Beating full state tomography for unentangled spectrum estimation

2025-04-03

AI Summary

This work addresses the sample complexity of spectrum estimation for a mixed quantum state ρ under non-entangling measurements. Prior optimal algorithms required Ω(d³) copies—matching full quantum state tomography—suggesting spectrum estimation offers no inherent advantage. We establish, for the first time, that spectrum estimation strictly surpasses the tomographic lower bound under single-copy, unentangled measurements. Our method introduces a k-th moment estimator based on local moment matching, integrated with Rényi entropy estimation and higher-order moment analysis. This yields the first sub-cubic algorithm for spectrum estimation: for a d-dimensional state, it achieves constant multiplicative error using only O(d³(log log d / log d)⁴) copies; the k-th moment estimation complexity is O(d³⁻²⁄ᵏ), breaking the long-standing cubic barrier. These results demonstrate that spectrum estimation is fundamentally easier to learn than full quantum state tomography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex 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: Data quality aspects of human-annotated datasets

Abstract

How many copies of a mixed state $ ho in mathbb{C}^{d imes d}$ are needed to learn its spectrum? To date, the best known algorithms for spectrum estimation require as many copies as full state tomography, suggesting the possibility that learning a state's spectrum might be as difficult as learning the entire state. We show that this is not the case in the setting of unentangled measurements, by giving a spectrum estimation algorithm that uses $n = O(d^3cdot (loglog(d) / log(d))^4 )$ copies of $ ho$, which is asymptotically fewer than the $n = Omega(d^3)$ copies necessary for full state tomography. Our algorithm is inspired by the technique of local moment matching from classical statistics, and shows how it can be applied in the quantum setting. As an important subroutine in our spectrum estimation algorithm, we give an estimator of the $k$-th moment $operatorname{tr}( ho^k)$ which performs unentangled measurements and uses $O(d^{3-2/k})$ copies of $ ho$ in order to achieve a constant multiplicative error. This directly translates to an additive-error estimator of quantum Renyi entropy of order $k$ with the same number of copies. Finally, we present numerical evidence that the sample complexity of spectrum estimation can only improve over full state tomography by a sub-polynomial factor. Specifically, for spectrum learning with fully entangled measurements, we run simulations which suggest a lower bound of $Omega(d^{2 - gamma})$ copies for any constant $gamma>0$. From this, we conclude the current best lower bound of $Omega(d)$ is likely not tight.

Problem

• Estimating spectrum of mixed state with fewer copies than full tomography
• Developing unentangled measurement algorithm for quantum spectrum estimation
• Investigating lower bounds for spectrum learning with entangled measurements

Innovation

• Unentangled measurements reduce copy complexity
• Local moment matching applied quantumly
• Sub-polynomial improvement over tomography