Table 1 Sizes of the k-qubit Clifford group and subgroups

From: Fault-tolerant quantum computation with few qubits

k

S k

GL(k, 2)

\(\left\langle {{\mathrm{CNOT}},H} \right\rangle \)

\({C}_k{\mathrm{/}}{P}_k\)

1

1

×1

×2

×3

2

2

×3

×12

×10

3

6

×28

×240

×36

4

24

×840

×17,280

×136

5

120

×83,328

×4,700,160

×528

6

720

×27,998,208

×4,963,368,960

×2080

7

5040

×3.2 · 1010

×2.1 · 1013

×8256

8

40,320

×1.3 · 1014

×3.4  · 1017

×32,896

  1. There are \(\left| {S_k} \right|\) = k! permutations of k qubits. CNOT gates generate a group of size \(\left| {\mathrm{GL}(k,2)} \right|\) = \(\mathop {\prod}\nolimits_{j = 0}^{k - 1} \left( {2^k - 2^j} \right)\), adding Hadamard gates generates a larger group, and finally the full Clifford group, up to the \(\left| {{\cal P}_k} \right| = 4^k\) Paulis, has size \(2^{k^2}\mathop {\prod}\nolimits_{j = 1}^k \left( {4^k - 1} \right)\). The sizes are given as multiples of the previous columns, e.g., \(\left| {{\cal C}_2{\mathrm{/}}{\cal P}_2} \right|\) = 2 × 3 × 12 × 10 = 720
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