the group generated by $X$, $\mathrm{SWAP}$, and arbitrary phase gates. Herein, we extend the theory of virtual $Z$ gates to the pulse-level, which underlies both gate design and the recent advancements of pulse-level quantum algorithms. These algorithms attempt to utilise the full potential of present-day noisy intermediate-scale quantum (NISQ) devices by removing overheads associated with the compilation and transpilation of gates. To extend the theory of virtual $Z$ gates, we derive a platform-agnostic theoretical framework for virtual $Z$ pulses by employing time dilations of the pulse sequences that control the quantum processor. Additionally, we provide worked examples of the implementation of virtual $Z$ pulses on both semiconductor spin qubit and superconducting quantum processor architectures. Moreover, we present a general overview of the hardware support for virtual $Z$ pulses. We find virtual $Z$ pulses (and thus, virtual $Z$ gates) can be used on hardware that, with previous methods, did not support the virtual $Z$ gate. Finally, we present two additional applications of virtual $Z$ pulses to pulse-level algorithms. First, broadening the class of Hamiltonians that can be natively simulated in an analogue manner. Second, increasing the expressibility of pulse-based variational quantum algorithms.

Citation

Christopher K. Long and Crispin H. W. Barnes. From virtual Z gates to virtual Z pulses, 2025, arXiv:2509.13453 [quant-ph].

BibTeX

@misc{long2025virtualzgatesvirtual,
    title={From virtual Z gates to virtual Z pulses}, 
    author={Christopher K. Long and Crispin H. W. Barnes},
    year={2025},
    eprint={2509.13453},
    archivePrefix={arXiv},
    primaryClass={quant-ph},
    url={https://arxiv.org/abs/2509.13453}, 
}

Software

• From-virtual-Z-gates-to-virtual-Z-pulses-source-code  [1]
• PySTE  [2]
• Suzuki-Trotter-Evolver  [3]

Data

• From-virtual-Z-gates-to-virtual-Z-pulses-data  [4]

Analytics

• Google Scholar
• SciRate
• INSPIRE-HEP
• Semantic Scholar

Social media posts

Bluesky

Thread

Ever wanted to use virtual Z gates with arbitrary powers of SWAP? Introducing the virtual Z pulse: arxiv.org/abs/2509.13453. In our new article, we present a method for distorting a pulse sequence to implement single-qubit Z controls virtually.

[image or embed]

— Christopher K. Long (@christopher-k-long.bsky.social) 18 September 2025 at 10:33

Example pulse distortions: (Top row) The modulation of a virtual Pauli-Z term in the system Hamiltonian. (2nd row) The time dilation required to implement the virtual Pauli-Z term without actually introducing it to the Hamiltonian. (3rd and 4th rows) The distorted two- and one-qubit control pulses.

[image or embed]

— Christopher K. Long (@christopher-k-long.bsky.social) 18 September 2025 at 10:33

LinkedIn

Ever wanted to use virtual Z gates with arbitrary powers of SWAP? Introducing the virtual Z pulse:

C. K. Long and Prof. Crispin H. W. Barnes, From virtual Z gates to virtual Z pulses, 2025. arXiv: 2509.13453 [quant-ph]. https://lnkd.in/eskd7xH6.

In our new article, we present a method for distorting a pulse sequence to implement single-qubit Z controls virtually.

Below are some example pulse distortions: The top row shows the modulation of a virtual Pauli-Z term in the system Hamiltonian. The 2nd row plots the time dilation required to implement the virtual Pauli-Z term without actually introducing it to the Hamiltonian. The 3rd and 4th rows present the distorted two- and single-qubit control pulses, respectively.

— Christopher K. Long (LinkedIn) 18 September 2025

X

Thread

Ever wanted to use virtual Z gates with arbitrary powers of SWAP? Introducing the virtual Z pulse: https://t.co/Pc6UdJBgav. In our new article, we present a method for distorting a pulse sequence to implement single-qubit Z controls virtually.

— Chris Long (@Chris_K_Long45) September 18, 2025

Example pulse distortions: (Top row) The modulation of a virtual Pauli-Z term in the Hamiltonian. (2nd row) The time dilation required to implement the virtual Z term without introducing it to the Hamiltonian. (3rd and 4th rows) The distorted two- and one-qubit control pulses. pic.twitter.com/ONgWLGOGR0

— Chris Long (@Chris_K_Long45) September 18, 2025

References

[1]

Christopher K. Long and Crispin H. W. Barnes. From virtual Z gates to virtual Z pulses source code, (Zenodo, 2025), DOI: 10.5281/zenodo.17123422. ↩

[2]

Christopher K. Long, Crispin H. W. Barnes, and Normann Mertig. PySTE, (Zenodo, 2025–2026), DOI: 10.5281/zenodo.17116431. ↩

[3]

Christopher K. Long, Crispin H. W. Barnes, and Normann Mertig. Suzuki-Trotter-Evolver, (Zenodo, 2025–2026), DOI: 10.5281/zenodo.17116329. ↩

[4]

Christopher K. Long and Crispin H. W. Barnes. From virtual Z gates to virtual Z pulses data, (Zenodo, 2025), DOI: 10.5281/zenodo.17113740. ↩