## 3.7 *Remarks and exercises*

### 3.7.1 Orthonormal Pauli basis

Show that

### 3.7.2 Pauli matrix expansion coefficients

Recall that any

- Show that the coefficients
a_k (fork=x,y,z ) are given by the inner producta_k = (\sigma_k|A) = \frac12\operatorname{tr}\sigma_k A .

In these notes, we usually deal with matrices that are Hermitian (

- Show that the unitarity condition implies that
u_0^2 + u_x^2 + u_y^2 + u_z^2 = 1 and show, using this parametrisation, that the determinant ofU ise^{i2\gamma} .

### 3.7.3 Linear algebra of the Pauli vector

Show that

\frac12\operatorname{tr}(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma}) = \vec{a}\cdot\vec{b} .*(You may find a certain identity involving the cross product helpful.)*Show that any

\vec{n}\cdot\vec{\sigma} has eigenvalues\pm|\vec{n}| .Show that, if

\vec{n}\cdot\vec{m}=0 , then the operators\vec{n}\cdot\vec{\sigma} and\vec{m}\cdot\vec{\sigma} anticommute.

### 3.7.4 Matrix Euler formula

Show that, if

A^2=\mathbf{1} , then we can manipulate the power series expansion ofe^{iA} into a simple expression: for any real\alpha ,e^{i\alpha A} = (\cos\alpha)\mathbf{1}+ (i\sin\alpha)A. Show that any

(2\times 2) unitary matrixU can be written, up to an overall multiplicative phase factor, asU = e^{i \theta \vec{n}\cdot\vec{\sigma}} = (\cos\theta)\mathbf{1}+ (i\sin\theta)\vec{n}\cdot\vec{\sigma}. (The argument here is the same as the argument thate^{i\theta}=\cos\theta +i\sin\theta ).

### 3.7.5 Special orthogonal matrix calculations

Show that

\operatorname{tr}\sigma_x\sigma_y\sigma_z = 2i .Consider

U(\vec e_k\cdot\sigma_k)U^\dagger=U\sigma_kU^\dagger={\vec f_k}\cdot\vec\sigma. SoU maps the unit vectors\vec e_x ,\vec e_y , and\vec z_z , (along thex -,y -, andz -axis, respectively), to new unit vectors\vec f_x ,\vec f_y , and\vec f_z . We already know that, in Euclidean space, this transformation is described by a(3\times 3) orthogonal matrixR_U . How are the three vectors\vec f_x ,\vec f_y , and\vec f_z related to the entries in matrixR_U ?Show that

\begin{aligned} \operatorname{tr}\sigma_x\sigma_y\sigma_z &= \operatorname{tr}({\vec f_x}\cdot\vec\sigma)( {\vec f_y}\cdot\vec\sigma)({\vec f_z}\cdot\vec\sigma) \\&= 2i\det R_U \end{aligned} (which implies that\det R_U=1 ).Use the orthonormality of the Pauli basis along with Equation (

\ddagger ) to show that the elements of the matrixR=R_U can be expressed in terms of those of the matrixU , in the formR_{ij}=\frac12\operatorname{tr}\left(\sigma_i U\sigma_j U^\dagger\right). Here,i andj take values in\{1,2,3\} , and\sigma_1\equiv\sigma_x ,\sigma_2\equiv\sigma_y ,\sigma_3\equiv\sigma_z .

### 3.7.6 Phase as rotation

Show that the phase gate

**Hint.** It might be helpful to start with the

### 3.7.7 Geometry of the Hadamard

Express the Hadamard gate

H in terms of\vec{n}\cdot\vec{\sigma} , and show that\begin{aligned} HZH&=X \\HXH&=Z \\HYH&=-Y. \end{aligned} Show that the Hadamard gate

H turns rotations about thex -axis into rotations about thez -axis, and vice versa. That is,\begin{aligned} H \left( e^{-i\frac{\varphi}{2}Z} \right) H &= e^{-i\frac{\varphi}{2}X} \\H \left( e^{-i\frac{\varphi}{2}X} \right) H &= e^{-i\frac{\varphi}{2}Z}. \end{aligned}

### 3.7.8 Swiss Granite Fountain

In the Singapore Botanic Gardens, there is a sculpture by Ueli Fausch called “Swiss Granite Fountain”. It is a spherical granite ball which measures 80cm in diameter and weighs 700kg, and is kept afloat by strong water pressure directed through the basal block. It is easy to set the ball in motion, and it keeps rotating in whatever way you start for a long time. Suppose you are given access to this ball only near the top, so that you can push it to make it rotate around any horizontal axis, but you don’t have enough of a grip to make it turn around the vertical axis. Can you make it rotate around the vertical axis anyway?

### 3.7.9 Dynamics in a magnetic field

A qubit (spin one-half particle) initially in state ^{61}
What is the state of the qubit after time

In Earth’s magnetic field, which is about

0.5 gauss, the value of\omega is of the order of10^6 cycles per second.↩︎