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Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC \"twist defects\" &#x2013;<\/mml:mtext><\/mml:math> anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and \"logical MZMs,\" with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.<\/jats:p>","DOI":"10.22331\/q-2024-07-10-1400","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T13:57:29Z","timestamp":1720619849000},"page":"1400","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation"],"prefix":"10.22331","volume":"8","author":[{"given":"Campbell","family":"McLauchlan","sequence":"first","affiliation":[{"name":"DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK"}]},{"given":"Benjamin","family":"B\u00e9ri","sequence":"additional","affiliation":[{"name":"DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK"},{"name":"T.C.M. 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Kane. ``Superconducting proximity effect and Majorana fermions at the surface of a topological insulator''. Phys. Rev. Lett. 100, 096407 (2008).","DOI":"10.1103\/physrevlett.100.096407"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Yuval Oreg, Gil Refael, and Felix von Oppen. ``Helical liquids and Majorana bound states in quantum wires''. Phys. Rev. Lett. 105, 177002 (2010).","DOI":"10.1103\/PhysRevLett.105.177002"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma. ``Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures''. Phys. Rev. Lett. 105, 077001 (2010).","DOI":"10.1103\/physrevlett.105.077001"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Jason Alicea. ``New directions in the pursuit of Majorana fermions in solid state systems''. Rep. Prog. 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Sci. 475, 20180427 (2019).","DOI":"10.1098\/rspa.2018.0427"},{"key":"73","doi-asserted-by":"publisher","unstructured":"Mitchell Chiew and Sergii Strelchuk. ``Discovering optimal fermion-qubit mappings through algorithmic enumeration''. Quantum 7, 1145 (2023).","DOI":"10.22331\/q-2023-10-18-1145"},{"key":"74","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi. ``Universal quantum computation with the ${\\nu}=5\/2$ fractional quantum hall state''. Phys. Rev. A 73, 042313 (2006).","DOI":"10.1103\/PhysRevA.73.042313"},{"key":"75","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi and Alexei Kitaev. ``Universal quantum computation with ideal Clifford gates and noisy ancillas''. Phys. Rev. A 71, 022316 (2005).","DOI":"10.1103\/physreva.71.022316"},{"key":"76","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi and Jeongwan Haah. ``Magic-state distillation with low overhead''. Phys. Rev. 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