Short-depth circuits for efficient expectation value estimation

The evaluation of expectation values \(Tr\left[\rho O\right]\) 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 \(N=\mathcal{O}\left(1/\epsilon\right)\) as a function of target accuracy \(\epsilon\). The use of Quantum Phase Estimation requires however long circuit depths \(C=\mathcal{O}\left(1/\epsilon\right)\) 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 \(N=\mathcal{O}\left(1/\epsilon^2\right)\) 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 \(C=\mathcal{O}\left(\epsilon^\mu\right)\) (with \(\mu\geq0\)) and to a number of measurements \(N=\mathcal{O}\left(1/\epsilon^{2+\nu}\right)\) for \(0