Direct Fidelity Estimation from Few Pauli Measurements

We describe a simple method for certifying that an experimental device prepares a desired quantum state $\rho$. Our method is applicable to any pure state $\rho$, and it provides an estimate of the fidelity between $\rho$ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of $d$, the dimension of the state space, and extends easily and naturally to quantum channels.

Figure 1

Left: The residual error has a standard deviation of 1.8%. Right: Most states use only a typical number of copies, with just 0.1% of trials using more than 4 times the expected number of copies, as shown in the inset.