Pauli measurements are not optimal for single-copy tomography

2025-02-25

AI Summary

This work studies the sample complexity of single-copy tomography for an unknown $N$-qubit quantum state $ ho$, aiming to reconstruct it with trace-distance accuracy $varepsilon$ and success probability at least $0.99$. Focusing on whether Pauli measurements are optimal, we introduce the first adaptive lower-bound framework tailored to measurement constraints—novelly integrating measurement-dependent hard-instance construction with eigenvalue characterization of information channels to quantify distinguishability. We provide the first rigorous proof that Pauli measurements are suboptimal: their sample complexity is upper-bounded by $O(10^N / varepsilon^2)$ and lower-bounded by $Omega(9.118^N / varepsilon^2)$, exhibiting an exponential separation from structured POVMs such as mutually unbiased bases (MUBs) or symmetric informationally complete (SIC) measurements. Our framework generalizes to $k$-outcome measurements and recovers and extends existing tight bounds.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Web measurements

Abstract

Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $ ho in mathbb{C}^{d imes d},d=2^N$, the goal is to learn the state up to an accuracy $epsilon$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $ ho$ needed to fulfill the task. Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(frac{N cdot 12^N}{epsilon^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $Omega(frac{8^n}{epsilon^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(frac{10^N}{epsilon^2})$. To complement it with a lower bound of $Omega(frac{9.118^N}{epsilon^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones.

Problem

• Improves quantum state tomography copy complexity.
• Demonstrates separation between Pauli and structured POVMs.
• Develops framework for adaptive quantum state tomography.

Innovation

• Novel adaptive quantum state tomography
• Measurement-dependent hard instances analysis
• Eigenvalues of measurement information channel