Quantum channel tomography: optimal bounds in the near-boundary regime

Abstract

Quantum channel tomography (also known as quantum process tomography) is a fundamental task in quantum information and quantum computing, aiming to learn the full description of an unknown quantum channel from black-box queries. For channels with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, [CGO+26] identifies the dilation rate $τ=rd_2/d_1\geq 1$ as a key parameter governing a Heisenberg-to-classical transition in query complexity. While the optimal bounds in the boundary regime ($τ=1$) and the away-from-boundary regime ($τ\geq 1+Ω(1)$) are now known, the precise behavior in the near-boundary regime ($1<τ<1+o(1)$) has remained unresolved. In this paper, we close this gap, thereby fully settling the query complexity of quantum channel tomography. Specifically, we show that the query complexity is - $Ω(rd_1d_2/\varepsilon^2)$ for diamond norm error $\varepsilon$, matching the upper bound by [MB26, CGO+26], - $Θ(d_1^2/\varepsilon+\min\{(τ-1)d_1^2,(τ-1)^2d_1^2d_2^2\}/\varepsilon^2)$ for Choi trace norm error $\varepsilon$. Our results reveal two qualitatively different forms of the Heisenberg-to-classical transition. For diamond norm tomography, the Heisenberg scaling disappears suddenly as soon as $τ$ leaves the boundary regime. In contrast, Choi trace norm tomography exhibits a gradual transition from Heisenberg to classical scaling as $τ$ moves from the boundary regime to the away-from-boundary regime.