Chaining Quantum Gates

If you want to learn to write quantum algorithms that solve hard problems with blazing speed, you will need to develop a gut feel for how gates adjust quantum states to arrive at the solution of your problem. To this end, in this section, you'll see how the qubelets concept can give you insight into what happens to the quantum states when you chain quantum gates together.

Consider the following quantum circuit that chains the H, Z and another H gate:

In this circuit, the $\ket{0}$ qubit in the quantum register $q[0]$ is acted on by a sequence of quantum gates — H, Z and H. The resulting state of the qubit is inspected by the Measure> gate on the right and recorded in the classical register $c[0]$.

The first $H$-gate on the left splits the $\ket{0}$ qubit:

Next, the $Z$-gate rotates the triangle $\ket{1}$ qubelet by $180^\circ$:

Finally, the $H$-gate on the right splits both the pentagon $\ket{0}$ and the triangle $\ket{1}$:

Since the $H$-gate splits a $\ket{1}$ into a pentagon $\ket{0}$ and an inverted triangle $\ket{1}$, it'll split an inverted triangle $\ket{1}$ into an inverted pentagon $\ket{0}$ and an inverted-inverted triangle $\ket{1}$. That is, a triangle that goes back to its non-inverted orientation.

Key Concept: Canceling Qubelets

According to the rules of quantum mechanics, a qubelet will cancel out with an inverted qubelet of the same type. So, the inverted pentagon cancels with the non-inverted one leaving a quantum state with 2 triangles:

So, after the canceling of the pentagon qublets, the quantum state only has 2 triangles. When you measure this quantum state, since it only has triangle $\ket{1}$ qubelets, the only classical state this quantum state can collapse to is a 1. In other words, no matter how many qubelets of the same type are in a quantum state, the qubit must always collapse to the classical bit corresponding to that qubelet.

Thus, starting out with a pentagon $\ket{0}$ qubit, the H-Z-H sequence of gates changes it to a triangle $\ket{1}$ qubit.

Remarkable Takeaway

What should strike you is that even though the H gates randomly collapse to 0 or 1, by understanding how the qubelets are acted upon by the gates, you can precisely figure out a chain of gates where the end result is deterministic.

Note that all of this canceling happens immediately without the need to invoke any special quantum instruction.

Of course, there's only so much you can do with a single qubit. To leverage the true power of quantum computers, in the next section you'll see the qubelets concept can help make sense of multi-qubit systems.

Where to learn more

The material in this section has been adapted from Chapter 7 of my book. In this chapter, you'll also learn how to write a quantum program for the concepts you've learned about here.