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Nontrivial Multiproduct Commutation Relation Toward Reducing T-Count in Sequential Pauli-Based Computation

Yusei Mori1,*, Hideaki Hakoshima1,2,†, and Keisuke Fujii1,2,3,‡

  • *Contact author: u839526i@ecs.osaka-u.ac.jp
  • †Contact author: hakoshima.hideaki.es@osaka-u.ac.jp
  • ‡Contact author: fujii.keisuke.es@osaka-u.ac.jp
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PRX Quantum 7, 020345 – Published 1 June, 2026

DOI: https://doi.org/10.1103/qw5z-3bwz

Abstract

Quantum compilers that reduce the number of T gates are essential for minimizing the overhead of fault-tolerant quantum computation. Achieving further T-count reduction calls for identifying equivalent circuit transformation rules beyond those utilized in existing tools. In this paper, we rewrite any given Clifford+T circuit using a Clifford block followed by a sequential Pauli-based computation and introduce a nontrivial, ancilla-free transformation rule, the multiproduct commutation relation (MCR). MCR constructs gate sequences based on specific commutation properties among multi-Pauli operators, yielding seemingly noncommutative instances that can be commuted, thereby enabling gate orderings that cannot be derived from pairwise commutation alone. We also propose the MCR Compiler, which incorporates MCR-based transformations as an optimization pass. To evaluate its effect, we use a benchmark circuit dataset generated through quantum circuit unoptimization. This approach intentionally adds redundancy to the circuit while keeping its equivalence, allowing a quantitative evaluation of compiler performance by comparison with the original circuit. Our numerical experiments reveal that the MCR Compiler achieves further T-count reduction beyond current compilers, establishing MCR-based transformations as a practical optimization primitive. These results highlight an untapped opportunity to enhance the optimization capabilities of quantum compilers.

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Popular Summary

Quantum compilers play a central role in enabling the efficient execution of quantum algorithms on physical quantum hardware. One of their key objectives is the reduction of the T-count as it largely determines the required number of qubits and execution time in fault-tolerant quantum computation. Despite substantial progress, conventional quantum compilers remain limited to local rewrite rules and pairwise commutation relations between gates. As a result, fundamental improvements in compiler performance require nonlocal transformation rules that go beyond existing circuit reasoning.

In this paper, we introduce the multiproduct commutation relation (MCR), a framework that substantially expands the range of equivalent circuit transformations. MCR enables the construction of gate sequences based on specific commutation properties among multi-Pauli rotations, resulting in cases where seemingly noncommutative gates can be commuted. This approach allows for gate orderings that are not accessible through conventional pairwise commutation rules. Numerical experiments indicate that state-of-the-art T-count optimization compilers struggle to simplify circuits that rely on these transformations, whereas an MCR-aware compiler successfully optimizes such circuits. Additionally, MCR enhances optimization performance for practical circuits, including those derived from physically motivated spin models.

Our results suggest that MCR establishes a new paradigm for quantum compilers. In particular, MCR extends the scope of compilation strategies, enabling the recognition and exploitation of hidden circuit transformations that were previously out of reach for existing optimization frameworks. Developing more efficient compilers that systematically leverage MCR-based principles can advance resource-efficient quantum computation toward practical implementation.

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Outline

Information

PRX Quantum 7, 020345– Published 1 June, 2026
  • Received 15 October 2025
  • Revised 19 March 2026
  • Accepted 22 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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