Solving lattice gauge theories using the quantum Krylov algorithm and qubitization

Lewis W. Anderson; Martin Kiffner; Tom O'Leary; Jason Crain; Dieter Jaksch

doi:10.22331/q-2025-03-25-1669

Solving lattice gauge theories using the quantum Krylov algorithm and qubitization

Lewis W. Anderson^1,2, Martin Kiffner^1,3, Tom O'Leary¹, Jason Crain^4,1, and Dieter Jaksch^5,1

¹Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK
²IBM Research Europe, Hursley, Winchester SO21 2JN, UK
³PlanQC GmbH, Lichtenbergstr. 8, 85748 Garching, Germany
⁴IBM Research Europe, The Hartree Centre STFC Laboratory, Sci-Tech Daresbury, Warrington WA4 4AD, UK
⁵Institut für Quantenphysik, Universität Hamburg, 22761 Hamburg, Germany

Published:	2025-03-25, volume 9, page 1669
Editor:	Carlo Beenakker
Eprint:	arXiv:2403.08859v4
Doi:	https://doi.org/10.22331/q-2025-03-25-1669
Citation:	Quantum 9, 1669 (2025).

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Abstract

Computing vacuum states of lattice gauge theories (LGTs) containing fermionic degrees of freedom can present significant challenges for classical computation using Monte-Carlo methods. Quantum algorithms may offer a pathway towards more scalable computation of groundstate properties of LGTs. However, a comprehensive understanding of the quantum computational resources required for such a problem is thus far lacking. In this work, we investigate using the quantum subspace expansion (QSE) algorithm to compute the groundstate of the Schwinger model, an archetypal LGT describing quantum electrodynamics in one spatial dimension. We perform numerical simulations, including the effect of measurement noise, to extrapolate the resources required for the QSE algorithm to achieve a desired accuracy for a range of system sizes. Using this, we present a full analysis of the resources required to compute LGT vacuum states using a quantum algorithm using qubitization within a fault tolerant framework. We develop of a novel method for performing qubitization of a LGT Hamiltonian based on a 'linear combination of unitaries' (LCU) approach. The cost of the corresponding block encoding operation scales as $\tilde{O}(N)$ with system size $N$. Including the corresponding prefactors, our method reduces the gate cost by multiple orders of magnitude when compared to previous LCU methods for the QSE algorithm, which scales as $\tilde{O}(N^2)$ when applied to the Schwinger model. While the qubit and single circuit T-gate cost resulting from our resource analysis is appealing to early fault-tolerant implementation, we find that the number of shots required to avoid numerical instability within the QSE procedure must be significantly reduced in order to improve the feasibility of the methodology we consider and discuss how this might be achieved.

Featured image: Diagram of Schwinger model showing physical state satisfying Gauss’s law (Top left), circuit implementing polynomial function of Hamiltonian (bottom left) and schematic of partitioned quantum subspace expansion algorithm (right).

Popular summary

Lattice gauge theories (LGTs) are quantum field theories defined on a discrete set of points. They are used model a broad range of phenomena within high energy particle physics and condensed matter physics. Despite historical success in computing properties of some lattice theories, many LGTs have features that mean classical computers have been unable to solve them. Quantum computing provides a possible avenue for being able to solve such models.

In this work, we consider the resources required for solving a simple LGT, namely the Schwinger model, representing quantum electrodynamics in one-spatial dimension using a quantum computer. We investigate how one could compute the vacuum state of the Schwinger model using an algorithm called quantum subspace expansion and calculate the size and number of resources required by a quantum computer to do so. As well as developing fundamental algorithmic improvements for studying LGTs, our work provides a benchmark by which future studies can compare the estimated resource requirements for studying LGTs and a stepping-stone towards quantum algorithms for more complex models.

► BibTeX data

@article{Anderson2025solvinglatticegauge,
  doi = {10.22331/q-2025-03-25-1669},
  url = {https://doi.org/10.22331/q-2025-03-25-1669},
  title = {Solving lattice gauge theories using the quantum {K}rylov algorithm and qubitization},
  author = {Anderson, Lewis W. and Kiffner, Martin and O'Leary, Tom and Crain, Jason and Jaksch, Dieter},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1669},
  month = mar,
  year = {2025}
}

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