For an \(N\times N\) unitary U, we can find N orthogonal eigenvectors \(\left| e_i\right\rangle \) which form an orthonormal basis and their corresponding eigenvalues \(\lambda _i=e^{i\phi _i}\) are complex numbers of modulus 1. Fig. 4 Variational quantum circuit for the eigenvec...
Let y^{(1)}, \cdots, y^{(n)} be an orthonormal basis of \mathbb R^n with V_1 = \text{span}(y^{(1)},\cdots, y^{(l)}), where l is the dimension of V_1. Let F = \sum_{t=1}^l Ay^{(t)}y^{(t)^T} \text{ and } \hat F = \sum_{t=1}^k Av^{(t)}...
one is often advised to use chebyshev polynomials instead of monomials. the point here is that ratfun automatically finds an appropriate basis for the particular problem given: if
This “state sieve” is an optimal search strategy in the sense that in general no shorter proposition systemexists which separates each individual state of the standard orthonormal basis. 3 The explicit formof the operators are (“diag” stands for a diagonal matrix) O 1 = diag 1, . . ....