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Quantum Max-Cut is NP hard to approximate

Stephen Piddock

TL;DR

The paper proves unconditional NP-hardness of approximating Quantum Max-Cut within a constant factor on unweighted graphs of bounded degree, by connecting Product-QMC to Quantum Max-Cut and embedding a classical Max-Cut instance via a two-stage reduction. It introduces a PTAS reduction framework and uses SU(2) representation theory and Lieb's inequality to relate product-state optima to full quantum optima. A rank-constrained Max-Cut construction with triangle gadgets and bipartite expanders yields APX-completeness for all constant k, including a direct hardness for Product-QMC with XY interactions. Overall, the work establishes strong hardness of approximation for a broad quantum local Hamiltonian family without relying on conjectures, and clarifies the computational landscape of quantum max-cut type problems.

Abstract

We unconditionally prove that it is NP-hard to compute a constant multiplicative approximation to the QUANTUM MAX-CUT problem on an unweighted graph of constant bounded degree. The proof works in two stages: first we demonstrate a generic reduction to computing the optimal value of a quantum problem, from the optimal value over product states. Then we prove an approximation preserving reduction from MAX-CUT to PRODUCT-QMC the product state version of QUANTUM MAX-CUT. More precisely, in the second part, we construct a PTAS reduction from MAX-CUT$_k$ (the rank-k constrained version of MAX-CUT) to MAX-CUT$_{k+1}$, where MAX-CUT and PRODUCT-QMC coincide with MAX-CUT$_1$ and MAX-CUT$_3$ respectively. We thus prove that Max-Cut$_k$ is APX-complete for all constant $k$.

Quantum Max-Cut is NP hard to approximate

TL;DR

The paper proves unconditional NP-hardness of approximating Quantum Max-Cut within a constant factor on unweighted graphs of bounded degree, by connecting Product-QMC to Quantum Max-Cut and embedding a classical Max-Cut instance via a two-stage reduction. It introduces a PTAS reduction framework and uses SU(2) representation theory and Lieb's inequality to relate product-state optima to full quantum optima. A rank-constrained Max-Cut construction with triangle gadgets and bipartite expanders yields APX-completeness for all constant k, including a direct hardness for Product-QMC with XY interactions. Overall, the work establishes strong hardness of approximation for a broad quantum local Hamiltonian family without relying on conjectures, and clarifies the computational landscape of quantum max-cut type problems.

Abstract

We unconditionally prove that it is NP-hard to compute a constant multiplicative approximation to the QUANTUM MAX-CUT problem on an unweighted graph of constant bounded degree. The proof works in two stages: first we demonstrate a generic reduction to computing the optimal value of a quantum problem, from the optimal value over product states. Then we prove an approximation preserving reduction from MAX-CUT to PRODUCT-QMC the product state version of QUANTUM MAX-CUT. More precisely, in the second part, we construct a PTAS reduction from MAX-CUT (the rank-k constrained version of MAX-CUT) to MAX-CUT, where MAX-CUT and PRODUCT-QMC coincide with MAX-CUT and MAX-CUT respectively. We thus prove that Max-Cut is APX-complete for all constant .

Paper Structure

This paper contains 11 sections, 7 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SU(2) representation theory
  4. Local Hamiltonians and Lieb's inequality
  5. The APX class and PTAS reductions
  6. Reduction from product state problem
  7. Rank constrained Max-Cut
  8. Construction
  9. Triangle gadget
  10. Bipartite graphs
  11. Proof of Lemma \ref{['lem:main']}

Key Result

Theorem 1

There is a constant $\alpha<1$ such that it is NP-hard to compute a value that approximates the optimal Quantum Max-Cut value within approximation ratio $\alpha$. This holds even when the graph $G$ is unweighted ($w_{ij}=1$ for all $ij\in E$) and each vertex has bounded (constant) degree.

Figures (2)

  • Figure 1: Example interaction graph of $H'$. At the top of the figure is a simple interaction graph of a 2-local Hamiltonian $H$ on $n=3$ qubits labelled $1,2,3$. Beneath is the interaction graph of the new Hamiltonian $H'$. Each qubit of $H$ has been replaced by a collection of $T=3$ qubits in $H'$.
  • Figure 2: Example of the construction of $G'$ described in Section \ref{['sec:construction']}. Here, the original graph $G$ is a path graph on three vertices $\{1,2,3\}$ and the parameter $\eta=2$. The original edges and vertices of $G$ are coloured black; the edges in the triangle gadgets are colooured blue, and the edges in the $d$-regular bipartite graph are coloured red.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 2 more