Fast quantum measurement tomography with dimension-optimal error bounds

• URL: http://arxiv.org/abs/2507.04500v1
• Date: Sun, 06 Jul 2025 18:35:07 GMT
• Title: Fast quantum measurement tomography with dimension-optimal error bounds
• Authors: Leonardo Zambrano, Sergi Ramos-Calderer, Richard Kueng,
• Abstract summary: We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost.<n>We show that the protocol requires $mathcalO(d3 L ln(d)/epsilon2)$ samples to achieve error $epsilon$ in worst-case distance.<n>We also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.
• Score: 0.6144680854063935
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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