Fast quantum measurement tomography with dimension-optimal error bounds

TL;DR

This paper introduces a two-step quantum measurement tomography protocol that achieves optimal sample complexity in system dimension, combining least-squares estimation and projection onto valid measurements, with proven error bounds and empirical validation.

Contribution

It proposes a dimension-optimal, low classical processing quantum measurement tomography method with rigorous error guarantees and empirical testing on superconducting qubits.

Findings

Achieves optimal sample complexity up to logarithmic factors.

Provides explicit analytic form for certain probe ensembles.

Demonstrates empirical performance on real quantum hardware.

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 in the system dimension. 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 $\mathcal{O}(d^3 L \ln(d)/\epsilon^2)$ samples to achieve error $\epsilon$ in worst-case distance, and $\mathcal{O}(d^2 L^2 \ln(dL)/\epsilon^2)$ samples in average-case distance. We further establish two almost matching sample complexity lower bounds of $\Omega(d^3/\epsilon^2)$ and $\Omega(d^2 L/\epsilon^2)$ for any non-adaptive, single-copy POVM tomography protocol. Hence, our projected least squares POVM tomography is sample-optimal in dimension $d$ up to logarithmic factors. 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.