Qubit basis states (2024)

Single-qubit computational basis states

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

$\vert 0\rangle$ ∣0⟩
$\vert 1\rangle$ ∣1⟩

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:
$\vert +\rangle =\frac{\vert 0\rangle + \vert 1\rangle}{\sqrt{2}}$ ∣+⟩=2∣0⟩+∣1⟩ $\vert -\rangle =\frac{\vert 0\rangle - \vert 1\rangle}{\sqrt{2}}$ ∣−⟩=2∣0⟩−∣1⟩

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

Multi-qubit computational basis states

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

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

Probability amplitudes

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

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

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

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

Initialization and measurement bases

By default, all qubits are initialized in the $|0\rangle$ ∣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 $\vert 0 \rangle$ ∣0⟩, $\vert R \rangle$ ∣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.

Declared states

When a qubit is in the $\vert 0 \rangle$ ∣0⟩ state ( $\vert 1 \rangle$ ∣1⟩ state), a measurement in the z-basis will result in 0 (1)
When a qubit is in the $\vert R \rangle$ ∣R⟩ state ( $\vert L \rangle$ ∣L⟩ state), a measurement in the y-basis will result in 0 (1)
When a qubit is in the $\vert + \rangle$ ∣+⟩ state ( $\vert - \rangle$ ∣−⟩ state), a measurement in the x-basis will result in 0 (1)

Notes

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