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Complexity Classification of Product State Problems for Local Hamiltonians

Authors John Kallaugher, Ojas Parekh , Kevin Thompson, Yipu Wang, Justin Yirka

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Author Details

John Kallaugher
  • Sandia National Laboratories, Albuquerque, NM, USA
Ojas Parekh
  • Sandia National Laboratories, Albuquerque, NM, USA
Kevin Thompson
  • Sandia National Laboratories, Albuquerque, NM, USA
Yipu Wang
  • Sandia National Laboratories, Albuquerque, NM, USA
Justin Yirka
  • Sandia National Laboratories, Albuquerque, NM, USA
  • The University of Texas at Austin, TX, USA

Funding

This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research for Quantum Computing. O.P. was also supported by U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. JY was partially supported by Scott Aaronson’s Simons Investigator Award.

Acknowledgements

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan.

Cite As Get BibTex

John Kallaugher, Ojas Parekh, Kevin Thompson, Yipu Wang, and Justin Yirka. Complexity Classification of Product State Problems for Local Hamiltonians. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 63:1-63:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.63

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 hardness of optimizing product states for Quantum Max-Cut (the quantum Heisenberg model) is NP-complete, even when every term is guaranteed to have positive unit weight.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum complexity
  • quantum algorithms
  • local hamiltonians

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