Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes
It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however...
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| Format: | Article |
| Language: | English |
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American Physical Society
2016
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| Online Access: | http://hdl.handle.net/1721.1/104631 https://orcid.org/0000-0001-9614-2836 https://orcid.org/0000-0001-7296-523X |
| _version_ | 1826192973969227776 |
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| author | Yoder, Theodore James Takagi, Ryuji Chuang, Isaac |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Yoder, Theodore James Takagi, Ryuji Chuang, Isaac |
| author_sort | Yoder, Theodore James |
| collection | MIT |
| description | It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching. |
| first_indexed | 2024-09-23T09:32:01Z |
| format | Article |
| id | mit-1721.1/104631 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:32:01Z |
| publishDate | 2016 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/1046312022-09-30T15:05:20Z Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes Yoder, Theodore James Takagi, Ryuji Chuang, Isaac Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Department of Physics Yoder, Theodore James Takagi, Ryuji Chuang, Isaac It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching. National Science Foundation (U.S.) (RQCC Project 1111337) United States. Army Research Office American Society for Engineering Education. National Defense Science and Engineering Graduate Fellowship Takenaka Scholarship MIT Department of Physics (Frank Fellowship) 2016-09-30T21:40:10Z 2016-09-30T21:40:10Z 2016-09 2016-03 2016-09-13T22:00:07Z Article http://purl.org/eprint/type/JournalArticle 2160-3308 http://hdl.handle.net/1721.1/104631 Yoder, Theodore J., Ryuji Takagi, and Isaac L. Chuang. “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes.” Physical Review X 6.3 (2016): n. pag. https://orcid.org/0000-0001-9614-2836 https://orcid.org/0000-0001-7296-523X en http://dx.doi.org/10.1103/PhysRevX.6.031039 Physical Review X Creative Commons Attribution http://creativecommons.org/licenses/by/3.0 authors application/pdf American Physical Society American Physical Society |
| spellingShingle | Yoder, Theodore James Takagi, Ryuji Chuang, Isaac Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title | Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title_full | Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title_fullStr | Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title_full_unstemmed | Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title_short | Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes |
| title_sort | universal fault tolerant gates on concatenated stabilizer codes |
| url | http://hdl.handle.net/1721.1/104631 https://orcid.org/0000-0001-9614-2836 https://orcid.org/0000-0001-7296-523X |
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