As a consequence of this definition, unitary matrices have a very particular structure. Upon seeing a $2 \times 2$ matrix of complex numbers, you might think that we need 8 real numbers to specify it (i.e., 1 complex number per entry, giving 2 real numbers per entry). By working through the exercises in this section, you'll see how consequences of this definition enable us to specify them with only 3 real numbers. This will provide insight into the form of some of the operations you'll see in the upcoming sections. ___ ***Exercise I.3.1.*** Suppose we write $$ U = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, $$ where $a, b, c, d$ are complex numbers. Evaluate the matrix product $U U^\dagger$ and write down the set of equations that must be satisfied for $U$ to be unitary. What does this tell you about the rows of $U?$ Do the same for $U^\dagger U,$ and check what this tells you about the columns of $U.$

Solution.

Evaluating $U U^\dagger$ gives us $$ U U^\dagger = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix} = \begin{pmatrix} \vert a\vert ^2 + \vert b\vert ^2 & a c^* + b d^* \\ a^*c + b^*d & \vert c\vert ^2 + \vert d\vert ^2 \end{pmatrix}. $$ Setting this equal to the identity gives 4 equations: $$ \begin{align*} \vert a\vert ^2 + \vert b\vert ^2 &= 1, \\ a c^* + b d^* &= 0, \\ a^*c + b^*d &= 0, \\ \vert c\vert ^2 + \vert d\vert ^2 &= 1. \end{align*} $$ This shows us that the lengths of both rows are 1. Doing the same for $U^\dagger U$ produces a similar set of equations that tells us the columns also have length 1. In both cases, products of rows with other rows gives a result of 0, thus different rows are orthogonal to each other. We say that $U$ has **orthonormal** rows and columns, i.e., its rows and columns are orthogonal and normalized (they have length 1 just like state vectors). ▢

___ Having learned about the rows and columns of $U,$ we can see that there are relationships between $a, b, c$ and $d$ that can simplify how we express unitary operations. ___ ***Exercise I.3.2.*** Starting from $$ U = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, $$ where $a, b, c, d$ are complex numbers, show that $U$ can be expressed as $$ U = \begin{pmatrix} a & -e^{i\beta} c^* \\ c & e^{i \beta} a^* \end{pmatrix}, $$ where $\beta$ is a real number (and we refer to the complex value $e^{i\beta}$ as a **phase**).

Hint.

The condition of unitarity, $U U^\dagger = I,$ tells you something about both the inverse and the determinant of $U.$ This can be applied to decrease the amount of complex numbers needed to describe $U.$

Solution.

Since $U U^\dagger = I,$ this means that $U^{-1} = U^\dagger.$ Inverting the matrix tells us that $$ U^\dagger = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix} = \frac{1}{\hbox{det}(U)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. $$ This allows us to re-express both $d$ and $b$ in terms of $a$ and $c$ respectively: $$ d = \hbox{det}(U)a^*, \quad b = -\hbox{det}(U) c^*. $$ This gives us the form $$ U = \begin{pmatrix} a & -\hbox{det}(U) c^* \\ c & \hbox{det}(U) a^* \end{pmatrix}. $$ Since determinants are multiplicative, $\hbox{det}(U U^\dagger) = \hbox{det}(U) \hbox{det}(U^\dagger) = 1.$ Therefore, the determinant of $U$ must be a complex number with modulus 1 (i.e., a unit complex number). Let's write $\hbox{det}(U) = e^{i \beta}$ so that $$ U = \begin{pmatrix} a & -e^{i\beta} c^* \\ c & e^{i \beta} a^* \end{pmatrix}. $$

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___ ***Challenge Exercise 1.3.3.*** Starting from the expression for $U$ obtained in the previous exercise, show that any unitary matrix can be expressed as $$ U(\phi, \theta, \omega) = \begin{pmatrix} e^{-i(\phi + \omega)/2} \cos(\theta/2) & -e^{i(\phi - \omega)/2} \sin(\theta/2) \\ e^{-i(\phi - \omega)/2} \sin(\theta/2) & e^{i(\phi + \omega)/2} \cos(\theta/2) \end{pmatrix}, $$ where $\phi,$ $\theta,$ and $\omega$ are real numbers.

Hint.

Start by expressing the complex numbers $a$ and $c$ in polar form to uncover the $\theta$ dependence. Then, find a way to re-express the phases involving $\phi,$ $\omega$ to match the result.

Hint.

In quantum computing, $U$ and $e^{i\theta} U$ are effectively the same operation. You can thus factor out complex numbers with modulus 1 from any unitary matrix.

Solution.

As per the first hint, let's write our complex numbers in polar form, $a = r e^{i\alpha}$ and $c = s e^{i\gamma}.$ Since $U$ is orthonormal, it must be that $\vert a\vert ^2 + \vert c\vert ^2 = r^2 + s^2 = 1.$ So without loss of generality, we can set $r = \cos(\theta / 2),$ $s = \sin(\theta / 2)$ to obtain $$ U = \begin{pmatrix} e^{i\alpha} \cos(\theta/2) & -e^{i(\beta-\gamma)} \sin (\theta/2) \\ e^{i\gamma} \sin(\theta/2) & e^{i (\beta - \alpha)} \cos(\theta/2) \end{pmatrix}. $$ The next step is to massage the phases. The second hint tells us that we can factor out complex numbers from our matrix. We start by removing a factor of $e^{i\beta/2}$ from each element: $$ U = \begin{pmatrix} e^{i \left( \alpha - \frac{\beta}{2} \right)} \cos(\theta/2) & -e^{i\left(\frac{\beta}{2}-\gamma\right)} \sin (\theta/2) \\ e^{i\left(\gamma - \frac{\beta}{2}\right)} \sin(\theta/2) & e^{i \left(\frac{\beta}{2} - \alpha\right)} \cos(\theta/2) \end{pmatrix}. $$ Next, let's adjust the signs to match: $$ U = \begin{pmatrix} e^{-i\left(\frac{\beta}{2} - \alpha\right)} \cos(\theta/2) & -e^{i\left(\frac{\beta}{2}-\gamma\right)} \sin (\theta/2) \\ e^{-i\left( \frac{\beta}{2} - \gamma\right)} \sin(\theta/2) & e^{i \left(\frac{\beta}{2} - \alpha\right)} \cos(\theta/2) \end{pmatrix}. $$ Now notice that in the desired expression, each phase contains both $\phi$ and $\omega.$ This suggests that all our phases should contain some combination of $\alpha,$ $\beta,$ and $\gamma.$ To ensure this, we both add and subtract $\frac{\alpha}{2}$ to the upper right and lower left corners, and similarly add and subtract $\frac{\gamma}{2}$ to the two diagonal elements. We'll also rewrite the existing $\alpha$ and $\gamma$ as a combination of two $\frac{\alpha}{2}$ (or $\frac{\gamma}{2}$): $$ U = \begin{pmatrix} e^{-i\left(\frac{\beta}{2} - \frac{\alpha}{2} - \frac{\alpha}{2} - \frac{\gamma}{2} + \frac{\gamma}{2}\right)} \cos(\theta/2) & -e^{i\left(\frac{\beta}{2}- \frac{\gamma}{2} - \frac{\gamma}{2} + \frac{\alpha}{2} - \frac{\alpha}{2}\right)} \sin (\theta/2) \\ e^{-i\left( \frac{\beta}{2} - \frac{\gamma}{2} - \frac{\gamma}{2} - \frac{\alpha}{2} + \frac{\alpha}{2}\right)} \sin(\theta/2) & e^{i \left(\frac{\beta}{2} - \frac{\alpha}{2} - \frac{\alpha}{2} - \frac{\gamma}{2} + \frac{\gamma}{2}\right)} \cos(\theta/2) \end{pmatrix}. $$ The final step is to make the substitution $$ \begin{align*} \phi &= \beta - \alpha - \gamma, \\ \omega &= \gamma - \alpha, \\ \end{align*} $$ to obtain $$ U(\phi, \theta, \omega) = \begin{pmatrix} e^{-i(\phi + \omega)/2} \cos(\theta/2) & -e^{i(\phi - \omega)/2} \sin(\theta/2) \\ e^{-i(\phi - \omega)/2} \sin(\theta/2) & e^{i(\phi + \omega)/2} \cos(\theta/2) \end{pmatrix}. $$ While this is the most general parametrization of a unitary matrix, many common single-qubit operations are actually much simpler than this. In the next few sections, we will explore a number of the most-used unitary operations and see how they affect qubit states. ▢

Codercise I.3.1 — Unitaries in PennyLane

Codercise I.3.2 — Parametrized unitaries

Operations on qubits

Unitary parametrization