Bloch Fear: I See Qubits

Welcome to Introduction to Quantum Computing. I am your guide, Associate Professor Chris Ferrie, a researcher in the UTS Centre for Quantum Software and Information. These are the notes for Lab 2. You should have already enjoyed Lecture 2. The syllabus is here:

In Lecture 2, you were introduced to qubits and the mathematical tools needed to work with them. Dirac notation is a powerful calculational tool to analyse quantum circuits, and perform linear algebraic manipulations more generally. We learned about quantum measurement and what it means to “look at” a qubit. But what does a qubit state “look like”? That is our first task.

Still Jenny from the Bloch

We have talked about qubits and states. We know qubits are abstract things that can be represented by 2-dimensional complex vectors, and the particular vector value the qubit is currently assigned is called its state. But just as the state of a bit can be represented by any binary alphabet (0 and 1 is just the most common), the state of a qubit can be represented by things other than 2-dimensional complex column vectors. One of the most popular representations is called the Bloch sphere.

The Bloch sphere is nice because it let’s you “see” the qubit. A bit, b, is b ϵ {0,1}. You can see both possible values laid out in front of you. Where as, a qubit is — I keep saying — is a vector in a 2-dimensional complex vector space. You can’t visualise this space because it has 4 real dimensions. And unless you are Tony Stark, we don’t have the ability to see in that many dimensions. But here comes this Bloch sphere and like magic you can now visualise the whole space of possibilities — well, almost.

Google “bloch sphere” and spend 15 minutes familiarising yourself with the concept. Seriously, go do it — I’ll wait.

Based on what you know from Lecture 2 — and in only 15 minutes of research — here is what I expected went through your mind. (1) oh, that’s pretty neat; (2) yeah, I like this a lot; (3) wait why are orthogonal vectors 180° apart; (4) there’s not even any complex numbers here; (5) this is all a lie!

The Bloch sphere is important for historical reasons — you should know it because it’s unavoidable. But, it is also an opportunity to talk about something you will hear about all the time — phase. Phase is the physics terminology for the argument of a complex number when written in polar coordinates. Often, the angle itself and the exponential are both referred to as the phase, and which we mean is almost always discernible by the context. Since a qubit state has two complex numbers, there are two phases. If we factor one out of the superposition, we identify one as the global phase and the other as the relative phase.

Suppose I have a qubit with the state |𝜓⟩. I can’t read |𝜓⟩ off the qubit — all I can do is measure it and get |0⟩ or |1⟩. Let’s see how a global and relative phase affect the chances of seeing the |0⟩ outcome.

So, it seems that global phases do not matter as far as observable consequences are concerned — all the action is in the relative phase! Okay, so we’ve gotten rid of 1 of the 4 real numbers. But a point on the sphere is specified by only two angles, so we need to get rid of one more. Recall that a state written in the computational basis as |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩ must satisfy the normalisation |𝛼|² + |𝛽|² = 1. This constrains the sum of magnitudes of two complex numbers. The two magnitudes are constrained to lie on a circle and can be specified by a single angle.

As an exercise, convince yourself that if we had not divided 𝜃 by 2 in the definition, we would end up with the Bloch dome instead of the Bloch sphere.

Working with Dirac

Have a look at the following video tutorial on how to work with Dirac notation. It introduces some new handy concepts such as the components of vectors and matrices in Dirac notation, which uses the ket-bra, or “outer product”.

But I don’t understand!

Did you know that you are able to convey more information as a speaker when you are talking to a good listener? In other words, effective teaching is only possible with active learners. Not listening, reading, or participating all week and then saying I didn’t understand the material is annoying as it is useless. A good litmus test for how well you are listening is your ability to ask good questions. How will you know if a question is good? No need. The community will tell you. Your task before the next lecture is to ask a question on Quantum Computing Stack Exchange that receives an answer.

Quantum theorist by day, father by night. Occasionally moonlighting as a author.