Bounds on approximating Max $k$XOR with quantum and classical local algorithms

• URL: http://arxiv.org/abs/2109.10833v3
• Date: Thu, 30 Jun 2022 21:19:35 GMT
• Title: Bounds on approximating Max $k$XOR with quantum and classical local algorithms
• Authors: Kunal Marwaha, Stuart Hadfield
• Abstract summary: We consider the power of local algorithms for approximately solving Max $k$XOR. In Max $k$XOR each constraint is the XOR of exactly $k$ variables and a parity bit. We show that the quantum algorithm outperforms the threshold algorithm for $k > 4$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $k > 4$. 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 $k$.

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