{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T00:23:43Z","timestamp":1773966223691,"version":"3.50.1"},"reference-count":56,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T00:00:00Z","timestamp":1493078400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits n<\/mml:mi><\/mml:math>, the number of encoded qubits k<\/mml:mi><\/mml:math>, the code distance d<\/mml:mi><\/mml:math>, the accuracy parameter &#x03B4;<\/mml:mi><\/mml:math> that quantifies how well the erasure channel can be reversed, and the locality parameter &#x2113;<\/mml:mi><\/mml:math> that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy &#x03B4;<\/mml:mi><\/mml:math> that is exponentially small in &#x2113;<\/mml:mi><\/mml:math>, which is the case for perturbations of local commuting projector codes, our bound reads k<\/mml:mi>d<\/mml:mi>2<\/mml:mn>D<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>&#x2264;<\/mml:mo>O<\/mml:mi>(<\/mml:mo><\/mml:mrow>n<\/mml:mi>(<\/mml:mo>log<\/mml:mi>&#x2061;<\/mml:mo>n<\/mml:mi>)<\/mml:mo>2<\/mml:mn>D<\/mml:mi><\/mml:mrow>D<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math> for codes on D<\/mml:mi><\/mml:math>-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed O<\/mml:mi>(<\/mml:mo><\/mml:mrow>&#x2113;<\/mml:mi>n<\/mml:mi>(<\/mml:mo>D<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn>)<\/mml:mo>\/<\/mml:mo><\/mml:mrow>D<\/mml:mi><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math> if &#x03B4;<\/mml:mi>&#x2264;<\/mml:mo>O<\/mml:mi>(<\/mml:mo>&#x2113;<\/mml:mi>n<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>D<\/mml:mi><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:math>. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance d<\/mml:mi><\/mml:math> and the size d<\/mml:mi>&#x007E;<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math> of a minimal region that can support all approximate logical operators satisfies d<\/mml:mi>&#x007E;<\/mml:mo><\/mml:mover><\/mml:mrow>d<\/mml:mi>1<\/mml:mn>D<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>&#x2264;<\/mml:mo>O<\/mml:mi>(<\/mml:mo><\/mml:mrow>n<\/mml:mi>&#x2113;<\/mml:mi>D<\/mml:mi>D<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math>, where the logical operators are accurate up to O<\/mml:mi>(<\/mml:mo><\/mml:mrow>(<\/mml:mo>n<\/mml:mi>&#x03B4;<\/mml:mi>\/<\/mml:mo><\/mml:mrow>d<\/mml:mi>)<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math> in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.<\/jats:p>","DOI":"10.22331\/q-2017-04-25-4","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:51:31Z","timestamp":1493110291000},"page":"4","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":27,"title":["Limits on the storage of quantum information in a volume of space"],"prefix":"10.22331","volume":"1","author":[{"given":"Steven T.","family":"Flammia","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington, USA"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Michael J.","family":"Kastoryano","sequence":"additional","affiliation":[{"name":"NBIA, Niels Bohr Institute, University of Copenhagen, Denmark"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM T. J. Watson Research Center, Yorktown Heights, New York, USA"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo ON N2L 3G1, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2017,4,25]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"B. M. Terhal, ``Quantum error correction for quantum memories,'' Rev. Mod. Phys. 87, 307 (2015), arXiv:1302.3428.","DOI":"10.1103\/RevModPhys.87.307"},{"key":"1","doi-asserted-by":"crossref","unstructured":"D. Gottesman, ``An introduction to quantum error correction and fault-tolerant quantum computation,'' in Quantum Information Science and Its Contributions to Mathematics, Vol. 68, edited by S. J. Lomonaco, Jr. (American Mathematical Society, 2010) pp. 24\u201369, arXiv:0904.2557.","DOI":"10.1090\/psapm\/068\/2762145"},{"key":"2","doi-asserted-by":"publisher","unstructured":"D. W. Leung, M. A. Nielsen, I. L. 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