Hamiltonian simulation meets holographic duality

Toby Cubitt
University College London

"Analogue" Hamiltonian simulation involves engineering a Hamiltonian of interest in the laboratory and studying its properties experimentally. Large-scale Hamiltonian simulation experiments have been carried out in optical lattices, ion traps and other systems for two decades. Despite this, the theoretical basis for Hamiltonian simulation is surprisingly sparse. Even a precise definition of what it means to simulate a Hamiltonian was lacking.

AdS/CFT duality postulates that quantum gravity in a d-dimensional anti-de-Sitter bulk space is equivalent to a strongly interacting field theory on its d-1 dimensional boundary. Recently, connections between AdS/CFT duality and quantum error-correcting codes have led (amongst other things) to tensor network toy models that capture important aspects of this holographic duality. However, these toy models struggle to encompass dualities between bulk and boundary energy scales and dynamics.

On the face of it, these two topics seem to have nothing whatsoever to do with one another.

In this talk, I will explain how we put analogue Hamiltonian simulation on a rigorous theoretical footing, by drawing on techniques from Hamiltonian complexity theory and Jordan algebras. I will show how this proved far more fruitful than a mere mathematical tidying-up exercise, leading to the discovery of universal quantum Hamiltonians [Science, 351:6 278, p.1180, 2016], [Proc. Natl. Acad. Sci. 115:38 p.9497, 2018]. And I will explain how this new Hamiltonian simulation formalism allowed us to extend the toy models of AdS/CFT to encompass energy scales, dynamics, and even (toy models of) black hole formation [J. High Energy Phys 2019:17; J. High Energy Phys 2022:52]. Finally, I will also discuss the significant limitations of tensor networks as toy models of AdS/CFT duality, and what these results suggest about possible ways forward.

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