Short-depth circuits for efficient expectation-value estimation

Abstract

The evaluation of expectation values $\text{Tr}\left[\ensuremath{\rho}\mathcal{O}\right]$ for some pure state $\ensuremath{\rho}$ and Hermitian operator $\mathcal{O}$ 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 $N=O\left(1/\ensuremath{\epsilon}\right)$ as a function of target accuracy $\ensuremath{\epsilon}$. The use of quantum phase estimation (QPE) requires, however, long circuit depths $C=O\left(1/\ensuremath{\epsilon}\right)$ 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 $N=O\left(1/{\ensuremath{\epsilon}}^{2}\right)$ 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 ${\ensuremath{\epsilon}}_{r}$. We propose to overcome this problem using a single step of QPE and classical postprocessing. This approach leads to a circuit depth $C=O\left({\ensuremath{\epsilon}}^{\ensuremath{\mu}}\right)$ (with $\ensuremath{\mu}\ensuremath{\ge}0$) and to a number of measurements $N=O\left(1/{\ensuremath{\epsilon}}^{2+\ensuremath{\nu}}\right)$ for $0<\ensuremath{\nu}\ensuremath{\le}1$ and a much smaller prefactor. We provide detailed descriptions of two implementations of our strategy for $\ensuremath{\nu}=1$ and $\ensuremath{\nu}\ensuremath{\approx}0.5$ and derive appropriate conditions that a particular problem instance has to satisfy in order for our method to provide an advantage.