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arXiv:2510.07995

Quantum Max-Cut is NP hard to approximate

Stephen Piddock

Published Oct 9, 2025• 0 citations
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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 $k$.

DOI
10.48550/arxiv.2510.07995

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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-CUTk_kk​ (the rank-k constrained version of MAX-CUT) to MAX-CUTk+1_{k+1}k+1​, where MAX-CUT and PRODUCT-QMC coincide with MAX-CUT1_11​ and MAX-CUT3_33​ respectively. We thus prove that Max-Cutk_kk​ is APX-complete for all constant kkk.
Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of α3+εα_3+εα3​+ε for every ε>0ε>0ε>0, where α3≈.83600811464α_3\approx.83600811464α3​≈.83600811464 is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in [−1/2,2/5][-1/2,2/5][−1/2,2/5] and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of αBOV+εα_{\rm BOV}+εαBOV​+ε for every ε>0ε>0ε>0, where αBOV≈0.9563372685α_{\rm BOV}\approx 0.9563372685αBOV​≈0.9563372685 is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their Sk−1S^{k-1}Sk−1-valued Borell inequality for correlations in [−.5843,.5843][-.5843,.5843][−.5843,.5843] for all k≥3k\geq3k≥3.
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 present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2)Θ(n^{3/2})Θ(n3/2) bound for finding a maximum-out-degree vertex in an nnn-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an nnn-vertex undirected graph requires Θ(n3/2)Θ(n^{3/2})Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)(k,n-k)(k,n−k)-Cut, which asks for the largest cut with one side of size kkk. For every constant k≥1k\geq 1k≥1, we give an O(n2−1/(k+1))O(n^{2-1/(k+1)})O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)(k,n-k)(k,n−k)-Cut involving kkk-Graph Collision, which asks whether the marked vertices of a known graph contain a kkk-clique. For k≥3k\geq 3k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/23/23/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for kkk-Graph Collision, the reduction yields a lower bound matching our upper bound above.
It is well-known in physics that the limit of large quantum spin SSS should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-SSS generalizations of Quantum Max-Cut (QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​), equivalent to the problem of finding the ground state energy of an arbitrary spin-SSS quantum Heisenberg antiferromagnet (QHAS\mathrm{QHA}_SQHAS​). We prove that approximating the value of QHAS\mathrm{QHA}_SQHAS​ to inverse polynomial accuracy is QMA-complete for all SSS, extending previous results for S=1/2S=1/2S=1/2. We also present two distinct families of classical approximation algorithms for QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with SSS and converge to the Briët-Oliveira-Vallentin approximation ratio αBOV≈0.956α_{\mathrm{BOV}} \approx 0.956αBOV​≈0.956 from below as S→∞S \to \inftyS→∞.
Evaluating expectation values is a critical task for variational quantum eigensolvers, parameterized quantum circuits, and many other quantum algorithms. We consider the well-studied case of the Quantum Approximate Optimization Algorithm (QAOA) for the MaxCut problem. Recent work of Wang et al. [arXiv:2511.20212] showed this task to be NP-hard in general for any QAOA depth p≥2p\geq 2p≥2, complementing past results showing efficiently computable formulas for p=1p=1p=1 with arbitrary problem graphs. We sharpen this dichotomy showing that for p≥2p\geq 2p≥2 exact or exponentially precise cost expectation value evaluation is #P-hard under deterministic polynomial-time Turing reductions. Hardness at p≥2p\geq 2p≥2 is shown to remain even for evaluating single pairwise correlators $\langle Z\otimes Z\rangle $, as well as for highly restricted sets of algorithm parameters. Our proof refines the NP-hardness construction of Wang et al. that recovers the maximum cut value from the largest exponent of a QAOA Laurent polynomial, utilizing a distinct and simpler construction that extracts a value proportional to the total number of maximum cuts, in addition to the optimal cut value. Thus we show that the QAOA expectation value hardness transition from p=1p=1p=1 to p=2p=2p=2 is not only from tractability to optimization hardness, but to that of counting optimal solutions. As an application we show our results imply analogous hardness results for computing gradients and Hessians of QAOA circuits.