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In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.<\/jats:p>","DOI":"10.22331\/q-2024-10-18-1501","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T12:48:31Z","timestamp":1729255711000},"page":"1501","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Quantum Locally Testable Code with Constant Soundness"],"prefix":"10.22331","volume":"8","author":[{"given":"Andrew","family":"Cross","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiyang","family":"He","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."},{"name":"Department of Mathematics, Massachusetts Institute of Technology."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anand","family":"Natarajan","sequence":"additional","affiliation":[{"name":"Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mario","family":"Szegedy","sequence":"additional","affiliation":[{"name":"Rutgers, The State University of New Jersey, New Brunswick, NJ, United States."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2024,10,18]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Dorit Aharonov, Itai Arad, and Thomas Vidick, ``Guest column: the quantum PCP conjecture'' SIGACT News 44, 47-79 (2013).","DOI":"10.1145\/2491533.2491549"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Anurag Anshu, Nikolas P. 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Rubinfeld, ``Self-testing\/correcting with applications to numerical problems'' Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing 73\u201383 (1990).","DOI":"10.1145\/100216.100225"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Earl T Campbell ``A theory of single-shot error correction for adversarial noise'' Quantum Science and Technology 4, 025006 (2019).","DOI":"10.1088\/2058-9565\/aafc8f"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes, ``Locally testable codes with constant rate, distance, and locality'' Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing 357\u2013374 (2022).","DOI":"10.1145\/3519935.3520024"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick, ``Good Quantum LDPC Codes with Linear Time Decoders'' Proceedings of the 55th Annual ACM Symposium on Theory of Computing 905\u2013918 (2023).","DOI":"10.1145\/3564246.3585101"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Irit Dinur ``The PCP theorem by gap amplification'' J. 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