Fast quantum measurement tomography with optimal error bounds

Abstract

We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost and still achieves optimal sample complexity. Given measurement data from a known probe state ensemble, we first apply least-squares estimation to produce an unconstrained approximation of the POVM, and then project this estimate onto the set of valid quantum measurements. For a POVM with $L$ outcomes acting on a $d$-dimensional system, we show that the protocol requires $\mathcal{O}\left((d^3+d^2L)/ε^2\right)$ samples to achieve error $ε$ in worst-case distance, and $\mathcal{O}(d^2 L/ε^2)$ samples in average-case distance. We further establish two matching sample complexity lower bounds of $Ω((d^3 + d^2 L) /ε^2)$ and $Ω(d^2 L/ε^2)$ for any non-adaptive, single-copy POVM tomography protocol. Hence, our projected least squares POVM tomography is sample-optimal in both the dimension and the number of outcomes for both distances. Our method admits an analytic form when using global or local 2-designs as probe ensembles and enables rigorous non-asymptotic error guarantees. Finally, we also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.