A353196 Number of stabilizer states on n qubits.

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

Offset

1,1

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.

Links

Table of n, a(n) for n=1..14.

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

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

Example

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

Mathematica

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

Prog

(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<<i)+1 for i in range(1, n+1)) << n # Chai Wah Wu, Jun 20 2022

Crossrefs

Cf. A000079, A003956, A028362.

Sequence in context: A061573 A248655 A092356 * A285955 A001416 A251184

Adjacent sequences: A353193 A353194 A353195 * A353197 A353198 A353199

Keyword

nonn,easy

Author

James Rayman, Apr 29 2022

Status

approved