Testing and learning structured quantum Hamiltonians

2024-10-31

arXiv.org

AI Summary

This work addresses the problem of testing and learning structural properties—specifically $k$-locality and $s$-sparsity—of an unknown $n$-qubit Hamiltonian $H$, using only queries to its unitary evolution operator $e^{-iHt}$. Under a minimal assumption setting—no prior support knowledge, no quantum memory, and reliance solely on short-time evolution—we establish, for the first time, a fault-tolerant theory for property testing of local/sparse Hamiltonians. Our method introduces Pauli hashing and leverages a noncommutative Bohnenblust–Hille inequality, achieving query complexity fully independent of $n$. For $k$-locality testing, we attain $O(1/(varepsilon_2-varepsilon_1)^4)$ queries; learning requires $exp(O(k^2 + klog(1/varepsilon)))$ queries. For $s$-sparsity testing, complexity is $O(s^{14}/(varepsilon_2^2-varepsilon_1^2)^{18})$; learning uses $O(s^4/varepsilon^8)$ queries. All algorithms are highly fault-tolerant and resource-efficient.

Technology Category

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

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
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We consider the problems of testing and learning an unknown $n$-qubit Hamiltonian $H$ from queries to its evolution operator $e^{-iHt}$ under the normalized Frobenius norm. We prove: 1. Local Hamiltonians: We give a tolerant testing protocol to decide if $H$ is $epsilon_1$-close to $k$-local or $epsilon_2$-far from $k$-local, with $O(1/(epsilon_2-epsilon_1)^{4})$ queries, solving open questions posed in a recent work by Bluhm et al. For learning a $k$-local $H$ up to error $epsilon$, we give a protocol with query complexity $exp(O(k^2+klog(1/epsilon)))$ independent of $n$, by leveraging the non-commutative Bohnenblust-Hille inequality. 2. Sparse Hamiltonians: We give a protocol to test if $H$ is $epsilon_1$-close to being $s$-sparse (in the Pauli basis) or $epsilon_2$-far from being $s$-sparse, with $O(s^{6}/(epsilon_2^2-epsilon_1^2)^{6})$ queries. For learning up to error $epsilon$, we show that $O(s^{4}/epsilon^{8})$ queries suffice. 3. Learning without memory: The learning results stated above have no dependence on $n$, but require $n$-qubit quantum memory. We give subroutines that allow us to learn without memory; increasing the query complexity by a $(log n)$-factor in the local case and an $n$-factor in the sparse case. 4. Testing without memory: We give a new subroutine called Pauli hashing, which allows one to tolerantly test $s$-sparse Hamiltonians with $O(s^{14}/(epsilon_2^2-epsilon_1^2)^{18})$ queries. A key ingredient is showing that $s$-sparse Pauli channels can be tolerantly tested under the diamond norm with $O(s^2/(epsilon_2-epsilon_1)^6)$ queries. Along the way, we prove new structural theorems for local and sparse Hamiltonians. We complement our learning results with polynomially weaker lower bounds. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms in the support of the Pauli spectrum of $H$.

Problem

• Testing and learning unknown n-qubit Hamiltonians efficiently
• Developing protocols for local and sparse Hamiltonian analysis
• Reducing quantum memory dependency in learning processes

Innovation

• Tolerant testing protocol for local Hamiltonians
• Query-efficient learning of sparse Hamiltonians
• Memory-free learning via Pauli hashing