Quantum algorithms for dynamics and dynamical observables

Alexander Kemper
North Carolina State University

Response functions are a fundamental aspect of physics; they represent the link between
experimental observations and the underlying quantum many-body state. In particular, dynamical response functions are part of the toolbox that physicists use to unravel the nature of correlated matter. In this talk, I will discuss some aspects of obtaining response functions on quantum computers.

First, I will introduce a new method for measuring response functions by
using a linear response framework and making the experiment an inextricable
part of the quantum simulation. This method can be frequency- and momentum-selective, avoids
limitations on operators that can be directly measured, and is ancilla-free. As prototypical
examples of response functions, we demonstrate that both bosonic and fermionic Green’s
functions can be obtained, and apply these ideas to the study of a charge-density-wave material. The linear response method provides a robust framework for using quantum
computers to study systems in physics and chemistry. It also provides new paradigms for
computing response functions on classical computers.

Second, 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.

Presentation (PDF File)

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