Short-depth circuits for efficient expectation-value estimation

The evaluation of expectation values TrρO for some pure state ρ and Hermitian operator O is of central importance in a variety of quantum algorithms. Near-optimal techniques have been developed in the past and require a number of measurements N approaching the Heisenberg limit N=O1/ϵ as a function of target accuracy ϵ. The use of quantum phase estimation (QPE) requires, however, long circuit depths C=O1/ϵ making its implementation difficult on near-term noisy devices. The more direct strategy of operator averaging is usually preferred as it can be performed using N=O1/ϵ2 measurements and no additional gates aside from 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) to show that the latter strategy can require an overly large number of measurements in order to achieve a prefixed relative target accuracy ϵr. We propose to overcome this problem using a single step of QPE and classical postprocessing. This approach leads to a circuit depth C=Oϵμ (with μ≥0) and to a number of measurements N=O1/ϵ2+ν for 0<ν≤1 and a much smaller prefactor. We provide detailed descriptions of two implementations of our strategy for ν=1 and ν≈0.5 and derive appropriate conditions that a particular problem instance has to satisfy in order for our method to provide an advantage.