Accelerated variational algorithms for digital quantum simulation of many-body ground states

Chufan Lyu, Victor Montenegro, and Abolfazl Bayat

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China

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Abstract

One of the key applications for the emerging quantum simulators is to emulate the ground state of many-body systems, as it is of great interest in various fields from condensed matter physics to material science. Traditionally, in an analog sense, adiabatic evolution has been proposed to slowly evolve a simple Hamiltonian, initialized in its ground state, to the Hamiltonian of interest such that the final state becomes the desired ground state. Recently, variational methods have also been proposed and realized in quantum simulators for emulating the ground state of many-body systems. Here, we first provide a quantitative comparison between the adiabatic and variational methods with respect to required quantum resources on digital quantum simulators, namely the depth of the circuit and the number of two-qubit quantum gates. Our results show that the variational methods are less demanding with respect to these resources. However, they need to be hybridized with a classical optimization which can converge slowly. Therefore, as the second result of the paper, we provide two different approaches for speeding the convergence of the classical optimizer by taking a good initial guess for the parameters of the variational circuit. We show that these approaches are applicable to a wide range of Hamiltonian and provide significant improvement in the optimization procedure.

Many features of complex systems can only be understood as collective behavior of several particles, the so called many-body systems. Low-energy, especially the ground state, properties are of high importance in physics as they are responsible for phenomena such as magnetism, molecular formation and the emergence of new phase of matter.
Simulating quantum behavior at the many-body level is beyond the capability of classical computer due to the requirement of exponentially large resources. The true solution is to exploit another quantum system with better controllability, namely quantum simulator, to emulate the behavior of the complex system of interest. One can exploit adiabatic approach for simulating the ground state of many-body systems on quantum simulators, which are now emerging in various physical setups. However, this demands high quality hardware which cannot be achieved in existing quantum simulators. Therefore, a hybrid variational method, called variational quantum eigensolver (VQE), has already been proposed to simplify the quantum hardware at the price of the addition of a classical optimizer.
In this paper, we first provide a quantitative comparison for the hardware needed in the adiabatic and the VQE approaches. Our results show that the VQE can indeed significantly simplify the quantum circuit. Then we focus on accelerating the convergence of the classical optimizer. We have developed two methods to improve the initial guess for the parameters of the optimizer. This tends to start the optimization procedure closer to the optimal case and thus significantly speedup the convergence. We have shown that our protocol is applicable over a wide range of physical models.

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