Quantum eigenvalue processing (IVADO seminar)

Abstract

In this talk, I will present quantum algorithms to estimate and transform eigenvalues of high dimensional matrices. Such an eigenvalue processing problem arises from applications such as simulating non-Hermitian physics, transcorrelated quantum chemistry and solving differential equations, but is out of the reach of existing quantum singular value algorithms, because eigenvalues are different from singular values for non-normal operators. I will introduce a natural reduction to the quantum linear system problem, whose solution produces a quantum superposition of Faber polynomials—a nearly-best polynomial basis for function approximation over the complex plane. I will also describe a circuit to generate $n$ Fourier coefficients in superposition with $O(\mathrm{polylog}(n))$ gates improving over the standard approach with gate complexity $\Theta(n)$. Based on arXiv:1806.01838 and arXiv:2401.06240.