We consider the power of local algorithms for approximately solving Max $k$XOR, a generalization of two constraint satisfaction problems previously studied with classical and quantum algorithms (MaxCut and Max E3LIN2). In Max $k$XOR each constraint is the XOR of exactly $k$ variables and a parity bit. On instances with either random signs (parities) or no overlapping clauses and $D+1$ clauses per variable, we calculate the expected satisfying fraction of the depth-1 QAOA from Farhi et al [arXiv:1411.4028] and compare with a generalization of the local threshold algorithm from Hirvonen et al [arXiv:1402.2543]. Notably, the quantum algorithm outperforms the threshold algorithm for $k$$\gt$$4$.

On the other hand, we highlight potential difficulties for the QAOA to achieve computational quantum advantage on this problem. We first compute a tight upper bound on the maximum satisfying fraction of nearly all large random regular Max $k$XOR instances by numerically calculating the ground state energy density $P(k)$ of a mean-field $k$-spin glass [arXiv:1606.02365]. The upper bound grows with $k$ much faster than the performance of both one-local algorithms. We also identify a new obstruction result for low-depth quantum circuits (including the QAOA) when $k=3$, generalizing a result of Bravyi et al [arXiv:1910.08980] when $k=2$. We conjecture that a similar obstruction exists for all $k$.

中文翻译：

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使用量子和经典局部算法逼近 Max $k$XOR 的界限

我们考虑局部算法近似求解 Max $k$XOR 的能力，这是先前使用经典算法和量子算法（MaxCut 和 Max E3LIN2）研究的两个约束满足问题的概括。在 Max $k$XOR 中，每个约束都是恰好 $k$ 变量和奇偶校验位的 XOR。在具有随机符号（奇偶校验）或没有重叠子句和每个变量 $D+1$ 子句的实例上，我们计算来自 Farhi 等人 [arXiv:1411.4028] 的深度 1 QAOA 的预期满足分数，并与Hirvonen 等人 [arXiv:1402.2543] 的局部阈值算法。值得注意的是，量子算法在$k$$\gt$$4$ 上优于阈值算法。

另一方面，我们强调了 QAOA 在这个问题上实现计算量子优势的潜在困难。我们首先通过数值计算平均场$k$-spin glass的基态能量密度$P(k)$来计算几乎所有大型随机常规Max$k$XOR实例的最大满足部分的严格上限[ arXiv:1606.02365]。上限以 $k$ 的速度增长，比这两种单局部算法的性能都要快得多。我们还确定了 $k=3$ 时低深度量子电路（包括 QAOA）的新阻碍结果，概括了 $k=2$ 时 Bravyi 等人 [arXiv:1910.08980] 的结果。我们推测所有$k$都存在类似的障碍。