Discrete toy models
Discrete quantum systems, such as infinite spin chains or other many-body systems, have a storied history in condensed matter theory. Such systems serve as good testbeds for ideas in quantum information theory involving multiple parties. Those interested in holography sometimes use them as toy models to better understand gravitational phenomenon from the quantum, nongravitational side of the holographic correspondence. Additionally, certain classes of discrete quantum-error-correcting codes have even been studied as toy models of holographic duality itself.
My interest in discrete toy models is motivated by my interest in holography. Specifically, I use such systems to better understand ideas that are known to be of great importance in AdS/CFT. These ideas include complexity, which measures how "hard" it is to prepare a particular quantum state, and quantum chaos, i.e. the quantization of highly sensitive dynamical systems and their evolution in time. Such concepts are believed to encode aspects of black hole physics. While they are notoriously tricky to define and describe in continuum QFT, doing so in discrete quantum systems is more tractable.
I am also interested in using these types of systems to understand the holographic emergence of spacetime. This has ramifications for what holography "means" and is pivotal to understanding how to apply it beyond the realm of AdS gravity.
Chaos in quantum systems is poorly understood compared to its classical counterpart, and so one way to better understand quantum chaos might be to study it in a semiclassical regime. With this in mind, we propose "Clifford quantum cellular automata" (CQCA) as classically simulable, yet non-trivial toy models of the scrambling dynamics underlying quantum-chaotic systems. CQCA furnish a tunable parameter N controlling the Planck constant, and so they allow for a semiclassical limit corresponding to taking large N. We see how "space-time plots" of the out-of-time-ordered correlator, a simple probe of operator growth in quantum many-body systems, change as we increase N. We find that CQCA exhibit quantum "scarring" (i.e. weak ergodicity-breaking) at small N, but this scarring gets "smoothed out" with large N, reflecting a restoration of ergodicity in the semiclassical regime.
We study circuit complexity in functorial topological quantum field theory (TQFT). Functorial QFT is a formal, mathematically rigorous approach to defining QFT which uses category theory. In this language, a QFT is a mapping from a class of d-dimensional manifolds to Hilbert spaces and (d+1)-dimensional cobordant manifolds (called "cobordisms") to operators, with TQFT being the most basic example. In this language, circuit complexity for a particular cobordism is naturally defined by defining "building blocks" and counting the least number needed to construct a particular cobordism.