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We identify the logical operations on this gauge covariant code and show that the corresponding Hamiltonian can be expressed in terms of these logical operations while preserving the locality of the interactions. Furthermore, we demonstrate that these substitutions actually give a new way of writing the LGT as an equivalent hardcore boson model. Finally we demonstrate a method to perform fault-tolerant time evolution of the Hamiltonian within the gauge covariant code using both product formulas and qubitization approaches. This opens up the possibility of inexpensive end to end dynamical simulations that save physical qubits by blurring the lines between simulation algorithms and quantum error correcting codes.\n                  <\/jats:p>","DOI":"10.22331\/q-2026-01-16-1968","type":"journal-article","created":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T13:58:26Z","timestamp":1768571906000},"page":"1968","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes"],"prefix":"10.22331","volume":"10","author":[{"given":"L.","family":"Spagnoli","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, University of Trento, via Sommarive 14, I\u201338123, Povo, Trento, Italy"},{"name":"INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Roggero","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, University of Trento, via Sommarive 14, I\u201338123, Povo, Trento, Italy"},{"name":"INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.","family":"Wiebe","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Toronto, Toronto, ON M5S 2E4, Canada"},{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"},{"name":"Department of Physics, University of Washington, Seattle, WA 98195, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,1,16]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"S. 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Zoller, ``Simulating lattice gauge theories within quantum technologies,&apos;&apos; The European Physical Journal D, vol. 74, no. 8, Aug. 2020. [Online]. Available: http:\/\/dx.doi.org\/10.1140\/epjd\/e2020-100571-8 0pt.","DOI":"10.1140\/epjd\/e2020-100571-8"},{"key":"3","doi-asserted-by":"publisher","unstructured":"E. Zohar, ``Quantum simulation of lattice gauge theories in more than one space dimension\u2014requirements, challenges and methods,&apos;&apos; Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 380, no. 2216, Dec. 2021. [Online]. Available: http:\/\/dx.doi.org\/10.1098\/rsta.2021.0069 0pt.","DOI":"10.1098\/rsta.2021.0069"},{"key":"4","doi-asserted-by":"publisher","unstructured":"N. Klco, A. Roggero, and M. J. Savage, ``Standard model physics and the digital quantum revolution: thoughts about the interface,&apos;&apos; Reports on Progress in Physics, vol. 85, no. 6, p. 064301, May 2022. [Online]. Available: http:\/\/dx.doi.org\/10.1088\/1361-6633\/ac58a4 0pt.","DOI":"10.1088\/1361-6633\/ac58a4"},{"key":"5","doi-asserted-by":"publisher","unstructured":"C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhattacharya, M. Carena, W. A. de Jong, P. Draper, A. El-Khadra, N. Gemelke, M. Hanada, D. Kharzeev, H. Lamm, Y.-Y. Li, J. Liu, M. Lukin, Y. Meurice, C. Monroe, B. Nachman, G. Pagano, J. Preskill, E. Rinaldi, A. Roggero, D. I. Santiago, M. J. Savage, I. Siddiqi, G. Siopsis, D. Van Zanten, N. Wiebe, Y. Yamauchi, K. Yeter-Aydeniz, and S. Zorzetti, ``Quantum simulation for high-energy physics,&apos;&apos; PRX Quantum, vol. 4, p. 027001, May 2023. [Online]. Available: https:\/\/doi.org\/10.1103\/PRXQuantum.4.027001 0pt.","DOI":"10.1103\/PRXQuantum.4.027001"},{"key":"6","doi-asserted-by":"publisher","unstructured":"P. W. Shor, ``Scheme for reducing decoherence in quantum computer memory,&apos;&apos; Phys. Rev. A, vol. 52, pp. R2493\u2013R2496, Oct 1995. [Online]. 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