# Singular value threshold projector In this example, We introduce the implementation of singular value threshold projector. ### What is singular value threshold projector Singular value threshold projectors project out singular vectors with singular values below/above a certain threshold. Suppose that we have access to a unitary $$U$$, its inverse $$U^\dagger$$ and the controlled reflection operators $$(2\Pi-I)$$, $$(2\tilde{\Pi}-I)$$. $$A:=\tilde{\Pi} U\Pi$$ is of our interest and has a singular value decomposition $$A=W\Sigma V$$. For $$S\subset R$$ let $$\Sigma\_S$$ be the matrix obtained from $$\Sigma$$ by replacing all diagonal entries $$\Sigma\_{ii}\in S$$ by 1 and all diagonal entries $$\Sigma\_{ii}\notin S$$ by 0. We define $$\Pi\_S:=\Pi V\Sigma\_S V^{\dagger}\Pi$$ and similarly $$\tilde{\Pi}\_S:=\tilde{\Pi} W\Sigma\_S W^{\dagger}\tilde{\Pi}$$. For example, set $$S=\[0,\delta]$$ and $$\Pi\_S$$ projects out right singular vectors with singular value at most $$\delta$$. These threshold projectors play a major role in quantum algorithms. ### The equivalent problem in function approximation As an example, we implement $$\Pi\_{\[0,0.5]}$$. We need to implement singular value transformation of $$A$$ for the following rectangle function. We call a subroutine to find the best even polynomial approximating $$f(x)$$ on the interval $$D\_{\delta}=\[0, 0.5-\delta]\cup \[0.5+\delta,1].$$ We solve the problem by convex optimization. Here are the parameters set for the subroutine. ### Set up parameters ```matlab opts.intervals=[0,0.5-delta,0.5+delta,1]; opts.objnorm = Inf; opts.epsil = 0.1; opts.npts = 500; opts.isplot= true; % scaling factor for the infinity norm opts.fscale = 0.9; opts.maxiter = 100; opts.criteria = 1e-12; opts.useReal = false; opts.targetPre = true; opts.method = 'Newton'; ``` ### Solving polynomial approximation ```matlab targ = @(x) rectangularPulse(-0.5,0.5,x); % Compute its Chebyshev coefficients. parity = 0; deg = 250; coef_full=cvx_poly_coef(targ, deg, opts); % The solver outputs all Chebyshev coefficients while we have to post-select % those of odd order due to the parity constraint. coef = coef_full(1+parity:2:end); ```

Polynomial approximation of the target function

Polynomial approximation error

### Solving phase factors by running the solver ```matlab [phi_proc,out] = QSP_solver(coef,parity,opts); ``` ### Verifying the solution ```matlab xlist1 = linspace(0,0.5-delta,500)'; xlist2 = linspace(0.5+delta,1,500)'; xlist = cat(1, xlist1,xlist2); 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') print(gcf,'singular_value_threshold_projectors_error.png','-dpng','-r500'); ```

The point-wise error of the solved phase factors.

### Reference 1. Gilyén, A., Su, Y., Low, G. H., & Wiebe, N. (2019, June). Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In *Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing* (pp. 193-204). 2. Dong, Y., Meng, X., Whaley, K. B., & Lin, L. (2021). Efficient phase-factor evaluation in quantum signal processing. *Physical Review A*, *103*(4), 042419.
Output of the code ``` norm error = 6.5514e-08 max of solution = 0.9 iter err 1 +4.8369e-01 2 +1.9791e-01 3 +2.7179e-02 4 +3.0099e-04 5 +3.2263e-08 Stop criteria satisfied. The residual error is 3.3529e-14 ```
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