Quantum entanglement is a correlation in which qubits form one system that cannot be represented as independent component states. In quantum function evaluation, auxiliary qubits may become entangled with input and output qubits, so they cannot be discarded like classical temporary variables. section 6.5.1
Account for entanglement because it can:
- Prevent later interference by retaining information in auxiliary qubits that the theoretical computation no longer tracks. section 6.5.1
- Produce incorrect algorithm results when auxiliary qubits remain entangled with qubits used later. section 6.5.1
Entanglement preventing interference
A Hadamard gate makes amplitudes cancel or amplify: applying it to |+⟩ yields |0⟩ because the |1⟩ amplitudes cancel. Entanglement with an unaccounted auxiliary qubit can prevent this interference from occurring correctly. section 6.5.1
Table 6.3 The steps of the Bennett’s trick
| Step | Input | Auxiliary qubits | Auxiliary output | Final output |
| Initialize | \(\ket{x}\) | \(\ket{0}\) | \(\ket{0}\) | \(\ket{y}\) |
| \(\ket{x}\) | \(\ket{\text{aux}}\) | \(\ket{f(x)}\) | \(\ket{y}\) | |
| \(\ket{x}\) | \(\ket{\text{aux}}\) | \(\ket{f(x)}\) | \(\ket{y \oplus f(x)}\) | |
| \(\ket{x}\) | \(\ket{0}\) | \(\ket{0}\) | \(\ket{y \oplus f(x)}\) |
Trace Bennett’s trick as auxiliary information is computed, used, and returned to |0⟩ before the final output remains. section 6.5.1
