Interpolation of Trotter data for eigenvalue and expectation value estimation

In this work, we provide a paradigm-shifting approach in which we achieve a Õ(log 1/ε) cost scaling when estimating observables with a single ancillary qubit, where ε is the error on the observable from approximating the Hamiltonian dynamics. This method relies on interpolating the observable estimated where Hamiltonian dynamics are simulated at different Trotter-step sizes. With these methods, we avoid having to simulate the Hamiltonian evolution with high precision in situ which incurs in extra quantum cost and qubits. Instead, we choose to trade those quantum costs for similar classical post-processing costs. We also compare the error propagation on the interpolant using hard bounds (confidence intervals) coming typically from semi-classical phase estimation, or IQAE, and also a probabilistic approach using Gaussian quantum phase estimation. We demonstrate that using the probabilistic approach has an improved cost scaling, for which a confidence interval can also be obtained after interpolation is done, with a super-exponentially decaying error rate as you increase the interval. Numerical tests of the probabilistic approach are provided using the transverse Ising model with a size of two sities.