Short-depth circuits for efficient expectation value estimation

The evaluation of expectation values $Trleft[rho Oright]$ for some pure state $rho$ and Hermitian operator $O$ is of central importance in a variety of quantum algorithms. Near optimal techniques developed in the past require a number of measurements $N$ approaching the Heisenberg limit $N=mathcal{O}left(1/epsilonright)$ as a function of target accuracy $epsilon$. The use of Quantum Phase Estimation requires however long circuit depths $C=mathcal{O}left(1/epsilonright)$ making their implementation difficult on near term noisy devices. The more direct strategy of Operator Averaging is usually preferred as it can be performed using $N=mathcal{O}left(1/epsilon^2right)$ measurements and no additional gates besides those needed for the state preparation. In this work we use a simple but realistic model to describe the bound state of a neutron and a proton (the deuteron) and show that the latter strategy can require an overly large number of measurement in order to achieve a reasonably small relative target accuracy $epsilon_r$. We propose to overcome this problem using a single step of QPE and classical post-processing. This approach leads to a circuit depth $C=mathcal{O}left(epsilon^muright)$ (with $mugeq0$) and to a number of measurements $N=mathcal{O}left(1/epsilon^{2+ u}right)$ for $0< uleq1$. We provide detailed descriptions of two implementations of our strategy for $ u=1$ and $ uapprox0.5$ and derive appropriate conditions that a particular problem instance has to satisfy in order for our method to provide an advantage.