We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.

arXiv:2206.08342Jun 16, 20226 citations

An Efficient Parallel Repetition Theorem

Johan Håstad, Rafael Pass, Douglas Wikström +1 more

Lecture notes in computer scienceJan 1, 201046 citations

A Counterexample to Strong Parallel Repetition

Ran Raz

The parallel repetition theorem states that, for any two-prover game with value $1-\epsilon$ (for, say, $\epsilon\leq1/2$ ), the value of the game repeated in parallel n times is at most $(1-\epsilon^c)^{\Omega(n/s)}$ , where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to $(1-\epsilon)^{\Omega(n/s)}$ ; this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value $1-1/2m$ . We show that the value of the odd cycle game repeated in parallel n times is at least $1-(1/m)\cdot O(\sqrt{n})$ . This implies that, for large enough n (say, $n\geq\Omega(m^2)$ ), the value of the odd cycle game repeated in parallel n times is at least $(1-1/4m^2)^{O(n)}$ . Thus the following hold. 1. For parallel repetition of general games, the bounds of $(1-\epsilon^c)^{\Omega(n/s)}$ given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant $c\geq2$ . 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of $(1-\epsilon^2)^{\Omega(n)}$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of $1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least $1-\epsilon^2$ satisfiable from instances that are at most $1-(2/\pi)\cdot\epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.The parallel repetition theorem states that, for any two-prover game with value $1-\epsilon$ (for, say, $\epsilon\leq1/2$), the value of the game repeated in parallel n times is at most $(1-\epsilon^c)^{\Omega(n/s)}$, where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to $(1-\epsilon)^{\Omega(n/s)}$; this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value $1-1/2m$. We show that the value of the odd cycle game repeated in parallel n times is at least $1-(1/m)\cdot O(\sqrt{n})$. This implies that, for large enough n (say, $n\geq\Omega(m^2)$), the value of the odd cycle game repeated in parallel n times is at least $(1-1/4m^2)^{O(n)}$. Thus the following hold. 1. For parallel repetition of general games, the bounds of $(1-\epsilon^c)^{\Omega(n/s)}$ given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant $c\geq2$. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of $(1-\epsilon^2)^{\Omega(n)}$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of $1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least $1-\epsilon^2$ satisfiable from instances that are at most $1-(2/\pi)\cdot\epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.

Annual Symposium on Foundations of Computer ScienceOct 1, 200882 citations

Almost optimal classical algorithms for a quantum generalization of Max Cut.

Ojas Parekh

Jul 30, 20200 citations

ParaQAOA: Efficient Parallel Divide-and-Conquer QAOA for Large-Scale Max-Cut Problems Beyond 10,000 Vertices

Po-Hsuan Huang, Xie-Ru Li, Chi Chuang +2 more

Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising solution for combinatorial optimization problems using a hybrid quantum-classical framework. Among combinatorial optimization problems, the Maximum Cut (Max-Cut) problem is particularly important due to its broad applicability in various domains. While QAOA-based Max-Cut solvers have been developed, they primarily favor solution accuracy over execution efficiency, which significantly limits their practicality for large-scale problems. To address the limitation, we propose ParaQAOA, a parallel divide-and-conquer QAOA framework that leverages parallel computing hardware to efficiently solve large Max-Cut problems. ParaQAOA significantly reduces runtime by partitioning large problems into subproblems and solving them in parallel while preserving solution quality. This design not only scales to graphs with tens of thousands of vertices but also provides tunable control over accuracy-efficiency trade-offs, making ParaQAOA adaptable to diverse performance requirements. Experimental results demonstrate that ParaQAOA achieves up to 1,600x speedup over state-of-the-art methods on Max-Cut problems with 400 vertices while maintaining solution accuracy within 2% of the best-known solutions. Furthermore, ParaQAOA solves a 16,000-vertex instance in 19 minutes, compared to over 13.6 days required by the best-known approach. These findings establish ParaQAOA as a practical and scalable framework for large-scale Max-Cut problems under stringent time constraints.Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising solution for combinatorial optimization problems using a hybrid quantum-classical framework. Among combinatorial optimization problems, the Maximum Cut (Max-Cut) problem is particularly important due to its broad applicability in various domains. While QAOA-based Max-Cut solvers have been developed, they primarily favor solution accuracy over execution efficiency, which significantly limits their practicality for large-scale problems. To address the limitation, we propose ParaQAOA, a parallel divide-and-conquer QAOA framework that leverages parallel computing hardware to efficiently solve large Max-Cut problems. ParaQAOA significantly reduces runtime by partitioning large problems into subproblems and solving them in parallel while preserving solution quality. This design not only scales to graphs with tens of thousands of vertices but also provides tunable control over accuracy-efficiency trade-offs, making ParaQAOA adaptable to diverse performance requirements. Experimental results demonstrate that ParaQAOA achieves up to 1,600x speedup over state-of-the-art methods on Max-Cut problems with 400 vertices while maintaining solution accuracy within 2% of the best-known solutions. Furthermore, ParaQAOA solves a 16,000-vertex instance in 19 minutes, compared to over 13.6 days required by the best-known approach. These findings establish ParaQAOA as a practical and scalable framework for large-scale Max-Cut problems under stringent time constraints.

cs.DCCross-listedarXiv:2603.26232Mar 30, 20260 citations

The parallel repetition theorem states that, for any two-prover game with value

1-\epsilon

(for, say,

\epsilon\leq1/2

), the value of the game repeated in parallel n times is at most

(1-\epsilon^c)^{\Omega(n/s)}

, where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to

(1-\epsilon)^{\Omega(n/s)}

; this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value

1-1/2m

. We show that the value of the odd cycle game repeated in parallel n times is at least

1-(1/m)\cdot O(\sqrt{n})

. This implies that, for large enough n (say,

n\geq\Omega(m^2)

), the value of the odd cycle game repeated in parallel n times is at least

(1-1/4m^2)^{O(n)}

. Thus the following hold. 1. For parallel repetition of general games, the bounds of

(1-\epsilon^c)^{\Omega(n/s)}

given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant

c\geq2

. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of

(1-\epsilon^2)^{\Omega(n)}

given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of

1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})

given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least

1-\epsilon^2

satisfiable from instances that are at most

1-(2/\pi)\cdot\epsilon

satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.