The Quantum Overlap Gap Property and Algorithmic Hardness for the Quantum Hypergraph Max-Cut Problem

• URL: http://arxiv.org/abs/2609.10838v1
• Date: Wed, 09 Sep 2026 21:18:05 GMT
• Title: The Quantum Overlap Gap Property and Algorithmic Hardness for the Quantum Hypergraph Max-Cut Problem
• Abstract summary: We analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem.<n>We show a weak hardness result, demonstrating that for any Lipschitz constant $L$, there is some $k$ such that $L$-stable algorithms cannot approximate the optimal solution.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem using the theoretical framework of the Quantum Overlap Gap Property (QOGP). We establish two main results. Our first result applies to a wide class of stable quantum algorithms, satisfying a Lipschitz property with respect to the quantum Wasserstein distance of order $2$. We show a weak hardness result, demonstrating that for any Lipschitz constant $L$, there is some $k$ such that $L$-stable algorithms cannot approximate the optimal solution to Quantum Hypergraph Max-Cut on $k$-uniform hypergraphs in the average case. Additionally, we establish a strong hardness result where $k$ is independent of $L$, but only for a more restricted class of local quantum algorithms defined using the quantum Wasserstein distance of order $\infty$. We apply these results to establish concrete depth lower bounds for popular quantum algorithms for preparing near-optimal states for this problem.

Related papers

• Verifiable quantum advantage in extremely low depth [52.51019642214249]
  We give a problem that is solvable by shallow quantum circuits, hard for computation in time under lattice-based assumptions.<n>We prove that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.
  arXiv  Detail & Related papers  (2026-09-01T15:54:34Z)
• Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
  We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
  arXiv  Detail & Related papers  (2026-07-28T09:29:37Z)
• Quantum Approximate $k$-Minimum Finding [2.810947654192424]
  We propose an optimal quantum $k$-minimum finding algorithm that works with approximate values for all $k geq 1$.<n>We present efficient quantum algorithms for identifying the $k$ smallest expectation values among multiple observables and for determining the $k$ lowest ground state energies of a Hamiltonian.
  arXiv  Detail & Related papers  (2024-12-21T11:21:15Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Approximation Algorithms for Quantum Max-$d$-Cut [42.248442410060946]
  The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. We develop an algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees.
  arXiv  Detail & Related papers  (2023-09-19T22:53:17Z)
• Alleviating the quantum Big-$M$ problem [0.22615818641180715]
  Classically known as the "Big-$M$" problem, it affects the physical energy scale.<n>We take a systematic, encompassing look at the quantum big-$M$ problem, revealing NP-hardness in finding the optimal $M$.<n>We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks.
  arXiv  Detail & Related papers  (2023-07-19T18:00:05Z)
• Sampling Frequency Thresholds for Quantum Advantage of Quantum Approximate Optimization Algorithm [0.7046417074932257]
  We compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers. We find that classical solvers are capable of producing high-quality approximate solutions in linear time complexity. Other problems, such as different graphs, weighted MaxCut, maximum independent set, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.
  arXiv  Detail & Related papers  (2022-06-07T20:59:19Z)
• The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut [0.07614628596146598]
  We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using $textrmo(n)$ space.
  arXiv  Detail & Related papers  (2022-06-01T03:40:56Z)
• QAOA-in-QAOA: solving large-scale MaxCut problems on small quantum machines [81.4597482536073]
  Quantum approximate optimization algorithms (QAOAs) utilize the power of quantum machines and inherit the spirit of adiabatic evolution. We propose QAOA-in-QAOA ($textQAOA2$) to solve arbitrary large-scale MaxCut problems using quantum machines. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale optimization problems.
  arXiv  Detail & Related papers  (2022-05-24T03:49:10Z)
• On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids [63.75363908696257]
  The lackadaisical quantum walk is an algorithm developed to search graph structures whose vertices have a self-loop of weight $l$. This paper addresses several issues related to applying the lackadaisical quantum walk to search for multiple solutions on grids successfully.
  arXiv  Detail & Related papers  (2021-06-11T09:43:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.