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However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of [<\/mml:mo>[<\/mml:mo>N<\/mml:mi>,<\/mml:mo>K<\/mml:mi>,<\/mml:mo>D<\/mml:mi>]<\/mml:mo>]<\/mml:mo><\/mml:math> codes, referred to as hierarchical codes, that encode a number of logical qubits K<\/mml:mi>=<\/mml:mo>&#x03A9;<\/mml:mi>(<\/mml:mo>N<\/mml:mi>\/<\/mml:mo><\/mml:mrow>log<\/mml:mi>&#x2061;<\/mml:mo>(<\/mml:mo>N<\/mml:mi>)<\/mml:mo>2<\/mml:mn><\/mml:msup>)<\/mml:mo><\/mml:math>. The N<\/mml:mi>t<\/mml:mi>h<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math> element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance D<\/mml:mi>(<\/mml:mo>N<\/mml:mi>)<\/mml:mo><\/mml:math>. We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.<\/jats:p>","DOI":"10.22331\/q-2025-05-05-1728","type":"journal-article","created":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T10:31:39Z","timestamp":1746441099000},"page":"1728","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":["Hierarchical memories: Simulating quantum LDPC codes with local gates"],"prefix":"10.22331","volume":"9","author":[{"given":"Christopher A.","family":"Pattison","sequence":"first","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Stanford University, Stanford, CA, 94305"},{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125"},{"name":"AWS Center for Quantum Computing, Pasadena CA 91125"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,5,5]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"F. 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