Complexity and entanglement in non-local computation and holography

Alex May

Stanford University

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Abstract

Does gravity constrain computation? We study this question using the AdS/CFT correspondence, where computation in the presence of gravity can be related to non-gravitational physics in the boundary theory. In AdS/CFT, computations which happen locally in the bulk are implemented in a particular non-local form in the boundary, which in general requires distributed entanglement. In more detail, we recall that for a large class of bulk subregions the area of a surface called the ridge is equal to the mutual information available in the boundary to perform the computation non-locally. We then argue the complexity of the local operation controls the amount of entanglement needed to implement it non-locally, and in particular complexity and entanglement cost are related by a polynomial. If this relationship holds, gravity constrains the complexity of operations within these regions to be polynomial in the area of the ridge.

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Cited by

[1] Alex May, Jonathan Sorce, and Beni Yoshida, "The connected wedge theorem and its consequences", Journal of High Energy Physics 2022 11, 153 (2022).

[2] Kfir Dolev and Sam Cree, "Holography as a resource for non-local quantum computation", arXiv:2210.13500, (2022).

[3] Felix Leditzky, "Optimality of the pretty good measurement for port-based teleportation", arXiv:2008.11194, (2020).

[4] Sam Cree and Alex May, "Code-routing: a new attack on position verification", arXiv:2202.07812, (2022).

[5] Kfir Dolev and Sam Cree, "Non-local computation of quantum circuits with small light cones", arXiv:2203.10106, (2022).

[6] Felix Leditzky, "Optimality of the pretty good measurement for port-based teleportation", Letters in Mathematical Physics 112 5, 98 (2022).

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