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Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic {<\/mml:mo>p<\/mml:mi>,<\/mml:mo>q<\/mml:mi>}<\/mml:mo><\/mml:math> geometry with even q<\/mml:mi>&#x2265;<\/mml:mo>4<\/mml:mn><\/mml:math>, yielding an infinitely large class of subsystem codes. We construct an example for a {<\/mml:mo>5<\/mml:mn>,<\/mml:mo>4<\/mml:mn>}<\/mml:mo><\/mml:math> manifold and describe strategies of logical gauge fixing that lead to different rates k<\/mml:mi>\/<\/mml:mo><\/mml:mrow>n<\/mml:mi><\/mml:math> and distances d<\/mml:mi><\/mml:math>, which we calculate analytically, finding distances which range from d<\/mml:mi>=<\/mml:mo>2<\/mml:mn><\/mml:math> to d<\/mml:mi>&#x223C;<\/mml:mo>n<\/mml:mi>2<\/mml:mn>\/<\/mml:mo><\/mml:mrow>3<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with k<\/mml:mi>\/<\/mml:mo><\/mml:mrow>n<\/mml:mi>=<\/mml:mo>0.125<\/mml:mn><\/mml:math>, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk\/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.<\/jats:p>","DOI":"10.22331\/q-2025-08-08-1826","type":"journal-article","created":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T07:31:42Z","timestamp":1756279902000},"page":"1826","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3348-7380","authenticated-orcid":false,"given":"Matthew","family":"Steinberg","sequence":"first","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-3765-2748","authenticated-orcid":false,"given":"Junyu","family":"Fan","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8838-6299","authenticated-orcid":false,"given":"Robert J.","family":"Harris","sequence":"additional","affiliation":[{"name":"ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2023-2768","authenticated-orcid":false,"given":"David","family":"Elkouss","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2782-1469","authenticated-orcid":false,"given":"Sebastian","family":"Feld","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7142-0059","authenticated-orcid":false,"given":"Alexander","family":"Jahn","sequence":"additional","affiliation":[{"name":"Department of Physics, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2025,8,8]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"John Preskill. ``Quantum Computing in the NISQ era and beyond''. 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