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

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 d₁, output dimension d₂, and Kraus rank at most r, [CGO+26] identifies the dilation rate τ=rd₂/d₁≥ 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≥ 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(rd₁d₂/ε²) for diamond norm error ε, matching the upper bound by [MB26, CGO+26], - Θ(d₁²/ε+min\(τ-1)d₁²,(τ-1)²d₁²d₂²\/ε²) for Choi trace norm error ε. 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.