The Exercise

In the quantum search algorithm seen as a quantum simulation, the Hamiltonian is the sum $|x\rangle\langle x| + |\psi\rangle\langle\psi|$ of the projectors onto the marked state $|x\rangle$ and onto the starting state $|\psi\rangle$, and it is simulated by short evolutions under each projector in turn. The two states span a two dimensional space, so the question is one about a qubit: take $|x\rangle = |0\rangle$ and $|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ for an angle $\theta$. One step evolves the qubit for a time $\Delta t$ under $|x\rangle\langle x|$ and then for a time $\Delta t$ under $|\psi\rangle\langle\psi|$.

Equation (6.25) of the book writes this step in closed form: a global phase $e^{-i\Delta t}$ times a combination of the identity and the Pauli matrices, with coefficients built from $\Delta t$ and from the dot and cross products of the Bloch vectors of the two states. The exercise is to verify it.

A Quantum System

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 $\mathbb{C}^2$, with standard basis $|0\rangle$, $|1\rangle$.^✦

A pure state $|\psi\rangle$ of $S$ is a vector of unit norm in the state space of $S$, defined physically up to a global phase.^✦ For a polar angle $\theta$ and an azimuth $\phi$, both real, the state $|\psi(\theta,\phi)\rangle$ of the qubit is the pure state with vector $$|\psi(\theta,\phi)\rangle = \cos(\theta/2)\,|0\rangle + e^{i\phi}\sin(\theta/2)\,|1\rangle .$$(1)^✦ The starting state of the exercise is $|\psi(\theta,0)\rangle$, and $|\psi(0,0)\rangle = |0\rangle$.

The Bloch vector $\vec r(\theta,\phi)$ of polar angle $\theta$ and azimuth $\phi$ is the real vector in three dimensions $$\vec r(\theta,\phi) = (\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta),$$(2)^✦ the point of the unit sphere at those angles, with the $z$ axis along $|0\rangle$. The two Bloch vectors of the exercise are $\vec r(\theta,0) = (\sin\theta, 0, \cos\theta)$ and $\vec r(0,0) = \hat z$, the unit vector $(0,0,1)$ along the $z$ axis.

Evolution Under a Projector

An observable $A$ of a system $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ The projector Hamiltonian of a pure state $|\psi\rangle$ of $S$ is the observable $|\psi\rangle\langle\psi|$, the operator that sends a vector $|\varphi\rangle$ to $\langle\psi|\varphi\rangle\,|\psi\rangle$.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ The propagator of an observable $H$ of $S$, with Planck’s constant $\hbar$, from time $t_1$ to time $t_2$, is the unitary $$U(t_1,t_2) = \exp\bigl[-iH(t_2 - t_1)/\hbar\bigr],$$(3)^✦ which carries a state at $t_1$ to the state at $t_2$ under the Schrödinger equation, $|\psi(t_2)\rangle = U(t_1,t_2)|\psi(t_1)\rangle$.

Pauli Matrices

The Pauli matrices are the complex $2\times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(4) and the Pauli vector $\vec\sigma = (X, Y, Z)$ collects them, so that for a real vector $\vec n = (n_x, n_y, n_z)$ in three dimensions the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$.

For a time step $\Delta t$ and real vectors $\vec a$ and $\vec b$ in three dimensions, the axis vector $\vec v(\Delta t;\vec a,\vec b)$ of (6.25) is the real vector in three dimensions $$\vec v(\Delta t;\vec a,\vec b) = \cos(\Delta t/2)\,\frac{\vec a + \vec b}{2} + \sin(\Delta t/2)\,\frac{\vec a \times \vec b}{2}.$$(5)^✦

The step matrix $M(\Delta t;\vec a,\vec b)$, the right side of (6.25) without its phase, is the complex $2\times 2$ matrix $$M(\Delta t;\vec a,\vec b) = \bigl(\cos^2(\Delta t/2) - \sin^2(\Delta t/2)\, \vec a\cdot\vec b\bigr)\, I - 2i\sin(\Delta t/2)\, \vec v(\Delta t;\vec a,\vec b)\cdot\vec\sigma ,$$(6) with $I$ the identity and $\vec v$ the axis vector (5).

The Claim

In the product of the two evolutions, the first one stands on the right.

What Is Checked

The claim is stated over the qubit, its pure states $|\psi(\theta,0)\rangle$ and $|0\rangle = |\psi(0,0)\rangle$ of (1), the propagators (3) of their projector Hamiltonians, and the step matrix (6), built from the Pauli matrices (4) and the Bloch vectors (2).

The library proves that the vector (1) has unit norm, since $\cos^2(\theta/2) + \sin^2(\theta/2) = 1$ and the phase $e^{i\phi}$ has modulus one.