Relaxations and exact solutions to Quantum Max Cut via the algebraic structure of swap operators

Bene Watts, Adam (author), Chowdhury, Anirban (author), Epperly, Aidan (author), Helton, J. William (author), Klep, Igor (author)

The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance $10^{-7}$) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.

• Language: English
• Keywords: Quantum Max Cut, swap operators, noncommutative polynomials, symmetric group, Gröbner bases
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2024
• Year: 2024
• Number of pages: 88 str.
• Numbering: Vol. 8, [article no.] 1352
• PID: 20.500.12556/RUL-158307
• UDC: 512
• ISSN on article: 2521-327X
• DOI: 10.22331/q-2024-05-22-1352
• COBISS.SI-ID: 197706499
• Publication date in RUL: 04.06.2024

Record is a part of a journal

• Title: Quantum
• Shortened title: Quantum
• Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
• ISSN: 2521-327X
• COBISS.SI-ID: 526900761

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Quantum Max Cut, Swap operatorji, nekomutativni polinomi, simetrična grupa, Gröbnerjeve baze