In this talk, I will present an overview of how to obtain such quantities from a quantum computer. I will give a brief overview of some hardware platforms, outline why producing the desired dynamics can be difficult, and how Lie algebras arise naturally in this context.
Following that, I will highlight some of our recent work using Lie algebraic methods to simulate dynamics on quantum computers. Synthesizing the corresponding quantum circuit is typically done by breaking the operator into small circuit elements, named Trotter decomposition, which leads to circuits whose depth often scales unfavorably. We present two algorithms to help overcome these difficulties. First, it is possible to synthesize exact quantum circuit representations of the desired time evolution unitaries via Cartan decomposition.Second, when the circuit elements of the Trotter decomposition are limited to a subset of SU(4), or equivalently, when the Hamiltonian may be mapped onto free fermionic models, several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the circuit elements into a single block of quantum gates, resulting in a fixed depth time evolution for certain classes of Hamiltonians.

Speaker’s webpage: https://physics.sciences.ncsu.edu/people/akemper/