Amplitude and Phase

The latest version of this notebook is available on https://github.com/QISKit/qiskit-tutorial.

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Contributors

Jay Gambetta, Antonio Córcoles

Introduction

In superposition we looked at the concept of computational states, and superpostion states. Here we continue the development of the understanding of a quantum state by introducing the concept of amplitude and phase.

A qubit

The fundamental unit of quantum information is a qubit. It is a two-dimensional vector space of the complex numbers $\mathbb{C}^2$. A qubit has some similarities to a classical bit, but is overall very different. Like a bit, a qubit can have two possible values - normally a 0 or a 1 which we denote $|0\rangle$ and $|1\rangle$. The difference is that whereas a bit must be either 0 or 1, a qubit can be 0, 1, or a superpostion of both. An arbitrary pure state of the qubit can be written as

$$ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle$$

where $\alpha$ and $\beta$ are complex numbers. When a qubit in the state $|\psi\rangle$ is measured, the result is not deterministic, and one obtains the state $|0\rangle$ with probability $|\alpha|^2$ and the state $|1\rangle$ with probability $|\beta|^2$. This imposes the constrain that $|\alpha|^2 + |\beta|^2=1$ from conservation of probability. We can now define a pure state for a qubit as

$$ |\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle$$

where $0\leq \phi < 2\pi$ and $0 \leq \theta \leq \pi$.

Quantum gates for a qubit are represented as a $2\times2$ unitary matrix $U$. The action on the quantum gate is found by

$$ |\psi'\rangle = U |\psi \rangle $$

and the most general unitary must be able to take $|0\rangle$ to the above state. This gives

$$U = \begin{pmatrix} \cos(\theta/2) & a \\ e^{i\phi}\sin(\theta/2) & b \end{pmatrix}$$

where $a$ and $b$ are complex numbers constrained such that $U^\dagger U = I$ for all $0\leq \theta \leq \pi$ and $0\leq\phi < 2\pi$. This gives 3 constraints and as such $a\rightarrow -e^{i\lambda} \sin(\theta/2)$ and $b \rightarrow e^{i\lambda + i \phi} \cos(\theta/2)$ where $0\leq \lambda < 2\pi$

$$U = \begin{pmatrix} \cos(\theta/2) & -e^{i\lambda} \sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i\lambda + i \phi} \cos(\theta/2) \end{pmatrix}$$

This allows us to define a general single-qubit rotation as a function of the three parameters $\theta$, $\phi$ and $\lambda$, $U(\theta, \phi, \lambda)$

It is generally useful to understand general single-qubit rotations in terms of a very special group of matrices called the Pauli matrices $X$, $Y$ and $Z$. We already saw in superposition that the $X$ Pauli matrix could be defined as

$$ X =\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Similarly, we can also define

$$ Y =\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$

and

$$ Z =\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

An important aspect of the Pauli matrices is that $X^2=Y^2=Z^2=I$. This allows us to simplify

$$e^{i \theta P}= \cos(\theta)I + i\sin(\theta)P$$

with $P$ a Pauli matrix. The Pauli matrices can be thus considered as the generators of rotations in the single-qubit space. We can then define the rotations

$$r_x(\theta) = e^{-i \theta/2 X} = \cos(\theta/2)I - i \sin(\theta/2)X = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$$$r_y(\theta) = e^{-i \theta/2 Y} = \cos(\theta/2)I - i \sin(\theta/2)Y = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$$$r_z(\theta) = e^{-i \theta/2 Z} = \cos(\theta/2)I - i \sin(\theta/2)Z = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix} := \begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{pmatrix}$$

And now, in terms of the previously defined $U(\theta, \phi, \lambda)$, we can define the parametrized rotations

$$u_3(\theta, \phi, \lambda) = U(\theta, \phi, \lambda) = r_z(\phi)r_y(\theta)r_z(\lambda) = \begin{pmatrix} e^{-i(\phi + \lambda)/2}\cos(\theta/2) & -e^{-i(\phi - \lambda)/2} \sin(\theta/2) \\ e^{i(\phi-\lambda)/2}\sin(\theta/2) & e^{i(\phi + \lambda)/2} \cos(\theta/2) \end{pmatrix}$$

where we have multiplied the above defined $U(\theta, \phi, \lambda)$ by a global phase $e^{-i(\phi + \lambda)/2}$

Similarly, we can define the parametrized rotations

$$u_2(\phi, \lambda) = U(\pi/2, \phi, \lambda)$$

and

$$u_1(\lambda) = U(0,0,\lambda) := r_z(\lambda)$$

Phase gates

Let's explore the action of some of the single-qubit gates we have just defined. We will start with the phase gate $u_1(\lambda)$. We can see from the gate matrix that this gate does not modify the state $|0\rangle$ and applies a phase of $\lambda$ to the state $|1\rangle$. In order for us to observe the effect of this gate on a qubit we need to make use of the Hadamard gate defined in superposition.

Following a Hadamard gate, we will perform a $u_1(\lambda)$ of varying angle $\lambda \in \{ - \pi, \pi \}$ and observe the qubit state oscillate between the states $|+\rangle$ ($X$ eigenvalue +1) and $|-\rangle$ ($X$ eigenvalue -1).

We can see the eigenstate of $X$ oscillating between +1 and -1. The six symbols in the plot correspond to special gates that we use as part of our universal set of gates. These are $T$ ($T^{\dagger}$) for the red dot (cross), $S$ ($S^{\dagger}$) for the black dot (cross) and $Z$ ($Z^{\dagger}$) for the green dot (cross). Therefore, we can see that $T=u_1(\pi/4)$, $S=u_1(\pi/2)$ and $Z=u_1(\pi)$, with their adjoints corresponding to the negative arguments.

Now let's have a look at how the unitaries $u_3$ operate on a qubit state.

In this case, we see that $u_3(\pi/2,0,\pi)$ corresponds to a $\pi/2$ rotation around $X$ (black dot) and $u_3(\pi,0,\pi)$ corresponds to a bit-flip around $X$ (green dot). The rotations in the negative directions are shown as crosses. We could also have implemented these rotations around the $Y$ axis by simply using $u_3(\lambda,\pi/2,\pi/2)$.

As an exercise, prove that the Hadamard gate can be implemented as $H = u_2(0, \pi)$.

Now you know about single qubit states, it's time to move onto multi-qubit states. Please continue to entanglement to explore further!