The Exercise

Three qubits evolve under the Hamiltonian $X \otimes Y \otimes Z$: the Pauli $X$ acts on the first qubit, $Y$ on the second and $Z$ on the third. Exercise 4.51 asks for this evolution over any interval of time, built from the evolution under $Z \otimes Z \otimes Z$ and from gates that each act on a single qubit. The library answers with one identity between unitaries of the three qubits.

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The qudit with $d$ levels is the quantum system whose state space is $\mathbb{C}^d$ with its standard inner product.^✦ A qubit is the qudit with two levels, and the qudit with $2^n$ levels is the register of $n$ qubits; the exercise takes place on the register of three, with state space $\mathbb{C}^8$.

An observable $A$ of $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$; its real spectrum is the set of values a measurement can return. A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$, an evolution of $S$ as a closed system.^✦ For unitaries $U$ and $V$ of $S$, the composite $UV$ is the unitary of $S$ that applies $V$ first and then $U$.^✦ The adjoint $U^\dagger$ of a unitary $U$ of $S$ is again a unitary of $S$.^✦

The propagator $U_A(t_1, t_2)$ of an observable $A$ of $S$, taken as the Hamiltonian, from time $t_1$ to time $t_2$ is the unitary $$U_A(t_1, t_2) = \exp\!\left[-\frac{i A (t_2 - t_1)}{\hbar}\right]$$(1)^✦ of $S$, with $\hbar$ Planck’s constant. It solves the Schrödinger equation: a state $|\psi(t_1)\rangle$ at time $t_1$ becomes $|\psi(t_2)\rangle = U_A(t_1, t_2) |\psi(t_1)\rangle$ at time $t_2$.

Pauli Strings

The Pauli matrices are the four complex $2 \times 2$ matrices $$\sigma_0 = I, \quad \sigma_1 = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2)^✦ with $I$ the identity and $i$ the imaginary unit.

The tensor product $M_1 \otimes \cdots \otimes M_n$ of $n$ complex $2 \times 2$ matrices is the operator on $n$ qubits whose matrix, with rows and columns labelled by strings $a$ and $b$ of $n$ bits, has the entries $$(M_1 \otimes \cdots \otimes M_n)_{ab} = \prod_{k=1}^{n} (M_k)_{a_k b_k}.$$(3)^✦ For a choice $g$ of one Pauli matrix $\sigma_{g_k}$ for each qubit $k$, the Pauli string $\sigma_{g_1} \otimes \cdots \otimes \sigma_{g_n}$ is the tensor product (3) of these matrices.^✦

For a choice $g$ of one Pauli matrix $\sigma_{g_k}$ for each of $n$ qubits and a real number $c$, the Pauli string Hamiltonian $c\, \sigma_{g_1} \otimes \cdots \otimes \sigma_{g_n}$ is the observable of the register of $n$ qubits that acts on its state vectors as $c$ times the matrix of the Pauli string. With $c = 1$, the choices $(1, 2, 3)$ and $(3, 3, 3)$ give the Hamiltonians $X \otimes Y \otimes Z$ and $Z \otimes Z \otimes Z$ of three qubits.

The Basis Change

The Hadamard gate $H$ and the phase gate $S$ are the complex $2 \times 2$ matrices $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}.$$(4) The basis change gate for $Y$ is $SH$, the Hadamard gate followed by the phase gate; for $X$ it is $H$, and for $Z$ it is $I$. The basis change family of a Pauli string on $n$ qubits is the list of $n$ gates whose $k$th entry is the basis change gate of the $k$th Pauli matrix of the string. The gate $SH$ takes the eigenbasis of $Z$ to the eigenbasis of $Y$, as $H$ takes it to the eigenbasis of $X$.

The basis change gate $H \otimes SH \otimes I$ is the unitary of the register of three qubits given by the tensor product (3) of the basis change family $(H, SH, I)$.

The Claim

What Is Checked

The claim is stated over the register of three qubits, the qudit with eight levels, and over three of its unitaries: the propagators (1) of the Pauli string Hamiltonians $X \otimes Y \otimes Z$ and $Z \otimes Z \otimes Z$, and the basis change gate $H \otimes SH \otimes I$, composed with its adjoint.