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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, ONK1S 5B6, Canadae-mail:[email protected]
Bruce W. Jordan
Affiliation:
Department of Mathematics, Box B-630, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY10010, USAe-mail:[email protected]
Allan Keeton*
Affiliation:
Center for Communications Research, 805 Bunn Drive, Princeton, NJ08540, USAe-mail:[email protected]
Adam Logan
Affiliation:
The Tutte Institute for Mathematics and Computation, P.O. Box 9703, Terminal, Ottawa, ONK1G 3Z4, Canada School of Mathematics and Statistics, Carleton University, Ottawa, ONK1S 5B6, Canadae-mail:[email protected]
Yevgeny Zaytman
Affiliation:
Center for Communications Research, 805 Bunn Drive, Princeton, NJ08540, USAe-mail:[email protected]
*
e-mail: [email protected]
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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
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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References

Bocharov, A., Roetteler, M., and Svore, K. M., Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A 91(2015), 052317.CrossRefGoogle Scholar
Brown, K. S., Cohomology of groups . Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.Google Scholar
Forest, S., Gosset, D., Kliuchnikov, V., and McKinnon, D., Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets. J. Math. Phys. 56(2015), no. 8, 082201. http://arxiv.org/abs/10.1063/1.4927100 CrossRefGoogle Scholar
Ingalls, C., Jordan, B. W., Keeton, A., Logan, A., and Zaytman, Y., The corank of unitary groups over cyclotomic rings. Preprint, 2019. arXiv:1911:02137.Google Scholar
Ingalls, C., Jordan, B. W., Keeton, A., Logan, A., and Zaytman, Y., Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings. Preprint, 2020. arXiv:2001:01695.CrossRefGoogle Scholar
Naber, G. L., Topology, geometry, and gauge fields. 2nd ed., Texts in Applied Mathematics, 25, Springer, New York, 2011. http://arxiv.org/abs/10.1007/978-1-4419-7254-5 Google Scholar
Neukirch, J., Algebraic number theory. Grundlehren der Mathematischen Wissenschaften, 322, Springer-Verlag, Berlin, 1999. http://arxiv.org/abs/10.1007/978-3-662-0398-0 Google Scholar
Nielsen, M. A. and Chuang, I. L., Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000.Google Scholar
Radin, C. and Sadun, L., On $2$ -generator subgroups of SO(3). Trans. Amer. Math. Soc. 351(1999), no. 11, 4469–4480. http://arxiv.org/abs/10.1090/S0002-9947-99-02397-1 CrossRefGoogle Scholar
Sarnak, P., Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev theorem and golden gates, 2015. http://publications.ias.edu/sarnak/paper/2637 Google Scholar
Serre, J.-P., Cohomologie des groups discrets. Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, NJ, 1970), Ann. of Math. Studies, 70, 1971, pp. 77–169.CrossRefGoogle Scholar
Serre, J.-P., Le groupe quaquaversal, vu comme groupe S-arithmétique. Oberwolfach Rep. 6 (2009), no. 2, 1421–1426.Google Scholar
Vignéras, M.-F., Arithmétique des algébres de quaternions. Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.CrossRefGoogle Scholar
Washington, L. C., Introduction to cyclotomic fields. Graduate Texts in Mathematics, 83, Springer-Verlag, Berlin, 1982. http://arxiv.org/abs/10.1007/978-1-4684-0133-2 Google Scholar
Weber, H., Lehrbuch der Algebra. Vol. II. Zweite Auflage, Vieweg, Braunschweig, 1899.Google Scholar