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

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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}<br /> }

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[1] Christian W. Bauer, Zohreh Davoudi, Natalie Klco, and Martin J. Savage, “Quantum simulation of fundamental particles and forces”, Nature Reviews Physics 5 7, 420 (2023).

[2] Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W. Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, Roland de Putter, Andrea Delgado, Vedran Dunjko, Daniel J. Egger, Elias Fernandez-Combarro, Elina Fuchs, Lena Funcke, Daniel Gonzalez-Cuadra, Michele Grossi, Jad C. Halimeh, Zoe Holmes, Stefan Kuhn, Denis Lacroix, Randy Lewis, Donatella Lucchesi, Miriam Lucio Martinez, Federico Meloni, Antonio Mezzacapo, Simone Montangero, Lento Nagano, Voica Radescu, Enrique Rico Ortega, Alessandro Roggero, Julian Schuhmacher, Joao Seixas, Pietro Silvi, Panagiotis Spentzouris, Francesco Tacchino, Kristan Temme, Koji Terashi, Jordi Tura, Cenk Tuysuz, Sofia Vallecorsa, Uwe-Jens Wiese, Shinjae Yoo, and Jinglei Zhang, “Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group”, arXiv:2307.03236, (2023).

[3] Niklas Mueller, Joseph A. Carolan, Andrew Connelly, Zohreh Davoudi, Eugene F. Dumitrescu, and Kübra Yeter-Aydeniz, “Quantum Computation of Dynamical Quantum Phase Transitions and Entanglement Tomography in a Lattice Gauge Theory”, PRX Quantum 4 3, 030323 (2023).

[4] Torsten V. Zache, Daniel González-Cuadra, and Peter Zoller, “Quantum and Classical Spin-Network Algorithms for q -Deformed Kogut-Susskind Gauge Theories”, Physical Review Letters 131 17, 171902 (2023).

[5] Simone Romiti and Carsten Urbach, “Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations”, arXiv:2311.11928, (2023).

[6] Tomoya Hayata and Yoshimasa Hidaka, “String-net formulation of Hamiltonian lattice Yang-Mills theories and quantum many-body scars in a nonabelian gauge theory”, Journal of High Energy Physics 2023 9, 126 (2023).

[7] Raghav G. Jha, Felix Ringer, George Siopsis, and Shane Thompson, “Continuous variable quantum computation of the $O(3)$ model in 1+1 dimensions”, arXiv:2310.12512, (2023).

[8] Lento Nagano, Aniruddha Bapat, and Christian W. Bauer, “Quench dynamics of the Schwinger model via variational quantum algorithms”, Physical Review D 108 3, 034501 (2023).

[9] Berndt Müller and Xiaojun Yao, “Simple Hamiltonian for quantum simulation of strongly coupled (2 +1 )D SU(2) lattice gauge theory on a honeycomb lattice”, Physical Review D 108 9, 094505 (2023).

[10] Anthony N. Ciavarella, “Quantum simulation of lattice QCD with improved Hamiltonians”, Physical Review D 108 9, 094513 (2023).

[11] Xiaojun Yao, “SU(2) gauge theory in 2 +1 dimensions on a plaquette chain obeys the eigenstate thermalization hypothesis”, Physical Review D 108 3, L031504 (2023).

[12] S. V. Kadam, I. Raychowdhury, and J. Stryker, “Loop-string-hadron formulation of an SU(3) gauge theory with dynamical quarks”, The 39th International Symposium on Lattice Field Theory, 373 (2023).

[13] Timo Jakobs, Marco Garofalo, Tobias Hartung, Karl Jansen, Johann Ostmeyer, Dominik Rolfes, Simone Romiti, and Carsten Urbach, “Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory”, European Physical Journal C 83 7, 669 (2023).

[14] Marco Rigobello, Giuseppe Magnifico, Pietro Silvi, and Simone Montangero, “Hadrons in (1+1)D Hamiltonian hardcore lattice QCD”, arXiv:2308.04488, (2023).

[15] Andrei Alexandru, Paulo F. Bedaque, Andrea Carosso, Michael J. Cervia, Edison M. Murairi, and Andy Sheng, “Fuzzy Gauge Theory for Quantum Computers”, arXiv:2308.05253, (2023).

[16] Saurabh V. Kadam, Indrakshi Raychowdhury, and Jesse R. Stryker, “Loop-string-hadron formulation of an SU(3) gauge theory with dynamical quarks”, Physical Review D 107 9, 094513 (2023).

[17] Kyle Lee, James Mulligan, Felix Ringer, and Xiaojun Yao, “Liouvillian dynamics of the open Schwinger model: String breaking and kinetic dissipation in a thermal medium”, Physical Review D 108 9, 094518 (2023).

[18] Manu Mathur and Atul Rathor, “Exact duality and local dynamics in SU(N) lattice gauge theory”, arXiv:2109.00992, (2021).

[19] Marco Garofalo, Tobias Hartung, Timo Jakobs, Karl Jansen, Johann Ostmeyer, Dominik Rolfes, Simone Romiti, and Carsten Urbach, “Testing the $mathrm{SU}(2)$ lattice Hamiltonian built from $S_3$ partitionings”, arXiv:2311.15926, (2023).

[20] Manu Mathur and Atul Rathor, “Exact duality and local dynamics in SU(N) lattice gauge theory”, Physical Review D 107 7, 074504 (2023).

[21] Christopher Brown, Michael Spannowsky, Alexander Tapper, Simon Williams, and Ioannis Xiotidis, “Quantum Pathways for Charged Track Finding in High-Energy Collisions”, arXiv:2311.00766, (2023).

[22] Saurabh V. Kadam, “Theoretical Developments in Lattice Gauge Theory for Applications in Double-beta Decay Processes and Quantum Simulation”, arXiv:2312.00780, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-12-21 04:00:36). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-12-21 04:00:34).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Source: https://quantum-journal.org/papers/q-2023-12-20-1213/

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

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