Table 1 The elements of \({\mathcal{S}}\) and the explicit row patterns they are associated with under FFP.
From: Optimal two-qubit circuits for universal fault-tolerant quantum computation
Generator | Associated patterns under first finer partition (FFP) |
|---|---|
R(X ⊗ I, I ⊗ X) | \(\left\{\left\{1,4\right\},\left\{2,3\right\},\left\{5,6\right\}\right\},\left\{\left\{1,4\right\},\left\{2,3,5,6\right\}\right\},\left\{\left\{2,3\right\},\left\{1,4,5,6\right\}\right\},\left\{\left\{5,6\right\},\left\{1,2,3,4\right\}\right\}\) |
R(Y ⊗ I, I ⊗ Y) | \(\left\{\left\{1,3\right\},\left\{2,5\right\},\left\{4,6\right\}\right\},\left\{\left\{1,3\right\},\left\{2,4,5,6\right\}\right\},\left\{\left\{2,5\right\},\left\{1,3,4,6\right\}\right\},\left\{\left\{4,6\right\},\left\{1,2,3,5\right\}\right\}\) |
R(Z ⊗ I, I ⊗ Z) | \(\left\{\left\{1,2\right\},\left\{3,6\right\},\left\{4,5\right\}\right\},\left\{\left\{1,2\right\},\left\{3,4,5,6\right\}\right\},\left\{\left\{3,6\right\},\left\{1,2,4,5\right\}\right\},\left\{\left\{4,5\right\},\left\{1,2,3,6\right\}\right\}\) |
R(Y ⊗ I, I ⊗ Z) | \(\left\{\left\{1,3\right\},\left\{2,6\right\},\left\{4,5\right\}\right\},\left\{\left\{2,6\right\},\left\{1,3,4,5\right\}\right\}\) |
R(Z ⊗ I, I ⊗ Y) | \(\left\{\left\{1,2\right\},\left\{3,5\right\},\left\{4,6\right\}\right\},\left\{\left\{3,5\right\},\left\{1,2,4,6\right\}\right\}\) |
R(Z ⊗ I, I ⊗ X) | \(\left\{\left\{1,2\right\},\left\{3,4\right\},\left\{5,6\right\}\right\},\left\{\left\{3,4\right\},\left\{1,2,5,6\right\}\right\}\) |
R(X ⊗ I, I ⊗ Z) | \(\left\{\left\{1,6\right\},\left\{2,3\right\},\left\{4,5\right\}\right\},\left\{\left\{1,6\right\},\left\{2,3,4,5\right\}\right\}\) |
R(X ⊗ I, I ⊗ Y) | \(\left\{\left\{1,5\right\},\left\{2,3\right\},\left\{4,6\right\}\right\},\left\{\left\{1,5\right\},\left\{2,3,4,6\right\}\right\}\) |
R(Y ⊗ I, I ⊗ X) | \(\left\{\left\{1,3\right\},\left\{2,4\right\},\left\{5,6\right\}\right\},\left\{\left\{2,4\right\},\left\{1,3,5,6\right\}\right\}\) |
R(X ⊗ X, Y ⊗ Y) | \(\left\{\left\{1,4\right\},\left\{2,5\right\},\left\{3,6\right\}\right\}\) |
R(X ⊗ X, Z ⊗ Y) | \(\left\{\left\{1,4\right\},\left\{2,6\right\},\left\{3,5\right\}\right\}\) |
R(Z ⊗ X, Y ⊗ Y) | \(\left\{\left\{1,6\right\},\left\{2,5\right\},\left\{3,4\right\}\right\}\) |
R(Y ⊗ X, X ⊗ Y) | \(\left\{\left\{1,5\right\},\left\{2,4\right\},\left\{3,6\right\}\right\}\) |
R(Z ⊗ X, X ⊗ Y) | \(\left\{\left\{1,5\right\},\left\{2,6\right\},\left\{3,4\right\}\right\}\) |
R(Y ⊗ X, Z ⊗ Y) | \(\left\{\left\{1,6\right\},\left\{2,4\right\},\left\{3,5\right\}\right\}\) |
