Single-qubit computational basis states
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:
The two orthogonal y-basis states are:
The basis states are located at opposite points on the Bloch sphere:
Multi-qubit computational basis states
A single-qubit has two computational basis states. In the z-basis these are and . A two-qubit system has 4 computational basis states denoted as , , , .
A multi-qubit system of N qubits has computational basis states denoted as , , ... .
Associated with each computational basis state is a probability amplitude , which is a complex number.
As an example, a system of three qubits is described by the expression:
where are the probability amplitudes associated to the computational basis states.
Initialization and measurement bases
By default, all qubits are initialized in the state in the z-basis.
State initialization in a specific basis can be done explicitly with the cQASM instructions
prep_x, which prepare qubits in the , and states respectively.
By default, qubits are measured with the
measure_all instruction in the z-basis.
Qubit measurement in a specific basis can be done explicitly with the cQASM instructions
- When a qubit is in the state ( state), a measurement in the z-basis will result in 0 (1)
- When a qubit is in the state ( state), a measurement in the y-basis will result in 0 (1)
- When a qubit is in the state ( state), a measurement in the x-basis will result in 0 (1)
and stand for Right and Left. Other notations that are often used for these states are and .