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

Kunal Marwaha1,2,3,4 and Stuart Hadfield1,2

1Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett Field, CA
2USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA
3Berkeley Center for Quantum Information and Computation, University of California, Berkeley, CA
4Department of Computer Science, University of Chicago, Chicago, IL

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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$$\gt$$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 $k$.

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Cited by

[1] Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou, "The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model", arXiv:2110.14206.

[2] Nicolas PD Sawaya, Albert T Schmitz, and Stuart Hadfield, "Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems", arXiv:2203.14432.

[3] Yash J. Patel, Sofiene Jerbi, Thomas Bäck, and Vedran Dunjko, "Reinforcement Learning Assisted Recursive QAOA", arXiv:2207.06294.

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