The Group Structure of Quantum Cellular Automata
We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of control space; this is a corollary of a general method of ancilla removal. Further, we show how to de...
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| Veröffentlicht in: | Communications in mathematical physics 2022-02, Vol.389 (3), p.1277-1302 |
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| container_title | Communications in mathematical physics |
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| creator | Freedman, Michael Haah, Jeongwan Hastings, Matthew B. |
| description | We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of control space; this is a corollary of a general method of ancilla removal. Further, we show how to define a group of QCA that is well-defined without needing to use families, by showing how to construct a coherent family containing an arbitrary finite QCA; the coherent family consists of QCA on progressively finer systems of qudits where any two members are related by a shallow quantum circuit. This construction applied to translation invariant QCA shows that all translation invariant QCA in three dimensions and all translation invariant Clifford QCA in any dimension are coherent. |
| doi_str_mv | 10.1007/s00220-022-04316-x |
| format | Article |
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Math. Phys</addtitle><description>We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of control space; this is a corollary of a general method of ancilla removal. Further, we show how to define a group of QCA that is well-defined without needing to use families, by showing how to construct a coherent family containing an arbitrary finite QCA; the coherent family consists of QCA on progressively finer systems of qudits where any two members are related by a shallow quantum circuit. This construction applied to translation invariant QCA shows that all translation invariant QCA in three dimensions and all translation invariant Clifford QCA in any dimension are coherent.</description><subject>Cellular automata</subject><subject>Cellular structure</subject><subject>Circuits</subject><subject>Classical and Quantum Gravitation</subject><subject>Coherence</subject><subject>Complex Systems</subject><subject>Invariants</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoWEf_gKuC6-hN0uaxHIqOwoCI4zpk0lQd-hjzgPHfG63gzs05XPjOuXAQuiRwTQDETQCgFHAWDBUjHB-OUEEqlk9F-DEqAAhgxgk_RWch7ABAUc4LBJs3V678lPblc_TJxuRdOXXlUzJjTEPZuL5PvfHlMsVpMNGco5PO9MFd_PoCvdzdbpp7vH5cPTTLNbaMqIiF5UISKWvmmGVt10pab42yjLatU1xsZUVr11HFnWBCGltbAaS1sqNCKkXZAl3NvXs_fSQXot5NyY_5paacSiF5JSFTdKasn0LwrtN7_z4Y_6kJ6O9l9LyMzqJ_ltGHHGJzKGR4fHX-r_qf1Be5d2Uc</recordid><startdate>20220201</startdate><enddate>20220201</enddate><creator>Freedman, Michael</creator><creator>Haah, Jeongwan</creator><creator>Hastings, Matthew B.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20220201</creationdate><title>The Group Structure of Quantum Cellular Automata</title><author>Freedman, Michael ; Haah, Jeongwan ; Hastings, Matthew B.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-7c67818853e3c3dfd825ba9c32dde967b8425ef296e7378ac5c701dc8f2789923</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Cellular automata</topic><topic>Cellular structure</topic><topic>Circuits</topic><topic>Classical and Quantum Gravitation</topic><topic>Coherence</topic><topic>Complex Systems</topic><topic>Invariants</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Freedman, Michael</creatorcontrib><creatorcontrib>Haah, Jeongwan</creatorcontrib><creatorcontrib>Hastings, Matthew B.</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Freedman, Michael</au><au>Haah, Jeongwan</au><au>Hastings, Matthew B.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Group Structure of Quantum Cellular Automata</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. 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| ispartof | Communications in mathematical physics, 2022-02, Vol.389 (3), p.1277-1302 |
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| language | eng |
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| source | SpringerNature Journals |
| subjects | Cellular automata Cellular structure Circuits Classical and Quantum Gravitation Coherence Complex Systems Invariants Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Quantum Physics Relativity Theory Theoretical |
| title | The Group Structure of Quantum Cellular Automata |
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