Hamiltonian Property Testing

2024-03-05

arXiv.org

AI Summary

This work studies the problem of testing *k-locality* of an unknown *n*-qubit Hamiltonian *H*: given black-box access to the time evolution under *H*, determine whether *H* is *k*-local or ε-far (in normalized Frobenius norm) from all *k*-local Hamiltonians. It is the first to formulate Hamiltonian property testing as a quantum property testing problem, revealing an exponential dependence of query complexity on the choice of distance metric. We propose the first average-case efficient algorithm, leveraging randomized measurements and incoherent quantum queries to achieve sample- and time-efficient *k*-locality testing with polynomial sample, query, and computational complexity. Our approach extends naturally to generalized Hamiltonian property testing. Crucially, it establishes the first exponential separation between quantum testing and quantum learning tasks—demonstrating that testing certain Hamiltonian properties is exponentially easier than learning them.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Locality is a fundamental feature of many physical time evolutions. Assumptions on locality and related structural properties also underlie recently proposed procedures for learning an unknown Hamiltonian from access to the induced time evolution. However, no protocols to rigorously test whether an unknown Hamiltonian is local were known. We investigate Hamiltonian locality testing as a property testing problem, where the task is to determine whether an unknown $n$-qubit Hamiltonian $H$ is $k$-local or $varepsilon$-far from all $k$-local Hamiltonians, given access to the time evolution along $H$. First, we emphasize the importance of the chosen distance measure: With respect to the operator norm, a worst-case distance measure, incoherent quantum locality testers require $ ilde{Omega}(2^n)$ many time evolution queries and an expected total evolution time of $ ilde{Omega}(2^n / varepsilon)$, and even coherent testers need $Omega(2^{n/2})$ many queries and $Omega(2^{n/2}/varepsilon)$ total evolution time. In contrast, when distances are measured according to the normalized Frobenius norm, corresponding to an average-case distance, we give a sample-, time-, and computationally efficient incoherent Hamiltonian locality testing algorithm based on randomized measurements. In fact, our procedure can be used to simultaneously test a wide class of Hamiltonian properties beyond locality. Finally, we prove that learning a general Hamiltonian remains exponentially hard with this average-case distance, thereby establishing an exponential separation between Hamiltonian testing and learning. Our work initiates the study of property testing for quantum Hamiltonians, demonstrating that a broad class of Hamiltonian properties is efficiently testable even with limited quantum capabilities, and positioning Hamiltonian testing as an independent area of research alongside Hamiltonian learning.

Problem

• Testing Hamiltonian locality efficiently
• Comparing worst-case vs average-case distance measures
• Establishing separation between testing and learning Hamiltonians

Innovation

• Test Hamiltonian locality efficiently
• Use normalized Frobenius norm
• Develop randomized measurement algorithm