Transformation of Superposed Quantum States Using Measurement Operators

Quantum computation based on a gate model is described. )is model initially creates a superposition |ψ0〉 consisting of N � 2 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. )us, |ψ1〉 � (1/ �� N √ )(􏽐 N−1 j�0 |j〉)⊗ |0〉wk0 ⊗ |0〉wk1 ⊗ |0〉mk. )e Hadamard transformation is applied to |0〉wk0 and |0〉wk1. |ψ2〉 � (1/ �� N √ )(1/ � 2 √ )2(􏽐 N−1 j�0 |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/ �� N √ )(1/ � 2 √ )2 (􏽐j∈B|j〉)⊗ 􏽮

Quantum computation based on a gate model is described. is 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〉 wk0 and |0〉 wk1 are added for a measurement. Moreover, one marking qubit |0〉 mk is added to discriminate between states in a superposition. us, e Hadamard transformation is applied to |0〉 wk0 and |0〉 wk1 .

Introduction
Quantum computation based on a gate model initially creates a superposition |ψ 0 〉 that comprises of N � 2 n states from n qubits. ese states are labeled by an n qubit index value j. (1) For the measurement, two working qubits |0〉 wk0 and |0〉 wk1 are added to |ψ 0 〉. Moreover, to discriminate between states in a superposition, a marking qubit |0〉 mk is added.
e Hadamard transformation H, is applied to two working qubits. en, Starting with a state |ψ 2 〉, a computation proceeds. After this computation, a superposition is divided into two subsets; one is a subset bad (B), and the other is a subset good (G) [1,2].
To map bad or good states to a qubit |0〉 mk , a mapping function f is introduced as follows: where B ∩ G � ∅ and card(B) + card(G) � N.
Using a mapping function f, |ψ 2 〉 is transformed into [3] erefore, a discrimination between bad and good states is equivalent to an amplitude transformation of |0〉 mk and |1〉 mk .
Next, a superposition |ψ 3 〉 is measured by POVM. e measurement M is described in a collection of four In Section 2, P, Q, R, and S are defined.
After the measurement, a superposition |ψ 3 〉 is transformed into e relation is derived from the completeness equation [4] which is described in Section 3. e states |0〉 mk and |1〉 mk before the measurement are transformed into p 0 sinθ|0〉 mk and p 1 cosθ|1〉 mk , respectively. On the amplitude transformation of superposed quantum states, Grover's algorithm [5] is well known, which makes use of unitary transformations. However, quantum measurements play an important role in a one-way computation model [6] and a quantum teleportation model [7,8]. ese models have a wide applicability to solve NP-hard problems [9].

Measurement Operators
In this section, the measurement operators P, Q, R, and S are described. P, Q, R, and S are defined as follows: Each operator is a tensor product of three components. From the left, two components control an occurrence of an operator by acting on two qubits wk0 and wk1, and the remainder transforms an amplitude of a qubit mk. When the state |ψ 3 〉 is given, the occurrence is limited to P.

POVM
Let the adjoint operators of P, Q, R, and S be equal to P † , Q † , R † , and S † , respectively. POVM requires the following two conditions [4]: (1) P † P, Q † Q, R † R, and S † S are positive operators (2) P † P + Q † Q + R † R + S † S � I, where I is an identity operator Condition 1 is easily proved. Condition 2 is the completeness equation and proved in the following. P, Q, R, and S belong to Kraus operators [10] because of Condition 2.
We search for the conditions where C � P † P + Q † Q + R † R + S † S is equal to an identity operator. Let C be equal to (15) en, e value of the matrix element that does not appear in the above expressions is unconditionally equal to 0. Expression (16) plus Expression (20) is equal to the following expression: p * 0 p 0 + q * 1 q 1 sin 2 θ + r * 0 r 0 + s * 1 s 1 sin 2 ϕ + p * 1 p 1 + q * 0 q 0 cos 2 θ + r * 1 r 1 + s * 0 s 0 cos 2 ϕ � 8.
Quantum Engineering 3 Expression (17) plus Expression (21) is equal to the following expression: Expression (18) plus Expression (22) is equal to the following expression: Expression (19) plus Expression (23) is equal to the following expression: We assume that en, Expressions (25) and (27) Expression (24) minus Expression (26) is equal to the following expression: By using Expression (28), Expressions (29) and (30) change as follows: Expressions (28), (31), and (32) are the conditions that the measurement operators P, Q, R, and S satisfy the completeness equation.

Conclusion
In quantum computation, computation begins with a superposition which consists of 2 n states. Usually, an equal amplitude is given between states. After computation, a set of states is divided into two subsets; one is a subset bad (B), and the other is a subset good (G). e discrimination between bad and good states is enhanced by the measurement which is POVM. en, amplitudes representing bad and good states correspond to p 0 sinθ and p 1 cosθ, respectively. e relation p * 0 p 0 sin 2 θ + p * 1 p 1 cos 2 θ � 2 is derived from the completeness equation.
us, the measurement well discriminate between bad and good states.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.