# Singular vector transformation In this example, We introduce the implementation of singular vector transformation. ### What is singular vector transformation Singular vector transformation implements a matrix that transforms the right singular vector to the corresponding left singular vector. 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 $$A=\sum\_k \sigma\_k |\phi\_k \rangle \langle \psi\_k|$$. Singular vector transformation algorithm performs the transformation $$ \sum\_k \alpha\_k|\psi\_k\rangle \mapsto \sum\_k \alpha\_k|\phi\_k\rangle. $$ This algorithm can be used for devising a new method for singular value estimation. It also has many applications, for example, efficient ground state preparation of certain local Hamiltonians. ### The equivalent problem in function approximation As shown in the proof of Theorem 1 in [\[GSLW\]](#reference), this algorithm aims at achieving the singular value transformation of $$A$$ for sign function $$f$$, that is $$f^{(SV)}=\sum\_k |\phi\_k\rangle\langle \psi\_k|.$$ In practice, we find an odd polynomial approximation to $$f$$, denoted as $$f\_{poly}$$, on the interval $$D\_{\kappa}=\[-1, -\delta]\cup \[\delta,1]$$. By applying the singular value transformation for $$f\_{poly}$$ instead, this algorithm maps the a right singular vector having singular value at least $$\delta$$ to the corresponding left singular vector. For numerical demonstration, we scale the target function by a factor of 0.8, to improve the numerical stability. We call a subroutine to find the best odd polynomial approximating $$f(x)$$ on the interval $$D\_{\kappa}$$, where we solves the problem by convex optimization. Here are the parameters set for the subroutine. ```matlab opts.intervals=[delta,1]; opts.objnorm = Inf; opts.epsil = 0.1; opts.npts = 500; opts.isplot= true; opts.fscale = 1; % disable further rescaling of f(x) delta = 0.1; targ = @(x) 0.8*sign(x); parity = 1; % agrees with parity deg = 251; 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

### Set up parameters ```matlab opts.maxiter = 100; opts.criteria = 1e-12; opts.useReal = false; opts.targetPre = true; opts.method = 'Newton'; ``` ### Solving phase factors by running the solver ```matlab [phi_proc,out] = QSP_solver(coef,parity,opts); ``` ### Verifying the solution ```matlab xlist1 = linspace(-1,-delta,500)'; xlist2 = linspace(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, 2); 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_vector_transformation_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 = 6.67183e-08 max of solution = 0.8 iter err 1 +4.0710e-01 2 +5.8374e-02 3 +1.8592e-03 4 +1.8744e-06 5 +1.7525e-12 Stop criteria satisfied. The residual error is 2.8070e-13 ```
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