A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.
8, 192, 92160, 743178240, 97029351014400, 203286581427673497600, 6819500449352277792129024000, 3660967964237442812098963052691456000, 31446995505814020383166371418359014222725120000
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..20
- Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
- G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
- G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- Edwin Pednault, An alternative approach to optimal wire cutting without ancilla qubits, arXiv:2303.08287 [quant-ph], 2023.
- Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, Binary Subspace Chirps, arXiv:2102.12384 [cs.IT], 2021.
- Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
- Peter Selinger, Generators and relations for n-qubit Clifford operators, arXiv:1310.6813 [quant-ph], 2013; Log. Methods Comput. Sci. 11 (2:10) (2015), 1-17, doi:10.2168/LMCS-11(2:10)2015.
- Index entries for sequences related to groups.
Programs
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GAP
List([0..10], n-> 2^((n+1)^2 +2)*Product([1..n], j-> 4^j -1) ); # G. C. Greubel, Sep 24 2019
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Magma
[n eq 0 select 8 else 2^((n+1)^2+2)*(&*[4^j-1: j in [1..n]]): n in [0..10]]; // G. C. Greubel, Sep 24 2019
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Maple
a(n):= 2^(n^2+2*n+3)*mul(4^j-1, j=1..n); seq(a(n), n=0..10); # modified by G. C. Greubel, Sep 24 2019
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Mathematica
Table[2^(n^2+2n+3) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Nov 03 2017 *) -
PARI
vector(11, n, 2^(n^2 +2)*prod(j=1,n-1, 4^j-1) ) \\ G. C. Greubel, Sep 24 2019
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Python
from math import prod def A003956(n): return prod((1<
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Sage
[2^((n+1)^2 +2)*product(4^j -1 for j in (1..n)) for n in (0..10)] # G. C. Greubel, Sep 24 2019
Formula
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+2*n+3) * Product_{k=1..n} (4^k-1).
a(n) ~ c * 2^(2*n^2+3*n+3), where c = A100221. (End)