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Measurement-Based Quantum Computation in Symmetry-Enriched Topological Phases

Paul Herringer1,2,3,*, Vir B. Bulchandani4,5, Younes Javanmard1, David T. Stephen6,7,†, and Robert Raussendorf1,3

  • *Contact author: paul.herringer@itp.uni-hannover.de
  • †Present Address: Quantinuum, 303 S Technology Ct, Broomfield, Colorado 80021, USA
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PRX Quantum 6, 040314 – Published 21 October, 2025

DOI: https://doi.org/10.1103/j2z3-s6d6

Abstract

We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation (MBQC). This is possible due to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which have been limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit noncomputationally universal, application of our framework. We then present a new model with topological order the ground states of which are universal resources for MBQC. Both topological models are enriched by subsystem symmetries and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.

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Popular Summary

Measurement-based quantum computation (MBQC) is a way to perform quantum computations by measuring single particles in an entangled many-particle quantum state. In MBQC, the quantum state being measured is the source of computational power, but understanding the link between this power and the physical properties of quantum states remains a significant challenge. Over the past decade, it has been discovered that the power of MBQC remains constant across certain quantum phases of matter, connecting MBQC with condensed-matter physics. However, these “computational phases” have previously been found only in systems with short-range entanglement, which is limiting because many interesting physical systems feature long-range entanglement, also known as topological order.

In this paper, we have developed a framework that allows us to study computational phases in systems with topological order, significantly widening the range of systems that can be analyzed from the MBQC perspective. Our paper presents the first examples of computational phases in topologically ordered systems, including a “universal” phase in which any quantum circuit can be realized using MBQC. We also explore how computational and topological properties are related and demonstrate that the computational properties of a quantum state tell us more about it than topological properties alone. Our work strengthens the connection between MBQC and condensed-matter physics and expands our understanding of how many-body quantum states can be harnessed for quantum computation.

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PRX Quantum 6, 040314– Published 21 October, 2025
  • Received 19 April 2025
  • Revised 11 July 2025
  • Accepted 13 August 2025
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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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