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Register a new userClassical algorithms and quantum limitations for maximum cut on high-girth graphs
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We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of classical algorithms. (1) We prove that every (quantum or classical) one-local algorithm achieves on $D$-regular graphs of girth $> 5$ a maximum cut of at most $1/2 + C/sqrt{D}$ for $C=1/sqrt{2} approx 0.7071$. This is the first such result showing that one-local algorithms achieve a value bounded away from the true optimum for random graphs, which is $1/2 + P_*/sqrt{D} + o(1/sqrt{D})$ for $P_* approx 0.7632$. (2) We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/sqrt{D} - O(1/sqrt{k})$ for $D$-regular graphs of girth $> 2k+1$, where $C = 2/pi approx 0.6366$. This is an algorithmic version of the existential bound of Lyons and is related to the algorithm of Aizenman, Lebowitz, and Ruelle (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-loc
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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.
We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a quartic speedup while using exponentially smaller space than the fastest classical spectral algorithm, and a super-polynomial speedup over classical algorithms that use only polynomial space. The classical algorithms that we present are related to, but slightly different from those presented recently in Ref. 1. In particular, we have an improved threshold for recovery and the algorithms we present work for both even and odd order tensors. These results suggest that large-scale inference problems are a promising future application for quantum computers.
We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix $Ainmathbb{R}^{ntimes d}$, sublinear algorithms for the matrix game $min_{xinmathcal{X}}max_{yinmathcal{Y}} y^{top} Ax$ were previously known only for two special cases: (1) $mathcal{Y}$ being the $ell_{1}$-norm unit ball, and (2) $mathcal{X}$ being either the $ell_{1}$- or the $ell_{2}$-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed $qin (1,2]$, we solve the matrix game where $mathcal{X}$ is a $ell_{q}$-norm unit ball within additive error $epsilon$ in time $tilde{O}((n+d)/{epsilon^{2}})$. We also provide a corresponding sublinear quantum algorithm that solves the same task in time $tilde{O}((sqrt{n}+sqrt{d})textrm{poly}(1/epsilon))$ with a quadratic improvement in both $n$ and $d$. Both our classical and quantum algorithms are optimal in the dimension parameters $n$ and $d$ up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Caratheodory problem and the $ell_{q}$-margin support vector machines as applications.
In this work, we study the limitations of the Quantum Approximate Optimization Algorithm (QAOA) through the lens of statistical physics and show that there exists $epsilon > 0$, such that $epsilonlog(n)$ depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted $k$-spin glasses when $kgeq4$ is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the kxors problem. We further extend the limitation to other boolean constraint satisfaction problems as long as the problem satisfies a combinatorial property called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. As a consequence of our techniques, we confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth---in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. Our results provide a new way to study the power and limit of QAOA through statistical physics methods and combinatorial properties.
We give an approximation algorithm for MaxCut and provide guarantees on the average fraction of edges cut on $d$-regular graphs of girth $geq 2k$. 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. Our algorithm constructs an explicit vector solution to the standard semidefinite relaxation of MaxCut and applies hyperplane rounding. It may be viewed as a simplification of the previously best known technique, which approximates Gaussian wave processes on the infinite $d$-regular tree.
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