Parity measurement

Parity measurement (also referred to as Operator measurement) is a procedure in Quantum information science that is used to project a state into an eigenstate of an operator and to acquire its eigenvalue.[1] For example, given a Hermitian and Unitary operator ${\displaystyle U}$ (whose eigenvalues are ${\displaystyle \pm 1}$) and a state ${\displaystyle |\psi \rangle }$, the circuit on the top right performs a Parity measurement on ${\displaystyle U}$. After the first Hadamard gate, the state of the circuit is

${\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |\psi \rangle +|1\rangle |\psi \rangle )}$

Quantum circuit that exhibits Parity measurement

After applying the controlled-U gate, the state of the circuit evolves to

${\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle |\psi \rangle +|1\rangle U|\psi \rangle )}$

After applying the second Hadamard gate, the state of the circuit turns into

${\displaystyle {\frac {1}{2}}|0\rangle (|\psi \rangle +U|\psi \rangle )+{\frac {1}{2}}|1\rangle (|\psi \rangle -U|\psi \rangle )}$

If the state of the top qubit after measurement is ${\displaystyle |0\rangle }$, then ${\displaystyle |\phi \rangle =|\psi \rangle +U|\psi \rangle }$; which is the ${\displaystyle +1}$ eigenstate of ${\displaystyle U}$. If the state of the top qubit is ${\displaystyle |1\rangle }$, then ${\displaystyle |\phi \rangle =|\psi \rangle -U|\psi \rangle }$; which is the ${\displaystyle -1}$ eigenstate of ${\displaystyle U}$. [2]