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2020-07-30 first commit

## The problem definition

QAOA^{1}

## Traverse field Ising model

## Adibatic quantum computation

## Circuit design

## Coding

## Trival state preparation

After several days of thinking and researching, I decided to answer my own question.

N.B. The tensor product symbol are omitted when there is no risk in confusion, especially when the index is different. In symbol,.

Firstly, for case , we only consider qubit , in which . As and act on different qubits,

(1)

Now,

(2)

Thus,

(3)

Hence,

(4)

For the Ising traverse field Hamiltonian, we only consider .

For , it can be evaluated as

(5)

We can calculate two-qubit operations independently, such that

(6)

Numerically, if you cannot convince yourself,

(7)

and

(8)

and

(9)

Now, for a more specific example,

(10)

Q.E.D.

### Reference

- 1.Choi J, Kim J. A Tutorial on Quantum Approximate Optimization Algorithm (QAOA): Fundamentals and Applications. In:
*2019 International Conference on Information and Communication Technology Convergence (ICTC)*. IEEE; 2019. doi:10.1109/ictc46691.2019.8939749