To describe a quantum computer mathematically, we can do math similar to the known topic of probabilistic systems. We therefore first look into describing a probabilistic system.
Deterministic possibilities
At first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be heads or tails and for a dice this could be the labels of the different sides. We call these possibilities deterministic possibilities. Note that we will only be using a finite number of possibilities.
Imagine you have a random number generator, which outputs 2-bit numbers. The deterministic possibilities of this generator are \(00\), \(01\), \(10\) and \(11\).
Probability distribution
Next, we need to assign each possibility a probability. We write this as \(\Pr[x]=p\) where \(p \in [0,1]\) is the probability of the deterministic possibility \(x\).
For a coin flip the probability of heads would be \(\Pr[\text{heads}] = \frac{1}{2}\) and the probability for tails would be \(\Pr[\text{tails}] = \frac{1}{2}\).
If we combine all probabilities for all the possible outcomes and write them as a vector, we get a probability distribution.
Definition 2.1 (Probability distribution) A vector \(d \in \mathbb{R}^n\) is a valid probability distribution iff \(\sum d_i = 1\) and \(\forall i\) \(d_i \geq 0\).
This vector has \(n\) entries, where each entry corresponds to a deterministic possibility \(X\) and the probability of \(X\) is \(\Pr[X] = d_i\). The sum over all probabilities has to be \(1\) and each entry needs to be nonnegative in order to be a valid probability.
For a coin flip the probability distribution would be \(d_{\text{coin}} \in \mathbb{R}^2\) with \(d = \begin{pmatrix}\frac{1}{2}\\ \frac{1}{2} \end{pmatrix}\)
Recall your random 2-bit number generator from above. Imagine your generator outputs each deterministic possibility with equal probability, except for the possibility \(00\), which is never generated. The corresponding probability distribution would be \[
d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix}
\]
Probabilistic processes
With a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a probabilistic process.
A probabilistic process is a collection of \(n\) probability distributions, where for each deterministic possibility \(i\) there is a probability distribution \(a_i\). This means, that if the system is in deterministic possibility \(i\) before the process is applied, the system will afterwards be distributed according to \(a_i\). We can write this as a matrix, where each column is a probability distribution \(a_i\).
Definition 2.2 (Probabilistic process) A matrix \(A \in \mathbb{R}^{n\times n}\) is a valid probabilistic process iff for every column \(a\) of \(A\), \(a\) is a valid probability distribution.
From Definition 2.1 we know that a valid probability distribution \(a\) has the properties \(\sum a_i = 1\) and \(\forall i\) \(a_i \geq 0\), therefore a matrix \(A\) is a probabilistic process iff \(A \in \mathbb{R}^{n \times n}\) with \(\sum a_i = 1\) and \(\forall i\) \(a_i \geq 0\) . Such a matrix is also called a stochastic matrix.
Imagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of \(\frac{1}{3}\). The probability distributions for each of the deterministic possibility would then be \[
a_{00} = \begin{pmatrix} \frac{2}{3} \\ 0 \\ 0 \\ \frac{1}{3} \end{pmatrix}, a_{01} =\begin{pmatrix} 0 \\ \frac{2}{3} \\ \frac{1}{3} \\ 0 \end{pmatrix}, a_{10} =\begin{pmatrix} 0 \\ \frac{1}{3} \\ \frac{2}{3} \\ 0 \end{pmatrix}, a_{11} = \begin{pmatrix} \frac{1}{3} \\ 0 \\ 0 \\ \frac{2}{3} \end{pmatrix}
\] From this we can construct the process as a matrix from these processes as follows: \[
A_{\text{flip}} = \begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \end{pmatrix} = \begin{pmatrix} \frac{2}{3} & 0 & 0 & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \end{pmatrix}
\]
Applying a probabilistic process
Having defined probability distributions and probabilistic processes, we can now combine these two elements and apply a probabilistic process on a probability distribution.
Definition 2.3 (Applying a probabilistic process) Given an initial probability distribution \(x \in \mathbb{R}^n\) and a probabilistic process \(A \in \mathbb{R}^{n\times n}\), the result \(y \in \mathbb{R}^n\) of applying the process \(A\) is defined as \[
y = Ax
\]
Recall the 2-bit number generator and the bit flip from above. Imagine you would first draw a random 2-bit number from the generator and then run the bit flip device. We already know that the probability distribution of the generator is \(d_\text{2-bit}\). Using \(A_\text{flip}\) we can calculate the final probability distribution: \[
A_\text{flip} \cdot d_\text{2-bit} = \begin{pmatrix} \frac{2}{3} & 0 & 0 & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \end{pmatrix}\begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix} = \begin{pmatrix} \frac{1}{9} \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{2}{9} \end{pmatrix}
\]