Transformation of Superposed Quantum States Using Measurement Operators. (21st September 2022)

• Record Type: Journal Article
• Authors: Ito, Hideaki; Iida, Saburou
• Other Names: Dong Shi Hai Academic Editor.
• Is Part Of: Quantum engineering. Volume 2022(2022)
• Publication Date: 2022-09-21
• Subjects: Quantum theory -- Periodicals; Engineering -- Periodicals; Quantum theory; Engineering; Electronic journals; Periodicals; 530.12
• DOI: 10.1155/2022/4819682
• Languages: English
• ISSNs: 2577-0470

Abstract

Abstract : Quantum computation based on a gate model is described. This model initially creates a superposition | ψ 0 〉 consisting of N = 2 n states, and these states are labeled by an n qubit index value j . Two working qubits | 0 〉 w k 0 and | 0 〉 w k 1 are added for a measurement. Moreover, one marking qubit | 0 〉 m k is added to discriminate between states in a superposition. Thus, | ψ 1 〉 = 1 / N ∑ j = 0 N − 1 | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ | 0 〉 m k . The Hadamard transformation is applied to | 0 〉 w k 0 and | 0 〉 w k 1 . | ψ 2 〉 = 1 / N 1 / 2 2 ∑ j = 0 N − 1 | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 0 〉 m k . 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). | ψ 3 〉 = 1 / N 1 / 2 2 ∑ j ∈ B | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 0 〉 m k + ∑ j ∈ G | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 1 〉 m k . After a marking, a superposition is measured by POVM. The measurement is described by a collection of four measurement operators. The measurement transforms | ψ 3 〉 into | ψ 4 〉 = 1 / N ∑ j ∈ B | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ p 0 sin θ | 0 〉 m k + ∑ j ∈ G | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ p 1 cos θ | 1 〉 m k / D ; here, D = card B p 0 ∗ p 0 sin 2 θ + card G p 1 ∗ p 1 cos 2 θ, and p 0 ∗ p 0 sin 2 θ + p 1 ∗ p 1 cos 2 θ = 2 which is derived from the completeness equation. The state | 0 〉 m k and the state | 1 〉 m k before the measurement are transformed into p 0 sin θ | 0 〉 m k and p 1 cos θ | 1 〉 m k, respectively. This paper describes these measurement operators.