Relative phase is the phase difference between quantum amplitudes. Unlike amplitude magnitudes, it may not appear in computational-basis measurement probabilities, yet it can change interference and distinguish quantum operations. section 5.1.1 section 2.2
Repeated measurements can estimate basis-state probabilities and amplitude magnitudes, but they do not preserve relative-phase information. Small noiseless simulations expose the complete internal state, while hardware measurements remain limited and potentially noisy. section 2.2
Use relative phase when designing a phase-sensitive distinction:
- Prepare an input state whose possible outputs become orthogonal after the candidate operations. section 5.1.1
- Apply the unknown operation or a longer gate sequence, then measure to identify which orthogonal state resulted. section 5.1.1
- Inspect a simulator when measurement statistics cannot reveal the phase relationship directly. section 2.2
Distinguishing X from Z
Apply the unknown single-qubit gate to |0⟩. Z leaves |0⟩ unchanged, whereas X produces |1⟩, so computational-basis measurement identifies the operation. The phase-changing action of Z becomes relevant when the input contains superposition. section 5.1.1
Table 5.1 Effects of controlled \(X\) and controlled \(-X\) gates on two-qubit basis states
| The basis state | After applying controlled \(X\) | After applying controlled \(-X\) |
| \(\ket{00}\) | \(\ket{00}\) | \(\ket{00}\) |
| \(\ket{01}\) | \(\ket{01}\) | \(\ket{01}\) |
| \(\ket{10}\) | \(\ket{11}\) | \(-\ket{11}\) |
| \(\ket{11}\) | \(\ket{10}\) | \(-\ket{10}\) |
Compare controlled X and controlled −X on two-qubit basis states to see how a sign, and therefore a relative phase, changes selected outputs. section 5.1.1
