The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model

• URL: http://arxiv.org/abs/2110.14206v3
• Date: Thu, 7 Jul 2022 13:35:52 GMT
• Title: The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model
• Authors: Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, Leo Zhou
• Abstract summary: The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to optimization problems. We give an iterative formula to evaluate performance for any $D$ at any depth $p$. We make an optimistic conjecture that the QAOA, as $p$ goes to infinity, will achieve the Parisi value.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth $p$. We apply the QAOA to MaxCut on large-girth $D$-regular graphs. We give an iterative formula to evaluate performance for any $D$ at any depth $p$. Looking at random $D$-regular graphs, at optimal parameters and as $D$ goes to infinity, we find that the $p=11$ QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these $D$-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-$q$-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as $O(p^2 4^p)$. This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to $p=20$. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as $p$ goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.

Related papers

• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Fast Maximum $k$-Plex Algorithms Parameterized by Small Degeneracy Gaps [30.06993761032835]
  The maximum $k$-plex problem is important but computationally challenging in applications such as graph mining and community detection. We present an exact algorithm parameterized by $g_k(G)$, which has the worst-case running time in the size of the input graph and exponential in $g_k(G)$. We further extend our discussion to an even smaller parameter $cg_k(G)$, the gap between the community-degeneracy bound and the size of the maximum $k$-plex.
  arXiv  Detail & Related papers  (2023-06-23T01:28:24Z)
• ReSQueing Parallel and Private Stochastic Convex Optimization [59.53297063174519]
  We introduce a new tool for BFG convex optimization (SCO): a Reweighted Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. We develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings.
  arXiv  Detail & Related papers  (2023-01-01T18:51:29Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models [15.857373057387669]
  We prove concentration properties at any constant level (number of layers) on ensembles of random optimization problems in the infinite size limit. Our analysis can be understood via a saddle-point approximation of a sum-over-paths integral. We show that the performance of the QAOA at constant levels is bounded away from optimality for pure $q$-spin models when $qge 4$ and is even.
  arXiv  Detail & Related papers  (2022-04-21T17:40:39Z)
• Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach [94.37521840642141]
  We study the correlation clustering problem using the quantum approximate optimization algorithm (QAOA) and qudits. Specifically, we consider a neutral atom quantum computer and propose a full stack approach for correlation clustering. We show the qudit implementation is superior to the qubit encoding as quantified by the gate count.
  arXiv  Detail & Related papers  (2021-06-22T11:07:38Z)
• Classical algorithms and quantum limitations for maximum cut on high-girth graphs [6.262125516926276]
  We show that every (quantum or classical) one-local algorithm achieves on $D$-regular graphs of $> 5$ a maximum cut of at most $1/2 + C/sqrtD$ for $C=1/sqrt2 approx 0.7071$. We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/sqrtD - O (1/sqrtk)$ for $D$-regular graphs of $> 2k+1$, where
  arXiv  Detail & Related papers  (2021-06-10T16:28:23Z)
• Bayesian Optimistic Optimisation with Exponentially Decaying Regret [58.02542541410322]
  The current practical BO algorithms have regret bounds ranging from $mathcalO(fraclogNsqrtN)$ to $mathcal O(e-sqrtN)$, where $N$ is the number of evaluations. This paper explores the possibility of improving the regret bound in the noiseless setting by intertwining concepts from BO and tree-based optimistic optimisation. We propose the BOO algorithm, a first practical approach which can achieve an exponential regret bound with order $mathcal O(N-sqrt
  arXiv  Detail & Related papers  (2021-05-10T13:07:44Z)
• Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs [0.0]
  We show that for all degrees $D ge 2$ of girth $> 5$, QAOA$ has a larger expected cut fraction than QAOA$$ on $G$. We conjecture that for every constant $p$, there exists a local classical MAX-CUT algorithm that performs as well as QAOA$_p$ on all graphs.
  arXiv  Detail & Related papers  (2021-01-14T09:17:20Z)
• Hybrid quantum-classical algorithms for approximate graph coloring [65.62256987706128]
  We show how to apply the quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring of a graph. We construct an efficient classical simulation algorithm which simulates level-$1$ QAOA and level-$1$ RQAOA for arbitrary graphs.
  arXiv  Detail & Related papers  (2020-11-26T18:22:21Z)
• Generating Target Graph Couplings for QAOA from Native Quantum Hardware Couplings [3.2622301272834524]
  We present methods for constructing any target coupling graph using limited global controls in an Ising-like quantum spin system. Our approach is motivated by implementing the quantum approximate optimization algorithm (QAOA) on trapped ion quantum hardware. Noisy simulations of Max-Cut QAOA show that our implementation is less susceptible to noise than the standard gate-based compilation.
  arXiv  Detail & Related papers  (2020-11-16T18:43:09Z)

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.