General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

Zohreh Davoudi; Alexander F. Shaw; Jesse R. Stryker

doi:10.22331/q-2023-12-20-1213

General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

Zohreh Davoudi^1,2,3,4, Alexander F. Shaw^1,3, and Jesse R. Stryker^1,2,5

¹Department of Physics, University of Maryland, College Park, MD 20742, USA
²Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA
³Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, College Park, MD 20742, USA
⁴The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA
⁵Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Published:	2023-12-20, volume 7, page 1213
Eprint:	arXiv:2212.14030v3
Doi:	https://doi.org/10.22331/q-2023-12-20-1213
Citation:	Quantum 7, 1213 (2023).

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Abstract

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The loop-string-hadron formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantumly simulating other complex theories of relevance to nature.

Featured image: Some ingredients of the quantum simulation of SU(2) gauge theory in 1+1 dimensions, in its loop-string-hadron formulation. (Left) The elementary degrees of freedom at each lattice site. (Right) A high-level representation of the subroutines involved with simulating the interaction of charged fermions via SU(2) gauge fields.

Popular summary

Non-Abelian gauge theories describe strong and weak interactions in nature. Simulating dynamics of strongly-interacting matter starting from such underlying gauge-theory frameworks is an exciting application of quantum simulators and quantum computers. With a focus on digital quantum computation, we have analyzed, in depth, the resource requirements for evolving a system of fermionic matter coupled to non-Abelian gauge bosons in 1+1 spacetime dimensions. To this end, we have taken into account efficient choices of the model's representation for mapping to discrete degrees of freedom of the quantum computer, proposed wiser decomposition of time evolution operation into smaller operations by preserving as many symmetries of the model as possible, and kept track of the systematic uncertainties introduced by algorithmic approximations. This has led to complete algorithms with bounded errors, and concrete circuit constructions, for both near- and far-term era of quantum computing. Importantly, our general strategies, including the subalgorithms developed, are shown to be applicable to a larger class of physical models, including more complex gauge theories and beyond.

► BibTeX data

@article{Davoudi2023generalquantum,
  doi = {10.22331/q-2023-12-20-1213},
  url = {https://doi.org/10.22331/q-2023-12-20-1213},
  title = {General quantum algorithms for {H}amiltonian simulation with applications to a non-{A}belian lattice gauge theory},
  author = {Davoudi, Zohreh and Shaw, Alexander F. and Stryker, Jesse R.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1213},
  month = dec,
  year = {2023}
}

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