Basics
In order to fully understand this document, the basics of quantum physics and dirac notation are required. A really quick overview is given. See Quantum Physics for more details.
The quantum state of a qubit is a unitary vector denoted with |v\rangle.
A single qubit state can be expressed as a linear combination of an orthonormal basis, e.g.
|v\rangle = a |0\rangle + b |1\rangle \mapsto \begin{pmatrix} a \\ b \end{pmatrix}
where \{ |0\rangle, |1\rangle \} is called the standard basis:
|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \in \mathbb{C}^2 \qquad |1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \in \mathbb{C}^2
a and b are called amplitudes while their quare are called probabilities.
Another useful basys is the Hadamard basis:
|+\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \in \mathbb{C}^2 \qquad |-\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix} \in \mathbb{C}^2
A qubit-measurment device always returns the measurement result in it’s own basis. After the measurement has been performed, the qubit state collapse to the measured state.
This means that quantum mechanics allows us to perform fun party tricks like this one.
As we saw earlier, a qubit can be described with two complex numbers (four different real numbers): with this property, it is possible to encode lots of information (a.k.a. prepare the state) into a qubit amplitudes. The problem lies in the fact that it is impossible to measure them and that we cannot clone a qubit.
Two qubits that only differs by a global phase factor carry the same information.
The Bloch sphere is the space where all possible qubits lies.
On the Bloch sphere, two orthogonal qubits are aligned on the same direction with opposite orientations. On the z axis there is the standard basis, on the x axis there is the Hadamard basis and on the y axis there is the \{ |i\rangle, |-i\rangle } basis.
In spherical coordinates, a qubit can be rewritten as
|v\rangle = \cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \varphi} \sin \left( \frac{\theta}{2} \right) |1\rangle
Qubit operations
State preserved by qubits can be modified with gates. All operations performed by gates are reversible and deterministic (except for the measurements, that are neither reversible nor deterministic).
Gates may operate either on a single qubit or on multiple ones.
Gates can be represented as matrices that left-multiplies the qubit state.
Single qubit gates
Identity gate
The identity gate is the NOP of quantum
computers: it may look useless but will become useful when building
multi-qubit gates. It is represented by
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
Pauli-X gate
The Pauli-X gate (a.k.a. NOT gate) performs a rotation around the x axis by \pi radians. Practically, the amplitudes of the qubit are flipped. It is represented by
X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
Pauli-Z gate
The Pauli-Z gate (a.k.a. phase flip gate) performs a rotation around the z axisby \pi radians. Practically, the phase \varphi of the qubit is flipped. It is represented by
Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
Pauli-Y gate
The Pauli-Y gate performs a rotation around the y axis by \pi radians. It is represented by
Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}
Phase gate
The phase gate performs a rotation around the z axis by \pi / 2 radians. It is represented by
Y = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}
Hadamard gate
The Hadamard gate performs a rotation around the y axis by \pi / 2 radians followed by a rotation around the x axis by \pi radians. It maps |0\rangle \mapsto |+\rangle and |1\rangle \mapsto |-\rangle. It is represented by
H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}
Eigenvectors of single qubit gates.
The eigenvectors of a gate are always aligned to the direction around which the rotation is applied by the same gate.