Phase-Flip Errors
A phase-flip error applies the Pauli Z operator, flipping the sign of |1> and thereby scrambling the relative phase that carries quantum information.
The Z error
The phase-flip is the Pauli Z operator. It leaves |0> alone and multiplies |1> by -1: Z|0> = |0>, Z|1> = -|1>. On a superposition Z(a|0> + b|1>) = a|0> - b|1>. The magnitudes are untouched but the relative phase is inverted, which for many quantum algorithms is just as damaging as a bit flip.
Z has no classical counterpart, and phase errors are often the dominant error channel in real hardware because dephasing (T2 processes) tends to be faster than energy relaxation (T1). A code that ignores phase errors would be useless.
Duality with bit flips
Under the Hadamard transform, Z and X swap roles: HZH = X. In the |+>, |-> basis a Z error is just a bit flip, because Z|+> = |-> and Z|-> = |+>. This means the machinery for detecting bit flips can be reused for phase flips by working in the conjugate basis.
- Z|0> = |0>, Z|1> = -|1>: sign flip on |1>.
- Z|+> = |->, Z|-> = |+>: acts as a bit flip in the Hadamard basis.
- Detected by X-type parity operators X X, or equivalently by Z Z after Hadamards.
- Dephasing typically makes Z the most common physical error.
The three-qubit phase-flip code corrects a single Z error by encoding in the |+>, |-> basis. Combining a phase-flip code with a bit-flip code, one inside the other, yields the nine-qubit Shor code, which corrects an arbitrary single-qubit error.
Because any single-qubit error is a linear combination of I, X, Z, and Y = iXZ, correcting X and Z separately is enough to correct every single-qubit error, a fact known as the discretization of errors.