We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work~\cite{gamarnik2020lowFOCS}, we establish that any poly-size $n$-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least $\log n/(2\log\log n)$ as $n$ grows. This is stronger than the known state of the art bounds of the form $\Omega(\log n/(k(n)\log\log n))$ for similar combinatorial optimization problems, where $k(n)$ depends on the optimality value. For example, for the largest clique problem $k(n)$ corresponds to the square of the size of the clique~\cite{rossman2010average}. At the same time our results are not quite comparable since in our case the circuits are required to produce a \emph{solution} itself rather than solving the associated decision problem. As in our earlier work~\cite{gamarnik2020lowFOCS}, the approach is based on the overlap gap property (OGP) exhibited by random $p$-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.

中文翻译：

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p-Spin 优化问题的电路下界

我们考虑通过低深度电路找到具有 Rademacher 耦合的 $p$-spin 模型的近基态的问题。作为作者最近工作的直接扩展~\cite{gamarnik2020lowFOCS}，我们确定任何多尺寸 $n$ 输出电路在某个恒定最优因子内产生具有目标值的自旋分配，必须具有深度随着 $n$ 的增长，至少 $\log n/(2\log\log n)$。对于类似的组合优化问题，这比 $\Omega(\log n/(k(n)\log\log n))$ 形式的已知现有技术边界更强，其中 $k(n)$ 取决于最优值。例如，对于最大集团问题，$k(n)$ 对应于集团规模的平方~\cite{rossman2010average}。同时，我们的结果不太具有可比性，因为在我们的情况下，电路需要自己产生 \emph{solution} 而不是解决相关的决策问题。正如我们之前的工作~\cite{gamarnik2020lowFOCS}，该方法基于随机$p$-spin 模型展示的重叠间隙特性（OGP），但电路下界的推导进一步依赖于傅立叶分析的标准事实在布尔立方体上，特别是 Linial-Mansour-Nisan 定理。据我们所知，这是自旋玻璃理论的方法对电路复杂性产生影响的第一个例子。该方法基于随机 $p$-spin 模型展示的重叠间隙特性 (OGP)，但电路下界的推导进一步依赖于布尔立方体的傅立叶分析的标准事实，特别是 Linial-Mansour-尼散定理。据我们所知，这是自旋玻璃理论的方法对电路复杂性产生影响的第一个例子。该方法基于随机 $p$-spin 模型展示的重叠间隙特性 (OGP)，但电路下界的推导进一步依赖于布尔立方体的傅立叶分析的标准事实，特别是 Linial-Mansour-尼散定理。据我们所知，这是自旋玻璃理论的方法对电路复杂性产生影响的第一个例子。