Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition

2026-04-19

AI Summary

This work investigates the classical tomography of an unknown quantum channel with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, characterizing its optimal query complexity in terms of the expansion ratio $\tau = r d_2 / d_1$. By leveraging the Choi state representation, tools from quantum information theory, and the diamond norm as the error metric, the authors establish matching upper and lower bounds that reveal a sharp phase transition: when $\tau = 1$, the query complexity achieves the Heisenberg scaling $\Theta(r d_1 d_2 / \varepsilon)$; when $\tau \geq 1 + \Omega(1)$, it degrades to the classical scaling $\Theta(r d_1 d_2 / \varepsilon^2)$; and in the intermediate regime near the threshold, a hybrid scaling behavior emerges. This analysis provides the first complete characterization of this complexity phase transition.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood. In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, to within error $\varepsilon$. We identify the dilation rate $τ= r d_2 / d_1$ (which always satisfies $τ\geq 1$ due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of $τ$. - In the boundary regime ($τ= 1$): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon)$ for Choi trace norm error $\varepsilon$, and is upper bounded by $O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})$ and lower bounded by $Ω(r d_1 d_2/\varepsilon)$ for diamond norm error $\varepsilon$. - In the away-from-boundary regime ($τ\geq 1+Ω(1)$): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon^2)$ for both Choi trace norm and diamond norm errors $\varepsilon$. Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at $τ=1$, the optimal query complexity exhibits Heisenberg scaling $1/\varepsilon$, whereas for $τ\geq 1+Ω(1)$, it exhibits classical scaling $1/\varepsilon^2$. In addition, we show that in the near-boundary regime ($1< τ< 1+o(1)$), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.

Problem

• quantum channel tomography
• query complexity
• Heisenberg scaling
• classical scaling
• Kraus rank

Innovation

• quantum channel tomography
• query complexity
• Heisenberg-to-classical transition
• dilation rate
• phase transition