Shivan Mittal (UT-Austin)

"Scrambling from spectral gaps of local Hamiltonians"

Abstract: Random quantum circuits (RQCs) are the basic tools for quantum computational supremacy experiments (e.g. those performed by Google and USTC), decoupling theorems, information processing by black holes, growth of black hole volume, approach to equilibrium in condensed matter systems and more. One of the properties of these RQCs that facilitates the above phenomena is called unitary k-designs. Colloquially, the property characterizes how well random local interactions can create random global interactions. Specifically, if we have n qudits, then this property characterizes how well a RQC with few-qudit random unitaries at every step of the circuit could perform an n-qudit random unitary transformation. We make progress towards proving the unitary design property for any local RQC by studying the specific case of one-to-all interacting RQC (that is, at every step a fixed qudit interacts with one of the remaining qudits). Proving unitary k-design for an RQC boils down to lower bounding the spectral gaps of local Hamiltonians. We show 4 of our own results on lower bounds of spectral gaps of local Hamiltonians, along with numerical calculations of the same.