---
tags: Hamiltonian-level, Theory
sort-date: 2025-09-16
description: We develop a theory of virtual Z rotations at the Hamiltonian level. This allows Z terms to be added to the Hamiltonian at no additional cost.
---
[[/index|Christopher K. Long]]
and Crispin H. W. Barnes
Preprint published: 16 September 2025 18:37:59 UTC
DOI: [10.48550/arXiv.2509.13453](https://doi.org/10.48550/arXiv.2509.13453)
[[PDFs/2509.13453v1.pdf|PDF Download]], [[TeX_Source/2509.13453v1.tar.gz|TeX Source Download]]
# 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 $X$, $\operatorname{SWAP}$, 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
```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
- [[/Software/From-virtual-Z-gates-to-virtual-Z-pulses-source-code/index|From-virtual-Z-gates-to-virtual-Z-pulses-source-code]] [@long_2025_17149989]
- [[/Software/PySTE/index|PySTE]] [@PySTE]
- [[/Software/Suzuki-Trotter-Evolver/index|Suzuki-Trotter-Evolver]] [@SuzukiTrotterEvolver]
# Data
- [[/Datasets/From-virtual-Z-gates-to-virtual-Z-pulses-data]] [@long_2025_17113740]
# Analytics
- [Google Scholar](https://scholar.google.com/citations?view_op=view_citation&citation_for_view=GRSIcsEAAAAJ:_FxGoFyzp5QC)
- [SciRate](https://scirate.com/arxiv/2509.13453)
- [INSPIRE-HEP](https://inspirehep.net/literature/2970599)
- [Semantic Scholar](https://www.semanticscholar.org/paper/From-virtual-Z-gates-to-virtual-Z-pulses-Long-Barnes/cf6b7a932980a8ea37dfb9f0acf5b266c6260a10)
# 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]
>
>