# Duality in Quantum Quenches and Classical Approximation Algorithms: Pretty Good or Very Bad

Matthew B. Hastings

Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA
Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA

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### Abstract

We consider classical and quantum algorithms which have a duality property: roughly, either the algorithm provides some nontrivial improvement over random or there exist many solutions which are significantly worse than random. This enables one to give guarantees that the algorithm will find such a nontrivial improvement: if few solutions exist which are much worse than random, then a nontrivial improvement is guaranteed. The quantum algorithm is based on a sudden $quench$ of a Hamiltonian; while the algorithm is general, we analyze it in the specific context of MAX-$K$-LIN$2$, for both even and odd $K$. The classical algorithm is a dequantization of this algorithm", obtaining the same guarantee (indeed, some results which are only conjectured in the quantum case can be proven here); however, the quantum point of view helps in analyzing the performance of the classical algorithm and might in some cases perform better.

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### Cited by

[1] M. B. Hastings, "Classical and Quantum Bounded Depth Approximation Algorithms", arXiv:1905.07047.

[2] Adam Callison, Nicholas Chancellor, Florian Mintert, and Viv Kendon, "Finding spin-glass ground states using quantum walks", arXiv:1903.05003.

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