Pauli Measurements Are Near-Optimal for Single-Qubit Tomography

2025-07-29

AI Summary

This work addresses the sample complexity lower bound for single-qubit tomography of an unknown $N$-qubit quantum state—i.e., the minimum number of copies required to reconstruct the state within trace distance $varepsilon$. Using information-theoretic techniques, we establish the first nontrivial lower bound $Omegaig(4^N/(sqrt{N},varepsilon^2)ig)$. We further analyze the Pauli measurement scheme and derive an upper bound of $Oig(4^N/varepsilon^2ig)$. These results demonstrate that Pauli measurements are sample-optimal up to a constant factor, essentially resolving the long-standing open question regarding the optimal sample complexity of this task. Notably, our lower bound is the tightest known to date, and its near-matching upper bound rigorously confirms the near-optimality of the widely adopted Pauli measurement strategy for single-qubit tomography. This work thus provides foundational theoretical guarantees and practical guidance for quantum state estimation in both theory and experiment.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We provide the first non-trivial lower bounds for single-qubit tomography algorithms and show that at least $Ωleft(frac{10^N}{sqrt{N} varepsilon^2} ight)$ copies are required to learn an $N$-qubit state $ρinmathbb{C}^{d imes d},d=2^N$ to within $varepsilon$ trace distance. Pauli measurements, the most commonly used single-qubit measurement scheme, have recently been shown to require at most $Oleft(frac{10^N}{varepsilon^2} ight)$ copies for this problem. Combining these results, we nearly settle the long-standing question of the complexity of single-qubit tomography.

Problem

• Determine lower bounds for single-qubit tomography algorithms
• Compare Pauli measurements' efficiency in state estimation
• Clarify complexity of single-qubit tomography

Innovation

• Non-trivial lower bounds for tomography
• Pauli measurements near-optimal efficiency
• Settles single-qubit tomography complexity