Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates
We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized constr...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2018-06-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2018-06-07-71/pdf/ |
| _version_ | 1819043500343164928 |
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| author | Jeongwan Haah Matthew B. Hastings |
| author_facet | Jeongwan Haah Matthew B. Hastings |
| author_sort | Jeongwan Haah |
| collection | DOAJ |
| description | We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of
. Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes. |
| first_indexed | 2024-12-21T09:57:46Z |
| format | Article |
| id | doaj.art-7e0bde4d670a4accb32fec1f8013e48c |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-12-21T09:57:46Z |
| publishDate | 2018-06-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-7e0bde4d670a4accb32fec1f8013e48c2022-12-21T19:08:00ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2018-06-0127110.22331/q-2018-06-07-7110.22331/q-2018-06-07-71Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli GatesJeongwan HaahMatthew B. HastingsWe present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of . Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes.https://quantum-journal.org/papers/q-2018-06-07-71/pdf/ |
| spellingShingle | Jeongwan Haah Matthew B. Hastings Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates Quantum |
| title | Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates |
| title_full | Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates |
| title_fullStr | Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates |
| title_full_unstemmed | Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates |
| title_short | Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates |
| title_sort | codes and protocols for distilling t controlled s and toffoli gates |
| url | https://quantum-journal.org/papers/q-2018-06-07-71/pdf/ |
| work_keys_str_mv | AT jeongwanhaah codesandprotocolsfordistillingtcontrolledsandtoffoligates AT matthewbhastings codesandprotocolsfordistillingtcontrolledsandtoffoligates |