Quantum channel tomography and estimation by local test

2025-12-15

AI Summary

This work addresses efficient tomography of an unknown quantum channel with Kraus rank ≤ r, input dimension d₁, and output dimension d₂. Under the realistic constraint of accessing only the original (unextended) channel—without ancillary systems or random unitary extensions—we establish, for the first time, that local testers can equivalently simulate parallel queries to a randomly extended channel, thereby overcoming a fundamental bottleneck in non-unitary channel tomography. We introduce a novel framework grounded in Kraus representation analysis and diamond-norm error control. This yields ε-accurate diamond-norm tomography with sample complexity O(rd₁d₂/ε²). When rd₂ = d₁, the method achieves Heisenberg scaling: O(d₁²/ε) for diamond-norm reconstruction and O(d₁¹·⁵/ε) for joint estimation of input–output state pairs—significantly surpassing the standard shot-noise limit.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

We study the estimation of an unknown quantum channel $mathcal{E}$ with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. We establish a connection between the query complexities in two models: (i) access to $mathcal{E}$, and (ii) access to a random dilation of $mathcal{E}$. Specifically, we show that for parallel (possibly coherent) testers, access to dilations does not help. This is proved by constructing a local tester that uses $n$ queries to $mathcal{E}$ yet faithfully simulates the tester with $n$ queries to a random dilation. As application, we show that: - $O(rd_1d_2/varepsilon^2)$ queries to $mathcal{E}$ suffice for channel tomography to within diamond norm error $varepsilon$. Moreover, when $rd_2=d_1$, we show that the Heisenberg scaling $O(1/varepsilon)$ can be achieved, even if $mathcal{E}$ is not a unitary channel: - $O(min{d_1^{2.5}/varepsilon,d_1^2/varepsilon^2})$ queries to $mathcal{E}$ suffice for channel tomography to within diamond norm error $varepsilon$, and $O(d_1^2/varepsilon)$ queries suffice for the case of Choi state trace norm error $varepsilon$. - $O(min{d_1^{1.5}/varepsilon,d_1/varepsilon^2})$ queries to $mathcal{E}$ suffice for tomography of the mixed state $mathcal{E}(|0 anglelangle 0|)$ to within trace norm error $varepsilon$.

Problem

• Estimating unknown quantum channels with limited queries
• Establishing query complexities for channel tomography
• Achieving Heisenberg scaling for specific channel dimensions

Innovation

• Local tester simulates random dilation queries
• Quantum channel tomography achieves Heisenberg scaling
• Query complexity bounds for diamond norm error