Unfolding the color code
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d -dimensional closed manifold is equivalent to multiple decoupled copies of the d -dimensional toric code up to local unitary transformati...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2015-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/17/8/083026 |
| _version_ | 1797751167747358720 |
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| author | Aleksander Kubica Beni Yoshida Fernando Pastawski |
| author_facet | Aleksander Kubica Beni Yoshida Fernando Pastawski |
| author_sort | Aleksander Kubica |
| collection | DOAJ |
| description | The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d -dimensional closed manifold is equivalent to multiple decoupled copies of the d -dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d -dimensional color code with $d+1$ boundaries of $d+1$ distinct colors, we find that the code is equivalent to multiple copies of the d -dimensional toric code which are attached along a $(d-1)$ -dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d -dimensional toric code admits logical non-Pauli gates from the d th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d -qubit control- Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation. |
| first_indexed | 2024-03-12T16:43:32Z |
| format | Article |
| id | doaj.art-745f490a06844d81980921ac97d776c5 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:43:32Z |
| publishDate | 2015-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-745f490a06844d81980921ac97d776c52023-08-08T14:19:41ZengIOP PublishingNew Journal of Physics1367-26302015-01-0117808302610.1088/1367-2630/17/8/083026Unfolding the color codeAleksander Kubica0Beni Yoshida1Fernando Pastawski2Institute for Quantum Information & Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USAInstitute for Quantum Information & Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USAInstitute for Quantum Information & Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USAThe topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d -dimensional closed manifold is equivalent to multiple decoupled copies of the d -dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d -dimensional color code with $d+1$ boundaries of $d+1$ distinct colors, we find that the code is equivalent to multiple copies of the d -dimensional toric code which are attached along a $(d-1)$ -dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d -dimensional toric code admits logical non-Pauli gates from the d th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d -qubit control- Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.https://doi.org/10.1088/1367-2630/17/8/083026quantum codestransversal gatesfault-tolerant quantum computationcolor codetopological orderboundaries and domain walls |
| spellingShingle | Aleksander Kubica Beni Yoshida Fernando Pastawski Unfolding the color code New Journal of Physics quantum codes transversal gates fault-tolerant quantum computation color code topological order boundaries and domain walls |
| title | Unfolding the color code |
| title_full | Unfolding the color code |
| title_fullStr | Unfolding the color code |
| title_full_unstemmed | Unfolding the color code |
| title_short | Unfolding the color code |
| title_sort | unfolding the color code |
| topic | quantum codes transversal gates fault-tolerant quantum computation color code topological order boundaries and domain walls |
| url | https://doi.org/10.1088/1367-2630/17/8/083026 |
| work_keys_str_mv | AT aleksanderkubica unfoldingthecolorcode AT beniyoshida unfoldingthecolorcode AT fernandopastawski unfoldingthecolorcode |