{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T02:22:32Z","timestamp":1775874152748,"version":"3.50.1"},"reference-count":13,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,10,19]],"date-time":"2021-10-19T00:00:00Z","timestamp":1634601600000},"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 present a quantum error correcting code with dynamically generated logical qubits<\/mml:mtext><\/mml:mrow><\/mml:math>. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a t<\/mml:mi>w<\/mml:mi>o<\/mml:mi><\/mml:math>-qubit Pauli measurement.<\/jats:p>","DOI":"10.22331\/q-2021-10-19-564","type":"journal-article","created":{"date-parts":[[2021,10,19]],"date-time":"2021-10-19T14:48:25Z","timestamp":1634654905000},"page":"564","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":140,"title":["Dynamically Generated Logical Qubits"],"prefix":"10.22331","volume":"5","author":[{"given":"Matthew B.","family":"Hastings","sequence":"first","affiliation":[{"name":"Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA"},{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,10,19]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. Kitaev, ``Fault-tolerant quantum computation by anyons,'' Annals of Physics 303, 2\u201330 (2003), arXiv:quant-ph\/9707021.","DOI":"10.1016\/s0003-4916(02)00018-0"},{"key":"1","doi-asserted-by":"publisher","unstructured":"D. Poulin, ``Stabilizer formalism for operator quantum error correction,'' Physical Review Letters 95, 230504 (2005), arXiv:quant-ph\/0508131.","DOI":"10.1103\/physrevlett.95.230504"},{"key":"2","doi-asserted-by":"publisher","unstructured":"S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, ``Subsystem surface codes with three-qubit check operators,'' Quantum Information and Computation 13, 963\u2013985 (2013), arXiv:1207.1443.","DOI":"10.26421\/qic13.11-12-4"},{"key":"3","doi-asserted-by":"publisher","unstructured":"H. Bombin, ``Topological subsystem codes,'' Physical Review A 81, 032301 (2010), arXiv:0908.4246.","DOI":"10.1103\/physreva.81.032301"},{"key":"4","doi-asserted-by":"publisher","unstructured":"D. Bacon, ``Operator quantum error-correcting subsystems for self-correcting quantum memories,'' Physical Review A 73, 012340 (2006), arXiv:quant-ph\/0506023.","DOI":"10.1103\/physreva.73.012340"},{"key":"5","doi-asserted-by":"publisher","unstructured":"T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, ``Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majorana zero modes,'' Physical Review B 95, 235305 (2017), arXiv:1610.05289.","DOI":"10.1103\/physrevb.95.235305"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Y. Li, X. Chen, and M. P. A. Fisher, ``Quantum zeno effect and the many-body entanglement transition,'' Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134.","DOI":"10.1103\/PhysRevB.98.205136"},{"key":"7","doi-asserted-by":"publisher","unstructured":"B. Skinner, J. Ruhman, and A. 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Sethna, ``2e or not 2e : Flux quantization in the resonating valence bond state,'' Europhysics Letters (EPL) 6, 353\u2013358 (1988).","DOI":"10.1209\/0295-5075\/6\/4\/013"},{"key":"12","doi-asserted-by":"publisher","unstructured":"L. Fidkowski, J. Haah, and M. B. 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