Optimal learning of quantum channels in diamond distance

• URL: http://arxiv.org/abs/2512.10214v1
• Date: Thu, 11 Dec 2025 02:04:03 GMT
• Title: Optimal learning of quantum channels in diamond distance
• Authors: Antonio Anna Mele, Lennart Bittel,
• Abstract summary: We show that a quantum channel acting on a $d$-dimensional system can be estimated to accuracy $varepsilon$ in diamond distance.<n>We obtain, to the best of our knowledge, the first essentially optimal strategies for operator-norm learning of binary POVMs and isometries.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory and a key primitive for characterising noisy quantum devices. A long-standing open question is to determine the optimal number of uses of an unknown channel required to learn it in diamond distance, the standard measure of worst-case distinguishability between quantum processes. Here we show that a quantum channel acting on a $d$-dimensional system can be estimated to accuracy $\varepsilon$ in diamond distance using $O(d^4/\varepsilon^2)$ channel uses. This scaling is essentially optimal, as it matches lower bounds up to logarithmic factors. Our analysis extends to channels with input and output dimensions $d_{\mathrm{in}}$ and $d_{\mathrm{out}}$ and Kraus rank at most $k$, for which $O(d_{\mathrm{in}} d_{\mathrm{out}} k/\varepsilon^2)$ channel uses suffice, interpolating between unitary and fully generic channels. As by-products, we obtain, to the best of our knowledge, the first essentially optimal strategies for operator-norm learning of binary POVMs and isometries, and we recover optimal trace-distance tomography for fixed-rank states. Our approach consists of using the channel only non-adaptively to prepare copies of the Choi state, purify them in parallel, perform sample-optimal pure-state tomography on the purifications, and analyse the resulting estimator directly in diamond distance via its semidefinite-program characterisation. While the sample complexity of state tomography in trace distance is by now well understood, our results finally settle the corresponding problem for quantum channels in diamond distance.

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