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Parity 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. For example, given a Hermitian and Unitary operator U {\displaystyle U} and a state | ψ ⟩ {\displaystyle |\psi \rangle } , the circuit on the top right performs a Parity measurement on U {\displaystyle U}.After the first Hadamard gate, the state of the circuit is
After applying the controlled-U gate, the state of the circuit evolves to
After applying the second Hadamard gate, the state of the circuit turns into
If the state of the top qubit after measurement is | 0 ⟩ {\displaystyle |0\rangle } , then | ϕ ⟩ = | ψ ⟩ + U | ψ ⟩ {\displaystyle |\phi \rangle =|\psi \rangle +U|\psi \rangle } ; which is the + 1 {\displaystyle +1} eigenstate of U {\displaystyle U}. If the state of the top qubit is | 1 ⟩ {\displaystyle |1\rangle } , then | ϕ ⟩ = | ψ ⟩ − U | ψ ⟩ {\displaystyle |\phi \rangle =|\psi \rangle -U|\psi \rangle } ; which is the − 1 {\displaystyle -1} eigenstate of U {\displaystyle U}.