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

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.

Layerwise learning for quantum neural networks

“…This reduces the overall number of trained parameters and restricts optimization to a specific manifold, at the cost of requiring a deeper circuit for convergence (Kiani et al 2020). Aside from the context of barren plateaus, Lyu et al (2020) investigate a layer-by-layer training scheme to speed up the learning process of a variational quantum eigensolver.…”

Robust resource-efficient quantum variational ansatz through evolutionary algorithm

“…In such algorithms, the complexity of the system is divided between a quantum simulator and a classical optimizer, allowing an imperfect shallow NISQ circuit to eventually achieve quantum advantage over classical computers. The quantum-classical variational algorithms have been found useful for several applications in various fields, including computational chemistry [9][10][11][12][13][14], simulating strongly correlated systems [15][16][17][18] and their phase detection [19], optimization [20][21][22][23][24], solving linear [25][26][27] and nonlinear [28] equations, classification problems [29,30], generative models [31][32][33] and quantum neural networks [34,35]. Among these algorithms, the Variational Quantum Eigensolver (VQE) [10,36,37], as a special type of VQAs, has been developed for efficiently generating the ground state of many-body systems on quantum simulators.…”

“…Among these algorithms, the Variational Quantum Eigensolver (VQE) [10,36,37], as a special type of VQAs, has been developed for efficiently generating the ground state of many-body systems on quantum simulators. The VQE has been widely used in quantum chemistry [9][10][11][12][13][14] and condensed matter physics [15][16][17][18]. It has also been generalized to simulate higher-energy eigenstates [38][39][40][41], time evolution [42,43], Gibbs thermal states [44,45] and non-equilibrium steady states [46] of many-body systems.…”

“…From a resource perspective, all quantum-classical variational algorithms, including the VQE, demand two types of resources: (i) a classical resource which is quantified through the number of parameters and the iterations required for minimizing the average energy; and (ii) a quantum resource which is quantified through the depth of the circuit. Several proposals have been dedicated to accelerate the classical optimization through stochastic gradient decent [50], proper initial guess for the circuit parameters [16] and sequential minimal optimization [51]. In parallel, symmetries have been exploited for simplifying the quantum circuits to achieve their task with shallower depths [52].…”

“…The essential idea there is to construct an optimal control that will deal with random errors which vary over a certain range [58][59][60][61][62][63][64][65]. For variational quantum algorithms including VQE, there are two types of variations that can be made on quantum circuit: one is the variation of parameters in each elementary gate [16,51,66,67], and the other is the variation of the circuit structure [68][69][70][71][72]. Conventional discussions of robustness have focused on the optimization of parameters for a fixed circuit ansatz.…”