The Exercise

The circuit of Figure 10.25 applies the $\pi/8$ gate $T$ to a qubit with the help of an extra qubit in the state $|\Theta\rangle$. The exercise prepares that state by measuring the observable $M = e^{-i\pi/4}SX$, where $S$ is the phase gate and $X$ the bit flip: $|\Theta\rangle$ is the eigenstate of $M$ of eigenvalue $+1$, and $|\Theta^-\rangle$ its eigenstate of eigenvalue $-1$.

The measurement couples the data qubit, the one carrying the state to be measured, to an ancilla qubit prepared in $|0\rangle$. The ancilla passes through a Hadamard gate $H$, controls the gate $M$ on the data qubit, passes through a second Hadamard gate and is read in the computational basis. The exercise asks which outcome each eigenstate gives, and what a phase flip $Z$ on the ancilla, between the controlled gate and the last Hadamard gate, does to it.

In the fault tolerant form of the measurement the ancilla is a cat state of several qubits, the equal superposition of the all zeros and the all ones states, each qubit paired with a data qubit and controlling a gate on it. Two more questions belong to this form: what the $\pi/8$ gate on every qubit does to a cat state of seven qubits, and whether a phase flip on one ancilla qubit reaches the data through the controlled gates.

A Quantum System

A quantum system $A$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A pure state $\psi$ of $A$ is a vector of unit norm in the state space of $A$, $\lVert\psi\rVert = 1$.^✦ A unitary $U$ of $A$ is an operator on the state space of $A$ with $U^\dagger U = UU^\dagger = 1$.^✦ The unitary $U$ applied to a pure state $\psi$ of $A$ is the pure state $U\psi$^✦, and $U$ after a second unitary $V$ of $A$ is the unitary $UV$ of $A$, first $V$ then $U$.^✦

The composite system $A \otimes B$ of two systems $A$ and $B$ is the system whose state space is the tensor product of the state spaces of $A$ and $B$.^✦ Pure states $\psi$ of $A$ and $\varphi$ of $B$ give the product state $\psi \otimes \varphi$, the pure state of $A \otimes B$ whose vector is the tensor product of their vectors.^✦ Unitaries $U$ of $A$ and $V$ of $B$ give in the same way the product unitary $U \otimes V$ of $A \otimes B$, whose operator is the tensor product of their operators.^✦ With the identity evolution $1$ of $B$, the unitary whose operator is the identity, $U$ on the first wires of $A \otimes B$ is $U \otimes 1$.^✦

Gates on a Qubit

The qubit is the two level system whose state space is $\mathbb{C}^2$.^✦ The basis states $|0\rangle$ and $|1\rangle$ of the qubit, its computational basis, are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦ A $2 \times 2$ unitary matrix $U$ gives the single qubit gate of $U$, the unitary of the qubit that acts on $\mathbb{C}^2$ by the matrix $U$ in the computational basis.^✦

The Hadamard gate $H$ and the phase flip $Z$ are the single qubit gates of the matrices $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) and the bit flip $X$, the phase gate $S$ and the $\pi/8$ gate $T$ have the matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}.$$(2) The product $SX$, with $X$ acting first, is the matrix $$SX = \begin{pmatrix} 0 & 1 \\ i & 0 \end{pmatrix}.$$(3)^✦

The gate $M$ is the single qubit gate of the matrix $e^{-i\pi/4}SX$, $$M = \begin{pmatrix} 0 & e^{-i\pi/4} \\ e^{i\pi/4} & 0 \end{pmatrix},$$(4)^✦ and the gate $ZSX$ is the single qubit gate of the product of $Z$ with $SX$, $$ZSX = \begin{pmatrix} 0 & 1 \\ -i & 0 \end{pmatrix}.$$(5)^✦

The two eigenstates of $M$ are the pure states of the qubit $$|\Theta\rangle = \frac{1}{\sqrt{2}} \bigl( |0\rangle + e^{i\pi/4} |1\rangle \bigr),$$(6)^✦ $$|\Theta^-\rangle = \frac{1}{\sqrt{2}} \bigl( |0\rangle - e^{i\pi/4} |1\rangle \bigr).$$(7)^✦

The Measurement Circuit

For a unitary $U$ of a system $A$, the controlled gate $C(U)$ is the unitary of the composite system of a control qubit and a target $A$ with operator $$C(U) = |0\rangle\langle 0| \otimes 1 + |1\rangle\langle 1| \otimes U.$$(8)^✦

The measurement circuit of a unitary $U$ of a system $A$ is the unitary of the composite system of an ancilla qubit and $A$ with operator $$(H \otimes 1)\, C(U)\, (H \otimes 1).$$(9)^✦

For a unitary $U$ of a system $A$, the measurement circuit of $U$ with a $Z$ error on the ancilla is the unitary of the composite system of an ancilla qubit and $A$ with operator $$(H \otimes 1)(Z \otimes 1)\, C(U)\, (H \otimes 1).$$(12)^✦

The Transversal Circuit

A register of $n$ qubits, one for each of the $n$ elements of a finite set $\iota$, is the system whose state space has an orthonormal basis $|w\rangle$, its computational basis, with one vector for each bit string $w$ over $\iota$.^✦ The cat vector of the register is the unnormalized sum $|0\cdots0\rangle + |1\cdots1\rangle$ of its all zeros and all ones basis vectors.^✦ The transversal $T$ gate of the register is the unitary that is diagonal in the computational basis with $$T^{\otimes n}\, |w\rangle = e^{i\pi \operatorname{wt}(w)/4}\, |w\rangle,$$(14)^✦ where $\operatorname{wt}(w)$ is the number of ones in $w$.

For a unitary $U$ of a data system $D$, the transversal coupling of $U$ is the unitary of three pairs, each an ancilla qubit with a copy of $D$, that applies $C(U)$ to every pair with the ancilla as control, $C(U)^{\otimes 3} = C(U) \otimes C(U) \otimes C(U)$.^✦ For $i = 0, 1, 2$, the $Z$ error on ancilla $i$ is the unitary $Z_i$ of the three pairs that applies $Z$ to the ancilla of pair $i$ and the identity to its data and to the other two pairs.^✦

What Is Checked

The claims on a single ancilla are stated over the measurement circuit (9) of the gate $M$ and its form (12) with a $Z$ error, acting on the product state of the ancilla in $|0\rangle$ with the data qubit in $|\Theta\rangle$ or $|\Theta^-\rangle$. The transversal claims are stated over the transversal $T$ gate on seven qubits acting on the cat vector, and over the coupling of three pairs of qubits by $C(ZSX)$ with the $Z$ errors on its ancillas.

About these constructions the library proves that the matrices $M$ and $ZSX$ are unitary and that the controlled gate $C(U)$ of every unitary $U$ is unitary^✦; the product unitary $U \otimes V$ and $U$ after $V$ are unitary, and so is the transversal $T$ gate. It proves as well that $|\Theta\rangle$ and $|\Theta^-\rangle$ have unit norm, and that a unitary applied to a pure state, like the product of two pure states, is again a pure state.