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Wednesday, September 11th 2019

1:25 pm:

We will review the advances and challenges in the field of quantum combinatorial optimization

and closely related problem of low-energy eigenstates and coherent dynamics in transverse field

quantum spin glass models. We will discuss the role of collective spin tunneling that gives

rise to bands of delocalized non-ergodic quantum states providing the coherent pathway for the

population transfer (PT) algorithm: the quantum evolution under a constant transverse field that

starts at a low-energy spin configuration and ends up in a superposition of spin configurations

inside a narrow energy window. We study the transverse field induced quantum dynamics of the

following spin model: zero energy of all spin configurations except for a small fraction of spin

configuration that form a narrow band at large negative energy. We use the cavity method for

heavy-tailed random matrices to obtain the statistical properties of the low-energy eigenstates in

an explicit analytical form. In a broad interval of transverse fields, they are non-ergodic, albeit

extended giving rise to a qualitatively new type of quantum dynamics. For large transverse

fields the typical runtime of PT algorithm scales with n and as that of the Grover’s quantum

search, except for the small correction to the exponent . The model we consider is non-

integrable. As a result, our PT protocol does not require any fine-tuning of and may be

initialized in a computational basis state. We argue that our approach can be applied to study PT

protocol in other optimization problems with the potential quantum advantage over classical

algorithms.

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