The Computational Basis

The computational basis is the standard {|0>,|1>} reference frame for qubits, in which states are prepared, gates are defined, and measurements are read.

The default frame

The computational basis is the conventional orthonormal basis for describing qubits: {|0>, |1>} for one qubit, and the binary strings |00>, |01>, ... for many. It is the reference frame in which qubit states are written, gates are given as matrices, and measurements are usually performed.

One and many qubits

For n qubits the computational basis is the 2^n states |x> labelled by n-bit strings x. As vectors these are the standard unit vectors: |0> is (1,0), |1> is (0,1), and |x> is the vector with a single 1 in position x. This makes the mapping between binary integers and basis states direct.

Its role in measurement

A default measurement is in the computational basis: the qubit collapses to |0> or |1> and returns the corresponding bit. Hardware readout is built to distinguish these two states — for a superconducting qubit, the two lowest energy levels. Measuring in any other basis means rotating into it first, then measuring in the computational basis.

Not the only basis

Other bases are equally valid frames. The diagonal (X) basis {|+>, |->} and the circular (Y) basis are common alternatives. The same physical state has different coordinates in each; a state definite in one basis is a superposition in another. Choosing which basis to measure in is a design decision, because it determines what property of the state you extract.

Alignment with the Z axis

On the Bloch sphere the computational basis states are the north and south poles, aligned with the Z axis. This is why the Pauli Z operator is diagonal in this basis and a Z measurement is synonymous with a computational-basis measurement.