The Exercise

Two qubits are carried by two coupled spins, as in nuclear magnetic resonance (NMR): each spin can be rotated on its own, and between rotations the pair evolves under its coupling. The controlled NOT (CNOT) flips the second qubit, the target, exactly when the first, the control, is $|1\rangle$. The exercise is to show that a rotation of the target by $\pi/2$ about $\hat y$, a period of free evolution under the coupling and a rotation of the target by $\pi/2$ about $\hat x$ perform the CNOT on the computational basis up to phases, and that rotations about $\hat z$ and one global phase remove those phases.

The library states this for a coupling constant $J$ and a coupling time $t$, first on the basis states and then for the circuit with the added rotations about $\hat z$ and the global phase, as an equality of unitaries and as an evolution of states.

Two Qubits

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the quantum system whose state space is $\mathbb{C}^2$; its computational basis states $|0\rangle$ and $|1\rangle$ are the standard basis vectors $e_0$ and $e_1$. The composite system $S \otimes T$ of two systems $S$ and $T$ is the quantum system whose state space is the tensor product of their state spaces.^✦ Two qubits thus form $\mathrm{qubit} \otimes \mathrm{qubit}$, with basis states $|a\rangle|b\rangle = |a\rangle \otimes |b\rangle$ for bits $a$ and $b$.

A pure state $\psi$ of $S$ is a unit vector in the state space of $S$, and the basis states of the qubit are pure states.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ It evolves a state $\rho$ of $S$ into the state $U \rho U^\dagger$.^✦ The composite $UV$ of unitaries $U$ and $V$ of $S$ is the unitary of $S$ that applies $V$ first and then $U$.^✦ Unitaries $U$ of $S$ and $V$ of $T$ have a product $U \otimes V$, the unitary of $S \otimes T$ whose operator is the tensor product of the two.^✦ A unitary $U$ of $S$ acts on the first system of $S \otimes T$ as $U \otimes 1$, with $1$ the identity unitary of $T$; for a unitary $V$ of $T$, the gate $V$ on the second system is the unitary $1 \otimes V$ of $S \otimes T$, with $1$ the identity unitary of $S$. For a real angle $\alpha$, the global phase gate $e^{i\alpha}$ on a system $S$ is the unitary $e^{i\alpha}\, 1$ of $S$, which multiplies every vector of the state space by $e^{i\alpha}$.

Rotations and the Controlled NOT

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec n$ in three dimensions, $\vec n \cdot \vec\sigma = n_x X + n_y Y + n_z Z$, and the rotation by a real angle $\theta$ about $\vec n$ is the $2 \times 2$ matrix $$R_{\vec n}(\theta) = \exp\Bigl(-\frac{i\theta}{2}\, \vec n \cdot \vec\sigma\Bigr),$$(2) written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$. The rotations $R_x(\theta)$, $R_y(\theta)$ and $R_z(\theta)$ are those about $\hat x = (1,0,0)$, $\hat y = (0,1,0)$ and $\hat z = (0,0,1)$; thus $R_z(\theta) = e^{-i\theta Z/2}$. The gate $R_{\hat n}(\theta)$ is the unitary of the qubit whose matrix in the computational basis is $R_{\hat n}(\theta)$.

The Pauli $X$ gate, the quantum NOT, is the unitary of the qubit whose matrix is $X$.^✦ For a unitary $U$ of a system $S$, the controlled gate $C(U)$ is the unitary of $\mathrm{qubit} \otimes S$, a control qubit and the target $S$, given by $$C(U) = |0\rangle\langle 0| \otimes 1 + |1\rangle\langle 1| \otimes U .$$(3)^✦ The CNOT gate is the controlled gate $C(X)$ of the Pauli $X$, a unitary of two qubits.^✦

The Coupling

An observable $A$ on a system $S$ is a self adjoint operator on the state space of $S$.^✦ For a real vector $\vec n$ in three dimensions, the Pauli observable $\vec n \cdot \vec\sigma$ is the observable on the qubit whose operator has the matrix $\vec n \cdot \vec\sigma$; the observable $Z$ is the Pauli observable for $\hat z$. The tensor product $A \otimes B$ of observables $A$ on $S$ and $B$ on $T$ is the observable on $S \otimes T$ whose operator is the tensor product of theirs.^✦ The coupling Hamiltonian $H$ of two spins with real coupling constant $J$ is the observable on two qubits, written with $Z_1 = Z \otimes 1$ and $Z_2 = 1 \otimes Z$, the observable $Z$ on the first and on the second qubit, $$H = J\, Z_1 Z_2 = J\, Z \otimes Z .$$(4)^✦

The propagator $U(t_1, t_2)$ of a Hamiltonian $H$ on $S$ from time $t_1$ to time $t_2$, with Planck’s constant $\hbar$, is the unitary $\exp[-iH(t_2 - t_1)/\hbar]$ of $S$, which carries the state vector $|\psi(t_1)\rangle$ of $S$ at time $t_1$ to the state vector $|\psi(t_2)\rangle$ at time $t_2$.^✦ In units with $\hbar = 1$, the coupling acting from time $0$ to time $t$ gives $U(0, t) = e^{-iJt\, Z \otimes Z}$.

The Circuit

The NMR circuit $U_{J,t}$ is the unitary of two qubits that applies $R_y(\pi/2)$ to the target, lets the coupling act for a time $t$, and applies $R_x(\pi/2)$ to the target: $$U_{J,t} = \bigl(1 \otimes R_x(\pi/2)\bigr)\, e^{-iJt\, Z \otimes Z}\, \bigl(1 \otimes R_y(\pi/2)\bigr).$$(5)^✦

The phases $\varphi(a,b)$ of the circuit, one for each pair of bits $a$ and $b$, are the four complex numbers $$\begin{pmatrix} \varphi(0,0) & \varphi(0,1) \\ \varphi(1,0) & \varphi(1,1) \end{pmatrix} = \frac{\sqrt 2}{2} \begin{pmatrix} 1 - i & 1 + i \\ 1 - i & -1 - i \end{pmatrix} = \begin{pmatrix} e^{-i\pi/4} & e^{i\pi/4} \\ e^{-i\pi/4} & e^{-3i\pi/4} \end{pmatrix}.$$(6)^✦

The corrected circuit $\tilde U_{J,t}$ is the unitary of two qubits that applies $U_{J,t}$, then the gate $R_z(-\pi/2)$ to the target and the gate $R_z(\pi/2)$ to the control: $$\tilde U_{J,t} = \bigl(R_z(\pi/2) \otimes 1\bigr)\bigl(1 \otimes R_z(-\pi/2)\bigr)\, U_{J,t}.$$(7)

The Claim

What Is Checked

The claim is stated over the circuit $U_{J,t}$ of (5), built from the rotation gates $R_y(\pi/2)$ and $R_x(\pi/2)$ on the target and the propagator of the coupling Hamiltonian (4); over its phases (6), the corrected circuit (7) and the global phase gate; and over the CNOT gate $C(X)$ and the evolution $\rho \mapsto U \rho U^\dagger$ of the states of two qubits.

The controlled gate $C(U)$ of (3) is unitary for every unitary $U$.^✦ The same holds of the rotation gates, from the unitarity of their matrices, and of the propagator of an observable and the global phase gate. Products and tensor products of unitaries are again unitary, and a unitary carries a state to a state, so that $\tilde U_{J,t}\, \rho\, \tilde U_{J,t}^\dagger$ in (10) is a state of two qubits.