Quantum Circuits Are Just a Phase
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3776731.pdf
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880.99 KB
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Author(s)
Heunen, Chris
Lemonnier, Louis
McNally, Christopher
Rice, Alex
Date Issued
January 8, 2026
Journal
Proceedings of the ACM on Programming Languages
Publisher
ACM
Citation
Chris Heunen, Louis Lemonnier, Christopher McNally, and Alex Rice. 2026. Quantum Circuits Are Just a Phase. Proc. ACM Program. Lang. 10, POPL, Article 89 (January 2026), 28 pages.
Version
Final published version
Abstract
Quantum programs today are written at a low level of abstraction---quantum circuits akin to assembly languages - and the unitary parts of even advanced quantum programming languages essentially function as circuit description languages. This state of affairs impedes scalability, clarity, and support for higher-level reasoning. More abstract and expressive quantum programming constructs are needed.
To this end, we introduce a simple syntax for generating unitaries from "just a phase"; we combine a (global) phase operation that captures phase shifts with a quantum analogue of the "if let" construct that captures subspace selection via pattern matching. This minimal language lifts the focus from gates to eigendecomposition, conjugation, and controlled unitaries; common building blocks in quantum algorithm design.
We demonstrate several aspects of the expressive power of our language in several ways. Firstly, we establish that our representation is universal by deriving a universal quantum gate set. Secondly, we show that important quantum algorithms can be expressed naturally and concisely, including Grover's search algorithm, Hamiltonian simulation, Quantum Fourier Transform, Quantum Signal Processing, and the Quantum Eigenvalue Transformation. Furthermore, we give clean denotational semantics grounded in categorical quantum mechanics. Finally, we implement a prototype compiler that efficiently translates terms of our language to quantum circuits, and prove that it is sound with respect to these semantics. Collectively, these contributions show that this construct offers a principled and practical step toward more abstract and structured quantum programming.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3776731
