Speed Reading a Gate’s Operation from Its Matrix

In ​Classifying Quantum Gates​, you saw that single-qubit quantum gates can be classified by how they affect qubelets. In this section, you’ll learn to infer a gate’s matrix and understand how it affects the qubelets. We’ll not be concerned with whether the given matrix is unitary. Our interest is merely to figure out how the gate acts on the pentagon and triangle qubelets. Thus, we’ll assume valid gate matrices.

Single-Qubit Gate Matrices

To fix these ideas in your mind, consider the following matrix:

The letters , , , and denote complex numbers. Their complex conjugates are , , , and , respectively. So, if is , then . Equivalently, in polar coordinates using Euler’s formula:

Even though this matrix encodes how any quantum state, including blended ones, are modified by the corresponding gate, we’ll determine how the gate affects the qubelets by focusing on how the matrix affects the idealized states and , respectively.

To this end, as described in ​Quantum Gates as Matrices​, the first column lists the amplitudes of the quantum state obtained when the gate operates on .

In other words, when the gate corresponding to this matrix acts on , the gate puts the qubelet in the quantum state defined by the first column, as shown here:

Similarly, when the gate acts on the qubelet, the gate puts the qubelet in the quantum state defined by the second column, as follows:

Let’s now cover how to look at the entries of the gate matrix and reason out the following ways that the gate changes the qubelets:

• ​Leave Qubelets Alone​
• ​Switch Qubelets​
• ​Split Qubelets​
• ​Rotate Qubelets​
• ​Compound Operations on Qubelets​

Leave Qubelets Alone

When a gate acts on the pentagon qubelets only and doesn’t affect the triangle qubelets, the element corresponding to the amplitude of in the first column is zero. That is, the second element in the first column is zero:

If the gate acts on the triangle qubelets only and leaves the pentagon qubelets alone, then the amplitude for is zero. That is, the first element in the second column is zero:

Switch Qubelets

When a gate switches a qubelet, it “changes” it to the other type. For example, a pentagon qubelet is switched to a triangle qubelet. The entry for the amplitude of becomes zero and the one for the amplitude of is nonzero:

In the same way, when a triangle qubelet is switched to a pentagon qubelet, the second element in the second column is zero:

For instance, the NOT gate, which switches a pentagon qubelet with a triangle qubelet, and a a triangle qubelet with a pentagon qubelet, has the following matrix:

Split Qubelets

When a gate splits either a pentagon or a triangle qubelet, it creates qubelets of both types. For example, splitting a pentagon qubelet creates both a pentagon and a triangle qubelet. In other words, both entries in the respective column are nonzero.

If a gate splits both pentagon and triangle qubelets, then the matrix has all nonzero entries like the H gate:

Rotate Qubelets

When a gate rotates a qubelet, it shows up as a complex number in the entry associated with that qubelet. For example, when a gate acts on a triangle qubelet and rotates it by, say, radians or 45° anticlockwise (and leaves the pentagon qubelet alone), the quantum state is expressed as:

Using Euler’s formula, write the previous equation as:

Write this operation in the bottom right element of the matrix:

Thus, any time you see a complex number as an entry in the gate matrix, it means the corresponding qubelet is rotated.

Compound Operations on Qubelets

Knowing how to recognize the basic operations we’ve listed will let you infer when a gate combines them when modifying the qubelets. For example, consider the gate matrix associated with the ​Y Gate​, as follows:

Looking at the first column associated with the gate acting on the qubit, the complex number in the bottom element indicates that the pentagon qubelet is switched to the triangle qubelet and is rotated.

You can calculate the exact angle of rotation by noticing that can be expressed as follows, again using Euler’s formula:

Thus, the triangle qubelet is rotated radians, or a quarter turn anticlockwise.

You can reason out the action on the qubit in a similar way by looking at the second column.