10.4 Phase kickback

In section 9.8.2, we saw an example of phase kickback while working with oracles. In this section, we look at the concept again in the context of controlled z-rotation gates. We need this within Shor’s factoring algorithm to estimate eigenvalues. phase$kickback

Consider the circuit in Figure 10.8.

Figure 10.8: A circuit with a Hadamard and controlled z-rotation gate

As we saw in section 7.6.6, |1⟩ is an eigenvector of R_φ^z with eigenvalue e ^φi:

The initial quantum state in Figure 10.8 is |00⟩ = |0⟩⊗|0⟩. We apply H⊗X, yielding

We apply the CR_φ^z:

We began with q₀ as the control qubit. The phase from the R_φ^z gate conditionally applied to q₁ ends up causing the phase e ^φi to appear in the state of q₀. The state of q₁ was not changed from |1⟩. This is the phase...