A353196 Number of stabilizer states on n qubits.

6, 60, 1080, 36720, 2423520, 315057600, 81284860800, 41780418451200, 42866709330931200, 87876754128408960000, 360118938418219918080000, 2950814581398894008747520000, 48352047730802277227336862720000, 1584496604138390624739828991334400000

Offset: 1

Author

James Rayman, Apr 29 2022

Keywords

• nonn
• easy

Comments

A stabilizer state is a quantum state on n qubits prepared by applying a series of Hadamard, CNOT, and S gates to the all-zero state. There are only a finite number of such states for any n.

Examples

			For n = 1, the a(1) = 6 states are |0>, |1>, |+>, |->, |i>, and |-i>.
		

Links

• D. Gross, Hudson's Theorem for finite-dimensional quantum systems, arXiv:quant-ph/0602001, 2006-2007.

Crossrefs

Cf. A000079, A003956, A028362.

Programs

Mathematica

Table[2^n * QPochhammer[-2, 2, n], {n, 13}] (* Amiram Eldar, Aug 17 2025 *)

Python

def a(n):
    ans = 2 ** n
    for i in range(1, n+1):
        ans *= 2 ** i + 1
    return ans

Python

from math import prod
def A353196(n): return prod((1<Chai Wah Wu, Jun 20 2022

Formula

a(n) = 2^n*Product_{i=1..n} (2^i+1).

a(n) = A000079(n)*A028362(n+1).

a(n) ~ c * 2^(n*(n+3)/2) where c = Product_{k>=1} (1 + 1/2^k) = A079555. - Amiram Eldar, Aug 17 2025