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The Clifford-cyclotomic group and Euler–Poincaré characteristics

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 September 2020

Colin Ingalls ,
Bruce W. Jordan ,
Allan Keeton ,
Adam Logan  and
Yevgeny Zaytman
Colin Ingalls
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada e-mail: cingalls@math.carleton.ca
Bruce W. Jordan
Affiliation:
Department of Mathematics, Box B-630, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY 10010, USA e-mail: bruce.jordan@baruch.cuny.edu
Allan Keeton*
Affiliation:
Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, USA e-mail: ykzaytm@idaccr.org
Adam Logan
Affiliation:
The Tutte Institute for Mathematics and Computation, P.O. Box 9703, Terminal, Ottawa, ON K1G 3Z4, Canada School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada e-mail: adam.m.logan@gmail.com
Yevgeny Zaytman
Affiliation:
Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, USA e-mail: ykzaytm@idaccr.org
*
e-mail: agk@idaccr.org
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Abstract

For an integer $n\geq 8$ divisible by $4$, let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$. Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$. We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$. We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$, $\operatorname {\mathrm {PSU_{2}}}(R_n)$, $\operatorname {\mathrm {PU_{2}}}(R_n)$, $\operatorname {\mathrm {PU_2^\zeta }}(R_n)$, and $\operatorname {\mathrm {SO_{3}}}(R_n^+)$.

Keywords

Clifford groupT gateClifford cyclotomicEuler–Poincaré characteristics

MSC classification

Primary: 81P45: Quantum information, communication, networks
Secondary: 20G30: Linear algebraic groups over global fields and their integers

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 651 - 666
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Copyright
© Canadian Mathematical Society 2020

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