# Kaiser window state
In this example, We introduce the preparation of Kaiser window state.
### What is Kaiser window state
The state is defined as
$$
|\Psi\rangle := \frac{1}{\sqrt{\sum\_x |f(x)|^2}} \sum\_{x=0}^{N-1} f(x) |x\rangle
$$
where the amplitude is described by the Kaiser window function $$f(x) = \frac{I\_0(\beta \sqrt{1-x^2})}{I\_0 (\beta)}$$. Here, $$I\_0$$ is the zeroth modified Bessel function of the first kind. The Kaiser window function can be used in quantum phase estimation to boost the success probability without (coherently) calculating the median of several phase evaluations.
### The equivalent problem in function approximation
One recent preprint Ref. [\[1\]](#reference) proposed a procedure for preparing Kaiser window 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 Kaiser window state is prepared. Consequently, the problem is reduced to finding the phase factors generating the following function
$$
h(z) = f(\arcsin(z)) \propto I\_0(\beta \sqrt{1 - \arcsin^2(z)}).
$$
where $$z = \sin(x)$$ transforms the domain to $$z\in \[0,\sin(1)]$$.
### Setup parameters
```matlab
beta = 8;
targ = @(x) besselj(0,1jbetasqrt(1-2asin(x).^2))/besselj(0,1jbeta);
deg = 100;
delta = 0.01;
opts.intervals = [0,sin(1)];
opts.objnorm = Inf;
opts.epsil = 0.01;
opts.npts = 500;
opts.fscale = 0.98; % scaling factor for the infinity norm
opts.isplot = true;
opts.maxiter = 100;
opts.criteria = 1e-12;
opts.useReal = true;
opts.targetPre = true;
opts.method = 'Newton';
```
### 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);
```
Polynomial approximation of Kaiser window function
