Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues exponentially-large sparse matrices.The maximum rate at which these may be learned, --known as Heisenberg limit--, constrained by bounds on circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants that do not require coherence between experiments have garnered interest recent years due lower depth and minimal overhead. In this work we show methods can achieve limit, {\em also} when one unable prepare eigenstates system. Given subroutine provides samples `phase function' $g(k)=\sum_j A_j e^{i \phi_j k}$ with unknown eigenphases $\phi_j$ overlaps $A_j$ cost $O(k)$, how estimate phases $\{\phi_j\}$ (root-mean-square) error $\delta$ total $T=O(\delta^{-1})$. Our scheme combines idea Heisenberg-limited multi-order single eigenvalue [Higgins et al (2009) Kimmel (2015)] subroutines so-called dense uses classical processing via time-series analysis QEEP problem [Somma (2019)] or matrix pencil method. For our adaptively fixes choice $k$ $g(k)$ prove scaling use time-series/QEEP subroutine. We present numerical evidence using technique well.