# 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); ```

Polynomial approximation of gaussian function

Polynomial approximation error

### Solving phase factors by running the solver ```matlab [phi_proc,out] = QSP_solver(coef,parity,opts); ``` ### Verifying the solution ```matlab xlist = linspace(0,sin(1),500)'; func = @(x) ChebyCoef2Func(x, coef, parity, true); targ_value = targ(xlist); func_value = func(xlist); QSP_value = QSPGetEntry(xlist, phi_proc, out); err= norm(QSP_value-func_value,Inf); disp('The residual error is'); disp(err); figure() plot(xlist,QSP_value-func_value) xlabel('$x$', 'Interpreter', 'latex') ylabel('$g(x,\Phi^*)-f_\mathrm{poly}(x)$', 'Interpreter', 'latex') ```

The point-wise error of the solved phase factors.

### Reference 1. McArdle, S., Gilyén, A., & Berta, M. (2022). Quantum state preparation without coherent arithmetic. *arXiv preprint arXiv:2210.14892*.
Output of the code ``` norm error = 1.34851e-09 max of solution = 0.99 L-BFGS solver started iter obj stepsize des_ratio 1 +1.1629e-02 +1.00e+00 +4.86e-01 2 +8.0537e-03 +1.00e+00 +7.10e-01 3 +3.8400e-03 +1.00e+00 +6.69e-01 4 +1.6803e-03 +1.00e+00 +7.00e-01 5 +6.2256e-04 +1.00e+00 +6.84e-01 6 +2.1175e-04 +1.00e+00 +6.84e-01 7 +6.3658e-05 +1.00e+00 +6.79e-01 8 +1.6662e-05 +1.00e+00 +6.75e-01 9 +9.0703e-06 +1.00e+00 +6.93e-01 10 +4.0572e-06 +1.00e+00 +7.32e-01 iter obj stepsize des_ratio 11 +1.3684e-06 +1.00e+00 +7.01e-01 12 +4.1191e-07 +1.00e+00 +6.36e-01 13 +1.6236e-07 +1.00e+00 +6.16e-01 14 +6.8715e-08 +1.00e+00 +6.64e-01 15 +2.1314e-08 +1.00e+00 +6.90e-01 16 +1.1814e-08 +1.00e+00 +6.88e-01 17 +1.5959e-09 +1.00e+00 +6.20e-01 18 +7.3575e-11 +1.00e+00 +5.69e-01 19 +2.3390e-12 +1.00e+00 +5.55e-01 20 +3.8866e-13 +1.00e+00 +6.20e-01 iter obj stepsize des_ratio 21 +1.5834e-14 +1.00e+00 +5.60e-01 22 +2.1621e-16 +1.00e+00 +5.45e-01 23 +6.4336e-19 +1.00e+00 +5.19e-01 24 +5.0662e-21 +1.00e+00 +5.23e-01 25 +2.1126e-22 +1.00e+00 +5.68e-01 26 +1.1727e-24 +1.00e+00 +5.29e-01 27 +1.1873e-26 +1.00e+00 +5.33e-01 Stop criteria satisfied. The residual error is 1.7719e-13 ```
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