3 Quantum systems
With the basics for a probabilistic system defined, we now look into describing a quantum computer mathematically. In the following table you can see the analogy from the quantum world to the probabilistic world.
Probabilistic world | Quantum world |
---|---|
Probability distributions | Quantum states |
Probabilities | Amplitudes |
Deterministic possibilities | Classical possibilities |
Stochastic matrix as process | Unitary matrix as process |
3.1 Classical possibilities
Like in the probabilistic systems we need to define all outcomes for a quantum system. For example, a photon can be in the state up or down. We call these possibilities classical possibilities. Note that we will only be using a finite number of possibilities.
We will always assume the classical possibilities to be ordered in some way (even if it is an arbitrary one), like the deterministic possibilities. In the example above, the classical possibilities are \(0\), \(1\) not \(1\), \(0\). We will need this to know the order of entries in vectors and matrices later.
3.2 Quantum states
One of the most important element of the quantum world is a quantum state. A quantum state describes the state of a quantum system as a vector. Each entry of the vector represents a classical possibility (similar to the deterministic possibilities in a probability distribution). The entries of a quantum state are called amplitude. In contrast to a probabilistic system, these entries can be negative and are also complex numbers.
These amplitudes tell us the probability of the quantum state being in the corresponding classical possibility. To calculate the probabilities from the amplitude, we can take the square of the absolute value of the amplitude.
This means that for the classical possibility \(x\) and a quantum state \(\psi\) the probability for \(x\) is \(\Pr[x] = |\psi|^2\). To have valid probabilities, the sum of all probabilities need to sum up to \(1\). From this we get the formal definition of a quantum state:
3.3 Unitary transformation
We now have defined quantum states and need a way to describe some processes, which we want to apply on the quantum states. In the probabilistic world, we have stochastic matrices for this, but unfortunately we can not use these matrices on quantum states, since the output of applying these on a quantum state is not guaranteed to be a quantum state again. We therefore look for a different property of a matrix for which the outcome of applying that matrix is guaranteed to be a quantum state. The following Lemma is therefore useful.
Then the evolution of a quantum state is always described by a unitary matrix. So if the current state is \(\psi\) and we apply the transformation matrix \(U\), the state is \(U\psi\) afterwards.
A unitary matrix is by definition invertible, therefore we can undo all unitary transformations by applying \(U^\dagger\).