Work

Hadamard-Pi: Equational Quantum Programming

Year: 2026

Type: article

Abstract: Quantum computing offers advantages over classical computation, yet the precise features that set the two apart remain unclear. In the standard quantum circuit model, adding a 1-qubit basis-changing gate—commonly chosen to be the Hadamard gate—to a universal set of classical reversible gates yields computationally universal quantum computation. However, the computational behaviours enabled by this addition are not fully characterised. We give such a characterisation by introducing a small quantum programming language extending the universal classical reversible programming language Π with a single primitive corresponding to the Hadamard gate. The language comes equipped with a sound and complete categorical semantics that is specified by a purely equational theory. Completeness is shown by means of a novel finite presentation, and a corresponding synthesis algorithm, for the groups of orthogonal matrices with entries in the ring z [ 1 2 ] .

Source: Proceedings of the ACM on Programming Languages

DOI: 10.1145/3776647

Authors: Wang Fang, Chris Heunen, Robin Kaarsgaard

Institutions: University of Edinburgh, University of Southern Denmark

Language: English

Topic: Quantum Computing Algorithms and Architecture

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

Funder: Engineering and Physical Sciences Research Council

Award: Rubber DUQ: Flexible Dynamic Universal Quantum programming