# Gaussian state
In this example, We introduce the preparation of Gaussian state.
### What is Gaussian state
Gaussian state is a $$n$$-qubit quantum state defined as
$$
|\Psi\rangle := \frac{1}{\sqrt{\sum\_x |f(x)|^2}} \sum\_{x=0}^{N-1} f(x) |x\rangle.
$$
where $$f(x) = \exp(-\beta x^2 / 2)$$ is the Gaussian function. Gaussian state is ubiquitous in the application of quantum algorithms in quantum chemistry, simulating quantum field theory, and quantum finance.
### The equivalent problem in function approximation
One recent preprint Ref. [\[1\]](#reference) proposed a procedure for preparing Gaussian state. It leverages the block encoding of the sine value of equally spaced sample points $$\sum\_x \sin(x/N)|x\rangle\langle x|$$. By applying quantum eigenvalue transformation on this block encoding, the Gaussian state is prepared. Consequently, the problem is reduced to finding the phase factors generating the following function
$$
h(z) = f(\arcsin(z)) \propto \exp(-\beta \arcsin^2(x) / 2).
$$
where $$z = \sin(x)$$ transforms the domain to $$z\in \[0,\sin(1)]$$.
### Setup parameters
```matlab
beta = 100;
targ = @(x) exp(-beta/2 *asin(x).^2);
deg = 100;
opts.intervals=[0,sin(1)];
opts.objnorm = Inf;
opts.epsil = 0.01;
opts.npts = 500;
opts.fscale = 0.99;
opts.isplot=true;
opts.maxiter = 100;
opts.criteria = 1e-12;
opts.useReal = true;
opts.targetPre = true;
opts.method = 'LBFGS';
```
### Approximating the target function by polynomials
```matlab
coef_full=cvx_poly_coef(targ, deg, opts);
parity = mod(deg, 2);
% only keep coefficients with consistent parity
coef = coef_full(1+parity:2:end);
```
