Quantum Bits

An introduction to quantum bits

Quantum Information

In the section about quantum mechanics, we talked about the idea of superposition, where more than one state is overlapped over top of each other. The system is in multiple distinguishable states at the same time - the information is superimposed. We don't know what going on in there, we can only talk in probabilities - what state are we likely to see if we measure (or observe) the system at a particular moment?

The Single Qubit System

A single quantum bit is the basic unit of quantum computation, but it's also the smallest (non-trivial) quantum system. Unlike a classical system, where a single bit can't do much at all, in the quantum system the individual bits can be used to do work alone (Nielsen, M. - p. 13).

Distinguishable States

Like a classical bit, the quantum bit has two observable states - 0 and 1. The difference here is very simple. In the quantum system, we have the potential for superposition. A quantum bit can be 1 and 0, at the same time. We measure this in between zone by talking about the probability of finding the qubit in a particular state when we measure it.

If you recall our discussion of vector spaces, the dimension of a system is defined by the number of basis vectors we have. So a single qubit system has two basis vectors - a 0 vector and a 1 vector.

This means that a single qubit system defines a plane. The combinations of those two basis vectors all live together on this plane. The vectors that live on the plane represent all the possible answers we can get to using a single qubit.

Notation

General Vector

A single qubit is represented as a 2-dimensional quantum vector. Recall that a quantum vector has as many dimensions as there are distinguishable states, so it makes sense that a single qubit vector has 2 elements. Like any quantum vector, we represent a qubit in an unknown state like this:

$$|\psi\rangle$$

Specific States

The two basis vectors, which represent the pure "0" state and the pure "1" state respectively, are represented like this:

$$|0\rangle = \begin{bmatrix}1\\0\end{bmatrix}, |1\rangle=\begin{bmatrix}0\\1\end{bmatrix}$$

Note here that the elements of this column vector represent the particular state a qubit is in. The first element tells us if a qubit exists in the first state (which is 0), and the second element tells us if a qubit exists in the second state (which is 1). As we add more qubits to the system, we offer more qubits and more combinations of states, and more elements.

Multiple Bits

As our system gets more complex and begins to contain more numbers, we can expand on this:

$$|00\rangle=\begin{bmatrix}1\\0\\0\\0\end{bmatrix},|01\rangle=\begin{bmatrix}0\\1\\0\\0\end{bmatrix},|10\rangle=\begin{bmatrix}0\\0\\1\\0\end{bmatrix},|11\rangle=\begin{bmatrix}0\\0\\0\\1\end{bmatrix}$$

For multi-qubit systems like this, we order qubits from right to left. So the rightmost qubit $|0\textcolor{red}1\rangle$ is the first, and the leftmost qubit is the last $|\textcolor{red}{0}1\rangle$.

You can imagine that if we have a whole bunch of qubits, this could get very confusing very quickly. So we might use the decimal version of these binary numbers streamline things:

$$|0\rangle=\begin{bmatrix}1\\0\\0\\0\end{bmatrix},|1\rangle=\begin{bmatrix}0\\1\\0\\0\end{bmatrix},|2\rangle=\begin{bmatrix}0\\0\\1\\0\end{bmatrix},|3\rangle=\begin{bmatrix}0\\0\\0\\1\end{bmatrix}$$

Binary number review

Combinations of the Basis Vectors

Going back to our earlier discussion of $|0\rangle$ and $|1\rangle$ as our basis vectors, this means that all of the possible measurements for $|\psi\rangle$ are represented by a linear combination of our basis vectors (Nielsen, M. - p. 13):

$$|\psi\rangle = a|0\rangle + b|1\rangle$$

Where $a$ and $b$ are complex numbers.

Recall that the inner product of our quantum vectors must equal 1, to allow for reversibility of operations: $\langle \psi|\psi\rangle = 1$

Measuring Qubits

When we measure our quantum bit, the information collapses into one state or the other. That means that this linear combination of states...

$$|\psi\rangle = a|0\rangle + b|1\rangle$$

...is a superposition of multiple possible states. We can solve this superposition two different ways - as a wave form, with the Schrodinger equation, or by literally looking at it.

When we measure our qubit (and it collapses into one state or the other), we store the result in a classical bit. That means that measuring $|0\rangle \rightarrow 0$ and $|1\rangle \rightarrow 1$.

The Probability of States

What will we get? It depends on what kind of superposition it is - that is to say it depends on the probabilities of arriving at each state.

The probability of getting any particular state, added to the probability of getting any other particular state has to add up to 1, but as long as that true the distribution could be anything (Nielsen, M. - p. 13):

$$\langle\psi|\psi\rangle=\text{prob}|0\rangle+\text{prob}|1\rangle=1$$

That is to say that we may be 50% likely to see $|0\rangle$ and 50% likely to see $|1\rangle$. The probability distribution then looks like this:

A graph showing an even probability distribution

The sum of the probabilities of finding the qubit in all possible states is therefore:

$$50\% + 50\% = 100\%$$

Or:

$$0.5+0.5=1$$

Quantum Results

Results from a classical system come in as a simple answer - for example, if we have two bits a and b:

$$a = 0, b = 1$$

And we add them together:

$$a + b = 1$$

This is the long and short of it, the answer is 1. I know the whole state of the system from top to bottom - I could measure it once, or ten thousand times, and I would always get the same answer.

Quantum systems are different - the state of the system is measured in probabilities. Thus, each time I measure it, I may get a different answer. The system may collapse into a different state. Therefore, I can't measure just once. If I do, I won't see the whole picture. The result that I'm looking for isn't a simple number. It's a probability distribution.

This also means we need to run our calculations multiple times - as many times as we can get away with in fact. This way we'll get as close as possible to the real truth of the system.