Start with \(\psi_0={\left|000\right\rangle}\) and allow the three bit flip operators \(X_1,X_2,X_3\). For a binary address \(a=(a_1,a_2,a_3)\), put
The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits.
Reading the drawing
Solid edges within each square change one of the last two bits. Dashed edges join the squares by changing the first bit. Three example operator words and the dimension of the addressed span appear on the right.
Drawing labels and data 13 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
- \({\left|000\right\rangle}\)
- \({\left|001\right\rangle}\)
- \({\left|010\right\rangle}\)
- \({\left|011\right\rangle}\)
- \({\left|100\right\rangle}\)
- \({\left|101\right\rangle}\)
- \({\left|110\right\rangle}\)
- \({\left|111\right\rangle}\)
- Three actual words
- \(W_{101}=X_1X_3\)
- \(W_{011}=X_2X_3\)
- \(W_{110}=X_1X_2\)
- \(\dim{\operatorname{span}}\{W_a{\left|000\right\rangle}\}=8\)
Two different meanings of “reach”
The eight vectors are orthonormal, so their linear span is all of \({\mathbb C}^8\). This proves cyclicity for the algebra generated by the flips. Preparing every normalized vector requires more than products of these flips alone.
For example, no such word sends \({\left|000\right\rangle}\) to \(({\left|000\right\rangle}+{\left|001\right\rangle})/\sqrt2\). A Hadamard gate, a suitable linear combination in the observable algebra, or a larger preparation grammar is required for that operation. Counting linearly independent addressed vectors and synthesizing a physical gate sequence are separate tasks.
The exact check constructs each \(8\times8\) word matrix, applies it to the seed, and verifies that the matrix of all eight outputs is the identity. It also tests the displayed superposition against every word output. No search over an unspecified gate set is involved.
Read this as
A cyclic seed generates a vector space under an algebra. That conclusion does not automatically provide a unitary preparation procedure for every vector in it.