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  • Open Access

Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits

Yuanjie Ren1, Nathanan Tantivasadakarn2, and Dominic J. Williamson3,*

  • *Present address: IBM Quantum, IBM Almaden Research Center, San Jose, California 95120, USA.
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Phys. Rev. X 15, 031060 – Published 29 August, 2025

DOI: https://doi.org/10.1103/b9hf-gx4f

Abstract

The classification of topological phases of matter is a fundamental challenge in quantum many-body physics, with applications to quantum technology. Recently, this classification has been extended to the setting of adaptive finite-depth local unitary (AFDLU) circuits, which allow global classical communication. In this setting, the trivial phase is the collection of all topological states that can be prepared via AFDLU. Here, we propose a complete classification of the trivial phase by showing how to prepare all solvable anyon theories that admit a gapped boundary via AFDLU, extending recent results on solvable groups. Our construction includes non-Abelian anyons with irrational quantum dimensions, such as Ising anyons, and more general acyclic anyons. Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase described by a solvable anyon theory with gapped boundary. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons and symmetry twist defects in solvable anyon theories. We apply our procedure to the quantum double of the group S3 and to several examples that are beyond solvable groups, including the doubled Ising theory, the Z3 Tambara-Yamagami string net, and doubled SU(2)4 anyons.

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Popular Summary

Topological phases of matter hold great promise for building robust quantum technologies, including fault-tolerant quantum computers. These phases host unusual particles called anyons, which can store and process information in ways that are naturally protected from errors. However, despite their potential, preparing and controlling these states on today’s quantum devices has remained a significant challenge. In this work, we present a universal method that uses adaptive quantum circuits to efficiently create and manipulate a broad class of anyons, known as solvable anyons, in constant time. We propose that this method offers the first comprehensive road map for generating all topological states accessible through such circuits.

Our approach builds on recent advances in classifying topological phases using adaptive, finite-depth quantum circuits—incorporating real-world capabilities like measurements and classical communication. Starting with basic Abelian topological states, we introduce a sequence of steps involving “gauging” and “ungauging” symmetries to design circuits that can handle more complex, non-Abelian anyons. These include string-net models—where particles emerge from fluctuating networks of strings—with rare and intricate features such as irrational quantum dimensions and cyclic fusion rules, properties beyond the reach of previous techniques.

This work provides a concrete theoretical foundation for simulating advanced topological phases on current quantum platforms, such as trapped-ion systems. It also introduces a key conjecture: Solvable anyons represent the complete set of topological phases that can be prepared in constant time. If true, this insight could guide future experiments and applications in quantum simulation, error correction, and computing, offering new ways to realize and control these exotic states of matter.

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Phys. Rev. X 15, 031060– Published 29 August, 2025
  • Received 15 January 2025
  • Revised 21 May 2025
  • Accepted 11 July 2025
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