Optimal learning of quantum channels in diamond distance

TL;DR

The paper resolves the optimal diamond-distance sample complexity for learning quantum channels by introducing a fully non-adaptive, parallel protocol that reduces channel learning to optimal pure-state tomography on purified Choi states, requiring $N = O(d_{in} d_{out} k / \varepsilon^2)$ uses. The method proceeds through parallel Choi-state preparation, a random purification step, covariant pure-state tomography, and a CPTP-projection regularisation, with coherence only during purification. This channel-centric framework unifies and sharpens results for states, isometries/unitaries, and POVMs, delivering essentially optimal bounds across these primitives and clarifying the roles of adaptivity and coherence. It thereby advances the understanding of quantum process tomography in diamond distance and provides practically relevant tomographic guarantees with clear scaling dependences on dimension, Kraus rank, and desired accuracy.

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.

Figures (1)

• Figure 1: Protocol for learning quantum channels. Repeated invocations of the unknown channel $\Lambda$, fed with maximally entangled inputs $\ket{\Omega}$, prepare multiple copies of its (normalised) Choi state $\Phi_c = J(\Lambda)$. A purification channel then maps $\Phi_c$ to a (random) purification $\ket{\Phi_c}$, which is reconstructed using an optimal pure-state tomography scheme. Finally, a classical post-processing step, implemented via a semidefinite program that projects onto the set of CPTP maps, returns a quantum channel estimate $\hat{\Lambda}$ satisfying $\|\hat{\Lambda} - \Lambda\|_\diamond \le \varepsilon$ with high probability.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2: Main theorem; informal
• Lemma 2.1: Dimension constraint from Kraus rank
• proof
• Lemma 2.2: SDP formulation of the diamond norm; cf. Eq. (7.23) in Ref. SkrzypczykCavalcantiSDP
• proof
• Lemma 2.3: Diamond norm for positive Choi operator
• proof
• Lemma 2.4: Diamond-norm Cauchy--Schwarz for purifications
• proof