Transformation of Superposed Quantum States Using Measurement Operators

Quantum computation based on a gate model is described. This model initially creates a superposition |ψ0〉 consisting of N=2n states, and these states are labeled by an n qubit index value j. Two working qubits |0〉wk0 and |0〉wk1 are added for a measurement. Moreover, one marking qubit |0〉mk is added to discriminate between states in a superposition. Thus, |ψ1〉=1/N∑j=0N-1|j〉⊗|0〉wk0⊗|0〉wk1⊗|0〉mk. The Hadamard transformation is applied to |0〉wk0 and |0〉wk1. |ψ2〉=1/N1/22∑j=0N-1|j〉⊗|0〉wk0+|1〉wk0⊗|0〉wk1+|1〉wk1⊗|0〉mk. After a computation, a set of states is divided into two subsets; one is a subset bad (B) and the other is a subset good (G). |ψ3〉=1/N1/22∑j∈B|j〉⊗|0〉wk0+|1〉wk0⊗|0〉wk1+|1〉wk1⊗|0〉mk+∑j∈G|j〉⊗|0〉wk0+|1〉wk0⊗|0〉wk1+|1〉wk1⊗|1〉mk. After a marking, a superposition is measured by POVM. The measurement is described by a collection of four measurement operators. The measurement transforms |ψ3〉 into |ψ4〉=1/N∑j∈B|j〉⊗|0〉wk0⊗|0〉wk1⊗p0sin θ|0〉mk+∑j∈G|j〉⊗|0〉wk0⊗|0〉wk1⊗p1cos θ|1〉mk/D; here, D=cardBp0∗p0sin2θ+cardGp1∗p1cos2θ, and p0∗p0sin2θ+p1∗p1cos2θ=2 which is derived from the completeness equation. The state |0〉mk and the state |1〉mk before the measurement are transformed into p0sinθ|0〉mk and p1cosθ|1〉mk, respectively. This paper describes these measurement operators. © 2022 Hideaki Ito and Saburou Iida.