Optimal lower bound for quantum channel tomography in away-from-boundary regime

2026-01-15

AI Summary

This work investigates the tomographic query complexity of quantum channels with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, under the non-boundary condition $r d_2 \geq 2 d_1$. By integrating tools from quantum information theory, channel representations, and diamond-norm error analysis, the authors establish—for the first time—a lower bound of $\Omega(r d_1 d_2 / \varepsilon^2)$ on the number of queries required to achieve reconstruction error $\varepsilon$, matching the best-known upper bound. This result fully characterizes the optimal query complexity in this regime as $\Theta(r d_1 d_2 / \varepsilon^2)$. Moreover, it highlights a fundamental distinction from the Heisenberg scaling observed in the special case of unitary channels ($r = 1$), underscoring how increased Kraus rank alters the intrinsic complexity of quantum channel tomography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Consider quantum channels with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. Any such channel must satisfy the constraint $rd_2\geq d_1$, and the parameter regime $rd_2=d_1$ is called the boundary regime. In this paper, we show an optimal query lower bound $\Omega(rd_1d_2/\varepsilon^2)$ for quantum channel tomography to within diamond norm error $\varepsilon$ in the away-from-boundary regime $rd_2\geq 2d_1$, matching the existing upper bound $O(rd_1d_2/\varepsilon^2)$. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions $d_1=d_2=d$ with $r\geq 2$, in sharp contrast to the unitary case $r=1$ where Heisenberg scaling $\Theta(d^2/\varepsilon)$ is achievable.

Problem

• quantum channel tomography
• query complexity
• diamond norm
• away-from-boundary regime
• Kraus rank

Innovation

• quantum channel tomography
• query complexity
• lower bound
• diamond norm
• Kraus rank