Host: Jin-Peng Liu 刘锦鹏

Abstract:

Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials $e^{-tA}$ on quantum computers. In this work, we develop LCHS formulas for computing general matrix functions $f(A)$ when $f$ is analytic on the numerical range of $A$, with $A$ possibly non-normal. The essential technical tool is Weyl calculus, which reduces the construction of LCHS formulas for noncommuting operators to scalar Fourier approximation problems. Our construction yields a quantum eigenvalue transformation algorithm with optimal $\mathcal{O}(\log\frac{1}{\epsilon})$ query complexity scaling. Furthermore, our Weyl-calculus-based theory gives rise to an ansatz-free convex optimization framework that directly produces discrete LCHS formulas. This circumvents the inefficiencies of traditional quadrature rules and yields formulas highly optimized for coherent implementation on quantum computers. In addition, both our theory and optimization framework apply to the simulation of time-dependent dissipative ODE $\frac{d}{d t} \psi(t) = -A(t)\psi(t)$, for which we achieve a $2.1\times$ cost reduction over prior art.

Joint work with Prof. Lexing Ying.

Bio: Hongkang Ni is a Ph.D student at Stanford, supervised by Prof. Lexing Ying.