Complexity Classification of Product State Problems for Local Hamiltonians

2024-01-12

arXiv.org

AI Summary

This work investigates the computational complexity of minimizing the ground-state energy of local Hamiltonians over product states (i.e., unentangled tensor products of single-qubit states), focusing on a complete classification for arbitrary fixed two-qubit interaction types. Methodologically, it introduces a novel framework combining constraint satisfaction problem classification with vector-cut reductions, constructs constant-strength coupling encodings, and defines a new variant of Vector Max-Cut—proven NP-complete via reduction from Max-Cut. The main contribution is the first rigorous dichotomy theorem: the problem lies in P if and only if all allowed interactions are 1-local; otherwise, it is NP-complete. Notably, this establishes NP-completeness even for the quantum Heisenberg model (quantum Max-Cut) under positive-weight product-state optimization. These results fully characterize the computational complexity boundary for estimating product-state ground energies across all families of two-qubit interactions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Applications of cryptography

Abstract

Product states, unentangled tensor products of single qubits, are a ubiquitous ansatz in quantum computation, including for state-of-the-art Hamiltonian approximation algorithms. A natural question is whether we should expect to efficiently solve product state problems on any interesting families of Hamiltonians. We completely classify the complexity of finding minimum-energy product states for Hamiltonians defined by any fixed set of allowed 2-qubit interactions. Our results follow a line of work classifying the complexity of solving Hamiltonian problems and classical constraint satisfaction problems based on the allowed constraints. We prove that estimating the minimum energy of a product state is in P if and only if all allowed interactions are 1-local, and NP-complete otherwise. Equivalently, any family of non-trivial two-body interactions generates Hamiltonians with NP-complete product-state problems. Our hardness constructions only require coupling strengths of constant magnitude. A crucial component of our proofs is a collection of hardness results for a new variant of the Vector Max-Cut problem, which should be of independent interest. Our definition involves sums of distances rather than squared distances and allows linear stretches. We similarly give a proof that the original Vector Max-Cut problem is NP-complete in 3 dimensions. This implies that optimizing over product states for Quantum Max-Cut (the quantum Heisenberg model) is NP-complete, even when every term is guaranteed to have positive unit weight.

Problem

• Quantum Computing
• Energy Minimization
• NP-Completeness

Innovation

• Quantum Interaction Rules
• NP-Completeness Proof
• Vector Max Cut Variant