Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs

• URL: http://arxiv.org/abs/2101.05513v2
• Date: Thu, 15 Apr 2021 05:08:13 GMT
• Title: Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs
• Authors: Kunal Marwaha
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The $p$-stage Quantum Approximate Optimization Algorithm (QAOA$_p$) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond $p=1$. We analyze QAOA$_2$ for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any $D$-regular graph of girth $> 5$ (i.e. without triangles, squares, or pentagons). We show that for all degrees $D \ge 2$ and every $D$-regular graph $G$ of girth $> 5$, QAOA$_2$ has a larger expected cut fraction than QAOA$_1$ on $G$. However, we also show that there exists a $2$-local randomized classical algorithm $A$ such that $A$ has a larger expected cut fraction than QAOA$_2$ on all $G$. This supports our conjecture that for every constant $p$, there exists a local classical MAX-CUT algorithm that performs as well as QAOA$_p$ on all graphs.

Related papers

• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• An Optimal Algorithm for Strongly Convex Min-min Optimization [79.11017157526815]
  Existing optimal first-order methods require $mathcalO(sqrtmaxkappa_x,kappa_y log 1/epsilon)$ of computations of both $nabla_x f(x,y)$ and $nabla_y f(x,y)$. We propose a new algorithm that only requires $mathcalO(sqrtkappa_x log 1/epsilon)$ of computations of $nabla_x f(x,
  arXiv  Detail & Related papers  (2022-12-29T19:26:12Z)
• 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)
• Finding the KT partition of a weighted graph in near-linear time [1.572727650614088]
  Kawarabayashi and Thorup gave a near-trivial time deterministic algorithm for minimum cut in a simple graph $G = (V,E)$. We give a near-linear time randomized algorithm to find the $(1+varepsilon)$-KT partition of a weighted graph.
  arXiv  Detail & Related papers  (2021-11-02T05:26:10Z)
• A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs [1.0435741631709405]
  An $soperatorname-t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects $s$ and $t$. In this work we describe a quantum algorithm for the minimum $soperatorname-t$ cut problem on undirected graphs.
  arXiv  Detail & Related papers  (2021-10-29T07:35:46Z)
• The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model [0.0]
  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.
  arXiv  Detail & Related papers  (2021-10-27T06:35:59Z)
• An explicit vector algorithm for high-girth MaxCut [0.0]
  For every $d geq 3$ and $k geq 4$, our approximation guarantees are better than those of all other classical and quantum algorithms known to the authors.
  arXiv  Detail & Related papers  (2021-08-27T19:47:56Z)
• 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)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• Maximizing Determinants under Matroid Constraints [69.25768526213689]
  We study the problem of finding a basis $S$ of $M$ such that $det(sum_i in Sv_i v_i v_itop)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning.
  arXiv  Detail & Related papers  (2020-04-16T19:16:38Z)
• Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity [94.37110094442136]
  We study the problem of agnostic $Q$-learning with function approximation in deterministic systems. We show that if $delta = Oleft(rho/sqrtdim_Eright)$, then one can find the optimal policy using $Oleft(dim_Eright)$.
  arXiv  Detail & Related papers  (2020-02-17T18:41:49Z)

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.