Quantum interference changes measurement outcomes when quantum amplitudes combine: some basis-state amplitudes cancel, while others reinforce. A Hadamard gate illustrates both effects directly. section 6.5.1
Entanglement with an unaccounted-for auxiliary qubit can prevent the intended interference, because entangled qubits form one inseparable system rather than independent groups. section 6.5.1
Hadamard-state test
Applying a Hadamard gate to |+⟩ cancels the |1⟩ amplitudes and reinforces |0⟩, so measurement returns 0. Applied to |−⟩, the reverse cancellation returns 1. Entanglement with another qubit changes this behavior. section 6.5.1
Table 6.3 The steps of the Bennett’s trick
| Step | Input | Auxiliary qubits | Auxiliary output | Final output |
| Initialize | \(\ket{x}\) | \(\ket{0}\) | \(\ket{0}\) | \(\ket{y}\) |
| \(\ket{x}\) | \(\ket{\text{aux}}\) | \(\ket{f(x)}\) | \(\ket{y}\) | |
| \(\ket{x}\) | \(\ket{\text{aux}}\) | \(\ket{f(x)}\) | \(\ket{y \oplus f(x)}\) | |
| \(\ket{x}\) | \(\ket{0}\) | \(\ket{0}\) | \(\ket{y \oplus f(x)}\) |
Use the Bennett’s trick table to trace auxiliary-qubit cleanup and avoid leaving entanglement that can obstruct later interference. section 6.5.1
