###### tags: `papers` `Quantum Calculation`

# Bravyi-Kitaev Superfast simulation of electronic structure on a quantum computer

[arXiv:1712.00446v3](https://arxiv.org/pdf/1712.00446.pdf)

This paper shows the results of hydrogen molevule simulations by BKSF and compare them to that of JW and BK transformations.

## Introduction
* Brief explantions about quantum simulations
* Mapping strategies used (e.g. JW, BK, BKSF) in quantum simulations
* BKSF is differnt from spin-fermion mappings (JW and BK) in sence that qubits are representing interactions between fermionnic-modes. (In JW and BK, qubit is representing fermionnic-modes itself.)
* The goal of this paper
* explain the methodology of BKSF
* consider the relation between BKSF and quantum error corrections
* simulate hydrogen molecule by BKSF

## Background for quantum simulation

### Fermionic Systems and Second Quantization
Explanation of second quantizations, that is seen in many textbooks
* definitions of anihilation and creation operators
* N body hamiltonian representations by anihilation and creation operators

### Implementing qubit Hamiltonians on quantum computers
* How to implement $e^{iHt}$ to represent time evolutions.
* If hamiltonian $H$ is devided into terms that are mutually comutable, this can be done easly.
* In general the terms are not comutable, thus we use trotter approzimations.

### Phase Estimation Algorithms
* Breif expalnations about PEA to calculate the ground energy of the hamiltonian.
* In molecular simulations $U = e^{iHt}$ and $v$ is at ground state. $U$ is not trival so prepared by suzuki-trotter decompositions.
* Measure the upper-resister to read the phase of $U$.

### State preparation
State preparation $v$ is not trivial. The solutions to this problem will be:
* instead prepare Hartree-Fock or post Hartree-Fock state (which has overlap between ground state) and use PEA for polynomial timses.
* Methods using adiabatic state preparation.
* VQE

### OpenFermion
Explanations about openfermion library.
* python based tool set, previously called fermilib
* open source project providing the classical routines necessary for simulating fermionic models and quantum chemistry problems on quantum computers
* support pis4 and projectQ
* can comlile the hamiltonian to hardwares

## Transformations from fermionic operators to qubit operators

### Jordan-Wigner and Bravyi-Kitaev Transformation
Explanation about JW and BK transformations.

* Mapping from spin to fermion
* JW and the alternative way (parity nethod) costs $O(n)$
* Bravyi-Kitaev striles the balance to cost only $O(\log n)$

where $n$ is number of qubits.

### Bravyi Kitaev Superfast (BKSF) Algorithm
* draws imspiration from Kitaev's honeycomb methods (where majorana modes are used to diagonalize the hamiltonian)
* pair the Majorana fermions which lead physically unpaired Majorana modes that cannot interact with environment
* Summary of BKSF
* map the modes to vertices of graphs
* get the number of edges
* find the representation of edge operator
* find the representation of quubit operator
* find independent loops in the graph and define stabilizers for each of them
* find the relevant subspace for quantum simulation and the vacuum stat
* stabilzer is a subspace where algebra of majorana modes satisfy

### Stabilizers and Vacuum State
How to construct stabilizers.

> Details will be summarized later. Need to learn about stabilaizers, majorana modes.

### BKSF representation of hamiltonian
How to construct number operators, coulomb-exchange operators, excitation operator, number-excitation operator, double excitation operator.
These operators are constructed using edge operators.

> Details will be summarized later.

## Electronic hamiltonian of hydrogen molecule

Construct hydrogen molecule using BKSF representation and compare to the JW and BK transformations.

* calculate the one-body and two-body terms individually
* construct the interaction hamiltonian as graph
* calculate the creation and anihilation operators in terms of edge operators using the integrals in table II


## Results
* The hamiltonian derived in the three differnt ways is presented at below.
* Calculated the propagetor $e^{iHt}$ by trotter approximations.

* Compared the ground energy wrt oredering of suzuki-trotter approximations. Simillar results were derived using BKSF, BK.

* Compared the "magnitude ordering" of three mappings. The three mapping acheived the same errors.

* The gate count were higher than the BK but lower than the JW.

* BKSF might be sufficient comparaed to JW if the interaction in the hamiltonian is localized
* Extend investigations to larger molecules
* Explore the effects of noise of the BKSF, which have close connections to quantum error corrections

## Conclusions
Compared the trotter errors on H$_2$ molecule using BKSF, JW and BK.
## My thoughts and impressions
* In abstract it saids that BKSF had lower trootter errors than JW and BK, but I rather think, according to this paper, that they're the same.
* I am also interested in the other molecule simulation.
* What does physically mean by hamiltonian is locallized?
* I haven't understand the details of BKSF and its physics so I want to try learning it someday. (Paper itself seems very helpful for understanding.)

## Applications
[openfermion BKSF by google](https://quantumai.google/reference/python/openfermion/transforms/bravyi_kitaev_fast)