Qubits, qubits, qubits: more is different

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. This is Lecture 3. It would probably be a good idea to have read the previous Lectures before continuing.

What did you learn last week?
Last week was your quantum trial by fire! You were introduced to the fundamental unit of quantum information: the qubit. Manipulating qubits is applied linear algebra, which is greatly facilitated by Dirac notation. Using Dirac notation, you modified the state of a qubit and calculated the probability of each results when “reading” the quantum information — the act of measurement.
What will you learn this week?
This week you will meet… more qubits! You’ll find out how multiple qubits are represented in the language of linear algebra and Dirac notation. You’ll be introduced to entanglement to get your first taste of a crucial difference between quantum and classical information. At the end of this week, you’ll have all the tools you need to analyse a quantum algorithm.
What will you be able to do at the end of this week 2?
At the end of this module, you should be able to answer these questions:
How is the state of two qubits represented?
How is the tensor product shorthand notation used?
How are quantum circuits of multiple qubits represented?
How do I prove a state is entangled?
How do I create an entangled state?

Before we get started, maybe you want to test your knowledge of qubits and Dirac notation first.

Qubits, qubits, qubits: More is the same

Back to bits. Remember that one bit is something that can be one of two values. A useful representation, especially if you have a standard keypad, is 0 and 1 for those values. If we have two bits, the state of them represents one of four values — 00,01,10, or 11 in binary notation. An analogous thing happens with qubits. The state of a single qubit can be represented by a 2-dimensional complex vector. The state of two qubits can be represented by a 4-dimensional complex vector.

As we know from linear algebra, this means any vector can be written as a linear combination of 4 basis vectors. The labels are arbitrary, but it is convenient to give the preferred one — the one we will call the computational basis in quantum computing — in binary. The state must still be a normalised vector, and so the magnitude of the coefficients sum to one.

The same correspondence between Dirac notation and the more familiar column vectors holds.

Matrices, usually written as 4×4 tables of numbers, can also be written in Dirac notation.

At this point it might seem, though, that Dirac notation is actually more cumbersome than writing out columns and tables. However, you will recall the power of summation notation!

Lastly, you will recall from the lecture on a single qubit that reading the quantum data amounts to performing a measurement in the computational basis. This is also the same for two, or more, qubits. However, we did oversimplify things. First, let’s review how to read a qubit using the language of projectors.

Now, we can more easily state the way quantum data is read for two qubits and later we will see how this can be generalised.

Qubits, qubits, qubits: More is different

Above, we looked at states of ququarts — that is, objects that can be represented by 4-dimensional complex vectors. We didn’t answer the following question. If I have one qubit in state |𝜓₀⟩ and another in state |𝜓₁⟩, which 4-dimensional complex vector represents the state of the pair? For that we use the tensor product, which for column vectors is sometimes called the kronecker product. There are some shorthand techniques for writing these things that you should be aware of. How you write it depends on what you want to achieve, but they all mean the same thing.

You can see that we have identified the preferred basis to be associated with the computational basis of each qubit. Now you may be thinking to yourself, can every vector in 4 dimensions be written as |𝜓₀⟩|𝜓₁⟩? No, consider the vector |𝜓⟩ = |00⟩+|11⟩ (suitably normalised). We can prove it cannot be written |𝜓⟩=|𝜓₀⟩|𝜓₁⟩ for any two single qubit vectors.

A vector that can be written |𝜓₀⟩|𝜓₁⟩, such as |00⟩ for example, is called a product state. A state that cannot be written this way is called entangled. Such states are a significant departure from the classical concept of information. With bits, to know the state of two bits I only need to tell the state of the first and the second, and then you know the state of both. The same is not true for qubits. That is, to know the state of qubits, it is not enough to know the state of each independently. More is different. The whole is greater than the sum of its parts. Turns out Aristotle understood more about quantum physics than many of its founders!

Let’s play: Find the missing gate

We know that when qubit 0 is in state |𝜓₀⟩ and qubit 1 is in state |𝜓₁⟩, the pair are in state |𝜓⟩=|𝜓₀⟩|𝜓₁⟩. But what about if we apply a gate to qubit 0? So, qubit 0 is now in state U₀|𝜓₀⟩, but what about the pair? Naively you might write U₀|𝜓⟩=U₀|𝜓₀⟩|𝜓₁⟩, and that would be fine, but it is ambiguous since U₀ is really a 2×2 matrix and |𝜓₀⟩|𝜓₁⟩ a 4×1 vector. This is where using the explicit notation of the tensor product comes in handy. The identity matrix means don’t do anything to this qubit.

Basically, whenever things don’t match up, add an identity matrix. An example is a measurement that only occurs on a single qubit.

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