The two orthogonal z-basis states of a qubit are defined as:

When we talk about the qubit basis states we implicitly refer to the z-basis states as the computational basis states.

The two orthogonal x-basis states are: ∣+⟩=∣0⟩+∣1⟩2\vert +\rangle =\frac{\vert 0\rangle + \vert 1\rangle}{\sqrt{2}}∣+⟩=2​∣0⟩+∣1⟩​ ∣−⟩=∣0⟩−∣1⟩2\vert -\rangle =\frac{\vert 0\rangle - \vert 1\rangle}{\sqrt{2}}∣−⟩=2​∣0⟩−∣1⟩​

The two orthogonal y-basis states are: ∣R⟩=∣0⟩+ı∣1⟩2\vert R\rangle =\frac{\vert 0\rangle + \imath \vert 1\rangle}{\sqrt{2}}∣R⟩=2​∣0⟩+ı∣1⟩​ ∣L⟩=∣0⟩−ı∣1⟩2\vert L\rangle =\frac{\vert 0\rangle - \imath \vert 1\rangle}{\sqrt{2}}∣L⟩=2​∣0⟩−ı∣1⟩​

The basis states are located at opposite points on the Bloch sphere:

A single-qubit has two computational basis states. In the z-basis these are ∣0⟩\vert 0 \rangle∣0⟩ and ∣1⟩\vert 1 \rangle∣1⟩. A two-qubit system has 4 computational basis states denoted as ∣00⟩\vert 00 \rangle∣00⟩, ∣01⟩\vert 01 \rangle∣01⟩, ∣10⟩\vert 10 \rangle∣10⟩, ∣11⟩\vert 11 \rangle∣11⟩.

A multi-qubit system of N qubits has 2N2 ^{N}2N computational basis states denoted as ∣00...00⟩\vert 00...00 \rangle∣00...00⟩, ∣00⋯01⟩\vert 00 \cdots 01 \rangle∣00⋯01⟩, ∣00⋯10⟩\vert 00 \cdots 10 \rangle∣00⋯10⟩ ... ∣11⋯11⟩\vert 11 \cdots 11 \rangle∣11⋯11⟩.

Associated with each computational basis state is a probability amplitude αi\alpha_{i}αi​, which is a complex number.

As an example, a system of three qubits is described by the expression:

∣Ψ⟩=α0∣000⟩+α1∣001⟩+α2∣010⟩+⋯+α7∣111⟩\lvert \Psi \rangle = \alpha_{0} \lvert 000 \rangle + \alpha_{1} \lvert 001 \rangle + \alpha_{2} \lvert 010 \rangle + \cdots + \alpha_{7} \lvert 111 \rangle∣Ψ⟩=α0​∣000⟩+α1​∣001⟩+α2​∣010⟩+⋯+α7​∣111⟩

where αi\alpha_{i}αi​ are the probability amplitudes associated to the computational basis states.

By default, all qubits are initialized in the ∣0⟩|0\rangle∣0⟩ state in the z-basis.

State initialization in a specific basis can be done explicitly with the cQASM instructions prep_z, prep_y and prep_x, which prepare qubits in the ∣0⟩\vert 0 \rangle∣0⟩, ∣R⟩\vert R \rangle∣R⟩ and ∣+⟩\vert + \rangle∣+⟩ states respectively.

By default, qubits are measured with the measure or measure_all instruction in the z-basis. Qubit measurement in a specific basis can be done explicitly with the cQASM instructions measure_x, measure_y and measure_z.

∣R⟩\vert R \rangle∣R⟩ and ∣L⟩\vert L \rangle∣L⟩ stand for Right and Left. Other notations that are often used for these states are ∣ı⟩\vert \imath \rangle∣ı⟩ and ∣−ı⟩\vert - \imath \rangle∣−ı⟩.