The Exercise

The Toffoli gate $U$ acts on three qubits and the $\pi/8$ gate $T$ on one. The exercise asks what conjugation by these gates does to the Pauli operators: for $U$, to $X$ and to $Z$ acting on any one of the three qubits with the identity on the other two, six operators in all; for $T$, to $X$ and to $Z$ on its single qubit.

The library gives the eight answers as identities between matrices in the computational basis, where it defines the gates and the Pauli operators.

Gates and Pauli Operators

The identity and the Pauli matrices are the $2 \times 2$ complex matrices $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) with rows and columns labelled by $0$ and $1$. The $\pi/8$ gate $T$ is the diagonal matrix $$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}.$$(2)^✦

The computational basis states $|abc\rangle$ of three qubits, with $a, b, c \in \{0, 1\}$, are numbered from $0$ to $7$ by reading $abc$ as a binary number, the first qubit the most significant: $$(a, b, c) \longmapsto 4a + 2b + c.$$(3)^✦ The $8 \times 8$ matrix of an operator $M$ on the three qubits has in row $4a + 2b + c$ and column $4a' + 2b' + c'$ the matrix element of $M$ between the corresponding basis states, $$M_{4a+2b+c,\; 4a'+2b'+c'} = \langle abc | M | a'b'c' \rangle,$$(4)^✦ and an operator on the three qubits is identified with this matrix.

For $k = 1, 2, 3$, the Pauli operator $X_k$ of the three qubits is the operator that acts as $X$ on qubit $k$ and as the identity on the other two: $$X_1 = X \otimes I \otimes I,$$(5)^✦ $$X_2 = I \otimes X \otimes I,$$(6)^✦ $$X_3 = I \otimes I \otimes X.$$(7)^✦ The Pauli operator $Z_k$ acts as $Z$ on qubit $k$ and as the identity on the other two: $$Z_1 = Z \otimes I \otimes I,$$(8)^✦ $$Z_2 = I \otimes Z \otimes I,$$(9)^✦ $$Z_3 = I \otimes I \otimes Z.$$(10)^✦

The Toffoli gate $U$ is the $8 \times 8$ permutation matrix that exchanges the basis states numbered $6$ and $7$, which are $|110\rangle$ and $|111\rangle$, and fixes the other six. With $\oplus$ denoting addition modulo $2$, it flips the third qubit exactly when the first two are $1$: $$U |a, b, c\rangle = |a, b, c \oplus ab\rangle.$$(11)^✦

The Claim

What Is Checked

The six identities for $U$ are stated over the Toffoli gate (11) and the operators $X_k$ and $Z_k$, tensor products of the matrices (1) written as $8 \times 8$ matrices by (4) in the numbering (3). The two for $T$ are stated over the matrices (1) and (2).