Basic Quantum Computing — Single Qubit Systems

We’re finally ready to start getting into some actual quantum computing. In our last post, we introduced the qubit. This was a quantum state of the form:

We said that it evolves in time according to some unitary matrix U.

In Quantum Computing, there is the idea of a quantum gate, which acts on a qubit. Just like a regular logic gate acts on a classical bit. Now a quantum gate changes a quantum state so it must be represented by a unitary matrix.

In this post, we’ll take a look at a couple of common gates that work on a single qubit.

The NOT Gate

Just like in classical computing, we have a NOT gate. This takes a qubit and sends 0 to 1 and visa versa. So if I have:

Applying a NOT gate to it gives me:

Because we can represent a qubit as a column vector:

We can represent the NOT gate by the matrix:

In a diagram we use the following symbols for it:

This gate is also known as the X gate because it is part of a family of three gates with the others called the Y and Z gates.

Pauli Gates

The other two gates that are in the same family as the NOT gate are the Y and Z gates. And they are given by the following matrices:

And we use the following symbols for it:

Hadamard Matrix

Finally, we are going to talk about the Hadamard gate. The Hadamard gate does the following:

At first, having a gate that does this might seem rather strange but it is in fact incredibly useful. This is because it allows us to put qubits into a state where they are equally likely to be 0 or 1.

It is represented by the matrix:

And we use the following symbol for it: