Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition
Kean Chen | University of Pennsylvania
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 d1, output dimension d2, and Kraus rank at most r, to within error ε. We identify the dilation rate τ=rd2/d1 (which always satisfies τ≥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(rd1d2/ε) for Choi trace norm error ε, and is upper bounded by O(min{rd1.51d2/ε,rd1d2/ε2}) and lower bounded by Ω(rd1d2/ε) for diamond norm error ε.
- In the away-from-boundary regime (τ≥1+Ω(1)): we show that the query complexity is Θ(rd1d2/ε2) for both Choi trace norm and diamond norm errors ε.
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/ε, whereas for τ≥1+Ω(1), it exhibits classical scaling 1/ε2. 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.
Location
- QNC 4104
- Online on Zoom
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Meeting ID: 912 8146 6256
Passcode
: 494237
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