Bit-Flip Errors

Bit-Flip Errors

A bit-flip error is the simplest quantum error: some time during your computation, the environment applies an unwanted X gate to one of your qubits. Before the error, the qubit was in α|0⟩ + β|1⟩; after the error, it is in β|0⟩ + α|1⟩. The two amplitudes have swapped. Viewed from the Bloch sphere, the state has been reflected through the X axis.

Mathematically this is exactly the same as a stray X gate appearing in your circuit where you did not want one. Physically it can be caused by thermal excitations, stray microwave photons, or cosmic rays passing through the chip. Whatever the cause, the effect is identical, a swap of the |0⟩ and |1⟩ amplitudes on the affected qubit, with no sign change.

The Gate Matrix

The Pauli X gate is simply:

X = [[0, 1],
     [1, 0]]

Its action on the computational basis is X|0⟩ = |1⟩ and X|1⟩ = |0⟩. X² = I, so two bit-flips in a row cancel. That means a single bit-flip is correctable if we can figure out which qubit flipped: just apply X to that qubit again.

The 3-Qubit Repetition Code

The cleanest way to see bit-flip protection in action is the 3-qubit repetition code. The idea is borrowed from classical error correction: store each logical bit in three physical bits and take a majority vote on readout. In the quantum version: