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

TL;DR

This paper investigates the optimal number of queries needed for quantum channel tomography, revealing a phase transition from quantum to classical scaling depending on the dilation rate parameter.

Contribution

It identifies the dilation rate as a key parameter and characterizes the distinct query complexity regimes, uncovering a Heisenberg-to-classical phase transition.

Findings

Query complexity scales as 1/ε in the boundary regime.

Query complexity scales as 1/ε^2 in the away-from-boundary regime.

Discovered a sharp phase transition in query complexity at τ=1.

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 $\tau = r d_2 / d_1$ (which always satisfies $\tau\geq 1$ 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 $\tau$. - In the boundary regime ($\tau = 1$): we show that the query complexity is $\Theta(r d_1 d_2/\varepsilon)$ 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 $\Omega(r d_1 d_2/\varepsilon)$ for diamond norm error $\varepsilon$. - In the away-from-boundary regime ($\tau \geq 1+\Omega(1)$): we show that the query complexity is $\Theta(r d_1 d_2/\varepsilon^2)$ 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 $\tau=1$, the optimal query complexity exhibits Heisenberg scaling $1/\varepsilon$, whereas for $\tau\geq 1+\Omega(1)$, it exhibits classical scaling $1/\varepsilon^2$. In addition, we show that in the near-boundary regime ($1< \tau < 1+o(1)$), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.