Quantum Krylov Algorithm for Szeg\"o Quadrature

We present a quantum algorithm to evaluate matrix elements of functions of unitary operators. The method is based on calculating quadrature nodes and weights using data collected from a quantum processor. Given a unitary $U$ and quantum states $|\psi_0\rangle$, $|\psi_1\rangle$, the resulting quadrature rules form a functional that can then be used to classically approximate $\langle\psi_1|f(U)|\psi_0\rangle$ for any function $f$. In particular, the algorithm calculates Szeg\"o quadrature rules, which, when $f$ is a Laurent polynomial, have the optimal relation between degree of $f$ and number of distinct quantum circuits required. The unitary operator $U$ could approximate a time evolution, opening the door to applications like estimating properties of Hamiltonian spectra and Gibbs states, but more generally could be any operator implementable via a quantum circuit. We expect this algorithm to be useful as a subroutine in other quantum algorithms, much like quantum signal processing or the quantum eigenvalue transformation of unitaries. Key advantages of our algorithm are that it does not require approximating $f$ directly, via a series expansion or in any other way, and once the output functional has been constructed using the quantum algorithm, it can be applied to any $f$ classically after the fact.