Title: Quantum Simulation of Many Body Systems: Time Evolution and Response Functions

Abstract:
Simulating many body systems/models has been a great tool for us to understand the nature of materials. It comes with three parts: state initialization, time evolution and data analysis. Since the Hilbert space dimension depends on the system size exponentially, conventional computation has been limited to brilliant approximations or small systems. Quantum computers however, in theory, do not have these limitations, and have potential to enable us to simulate relatively large many-body systems. In practice, current quantum computers are still very noisy, which requires us to compress/shorten our quantum circuits as much as possible. Our research is focused on generating efficient quantum algorithms for every aspect of quantum simulation, including time evolution and computation of response functions.

In the first part of this talk, I will discuss Lie algebraic methods to generate efficient quantum circuits for time evolution of certain models [1,2]. We have developed two methods that both use Lie algebraic methods based on the terms in a given Hamiltonian, to generate a fixed depth time evolution circuit. I will mainly focus on algebraic compression [1], which uses local operations among the quantum gates that form the Hamiltonian of the system to compress the time evolution circuit generated via Trotter-Suzuki formula into a fixed depth circuit. As I will show, this method can be applied to spin models such as 1-D transverse field XY and Ising models, and free fermion and controlled free fermion models on any lattice including 2 and 3-D.

The second part of this talk will be about our approach to generate efficient circuits for calculation of bosonic and fermionic response functions [3]. The information contained in response functions is vast. They show how a system would react to a small change caused by another system or environment, and most importantly, they are observable via experiments. Computing them connects theory to experiments, and allows us to understand the underlying physics. Inspired by this property of the response functions, our method is based on simulation of an  experiment in a quantum computer rather than the real world: time evolving a given state after a small perturbation. Then the response functions are calculated by the usage of linear response formalism. Advantages of this method over conventional methods are that it can be adjusted to be momentum and frequency selective, and it can calculate both bosonic and fermionic response functions without an ancilla qubit.

[1] E. Kökcü et al. Physical Review A 105.3 (2022): 032420.
[2] E. Kökcü et al. Physical Review Letters 129.7 (2022): 070501.
[3] E. Kökcü et al. arXiv:2302.10219 (2023).

Faculty Hosts: Eun-Ah Kim, Chao-Ming Jian, and Debanjan Chowdhury