# A full tutorial on QAOA algorithm without leave-it-as-an-exercise-for-reader

Many equations and formulas look intimidating. However, when you hunt them down, they are definitely not! Just papertigers!

Now let's hunt the papertiger.

This is not finished.
2020-07-30 first commit

QAOA​1​

## 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. 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

## Footnotes

There are many excellent tutorials out there. Some tutorials are too intuitive and it's helpful, but you cannot get it straight on the math details. Some focused on dymestifying math. Some focused on code. I found the best tutorials that give you the conceptual ideas and are possible for implementation without being blind to the math details. Drop a comment if I failed. It would be really appreciable.

Lachlan Chen, "A full tutorial on QAOA algorithm without leave-it-as-an-exercise-for-reader," in EarnFromScratch, July 30, 2020, https://www.earnfs.com/en/html/2021.htm.

or

@misc{lachlanchen2020tutorial,
title=A full tutorial on QAOA algorithm without leave-it-as-an-exercise-for-reader,
author={Chen, Lachlan},
year=July 30, 2020
}