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Register a new userOn the natural gradient for variational quantum eigensolver
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Naoki Yamamoto
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The variational quantum eigensolver is a hybrid algorithm composed of quantum state driving and classical parameter optimization, for finding the ground state of a given Hamiltonian. The natural gradient method is an optimization method taking into account the geometric structure of the parameter space. Very recently, Stokes et al. developed the general method for employing the natural gradient for the variational quantum eigensolver. This paper gives some simple case-studies of this optimization method, to see in detail how the natural gradient optimizer makes use of the geometric property to change and improve the ordinary gradient method.
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The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $epsilon$, QPE requires $O(1)$ repetitions of circuits with depth $O(1/epsilon)$, whereas each expectation estimation subroutine within VQE requires $O(1/epsilon^{2})$ samples from circuits with depth $O(1)$. We propose a generalised VQE algorithm that interpolates between these two regimes via a free parameter $alphain[0,1]$ which can exploit quantum coherence over a circuit depth of $O(1/epsilon^{alpha})$ to reduce the number of samples to $O(1/epsilon^{2(1-alpha)})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.
Hybrid quantum-classical algorithms have been proposed as a potentially viable application of quantum computers. A particular example - the variational quantum eigensolver, or VQE - is designed to determine a global minimum in an energy landscape specified by a quantum Hamiltonian, which makes it appealing for the needs of quantum chemistry. Experimental realizations have been reported in recent years and theoretical estimates of its efficiency are a subject of intense effort. Here we consider the performance of the VQE technique for a Hubbard-like model describing a one-dimensional chain of fermions with competing nearest- and next-nearest-neighbor interactions. We find that recovering the VQE solution allows one to obtain the correlation function of the ground state consistent with the exact result. We also study the barren plateau phenomenon for the Hamiltonian in question and find that the severity of this effect depends on the encoding of fermions to qubits. Our results are consistent with the current knowledge about the barren plateaus in quantum optimization.
The variational quantum eigensolver (VQE) is a promising algorithm to compute eigenstates and eigenenergies of a given quantum system that can be performed on a near-term quantum computer. Obtaining eigenstates and eigenenergies in a specific symmetry sector of the system is often necessary for practical applications of the VQE in various fields ranging from high energy physics to quantum chemistry. It is common to add a penalty term in the cost function of the VQE to calculate such a symmetry-resolving energy spectrum, but systematic analysis on the effect of the penalty term has been lacking, and the use of the penalty term in the VQE has not been justified rigorously. In this work, we investigate two major types of penalty terms for the VQE that were proposed in the previous studies. We show a penalty term in one of the two types works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector. We further give a convenient formula to determine the magnitude of the penalty term, which may lead to the faster convergence of the VQE. Meanwhile, we prove that the other type of penalty terms does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases. We finally provide numerical simulations to validate our analysis. Our results apply to general quantum systems and lay the theoretical foundation for the use of the VQE with the penalty terms to obtain the symmetry-resolving energy spectrum of the system, which fuels the application of a near-term quantum computer.
Establishing the nature of the ground state of the Heisenberg antiferromagnet (HAFM) on the kagome lattice is well known to be a prohibitively difficult problem for classical computers. Here, we give a detailed proposal for a Variational Quantum Eigensolver (VQE) with the aim of solving this physical problem on a quantum computer. At the same time, this VQE constitutes an explicit proposal for showing a useful quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices because of its natural hardware compatibility. We classically emulate a noiseless quantum computer with the connectivity of a 2D square lattice and show how the ground state energy of a 20-site patch of the kagome HAFM, as found by the VQE, approaches the true ground state energy exponentially as a function of the circuit depth. Besides indicating the potential of quantum computers to solve for the ground state of the kagome HAFM, the classical emulation of the VQE serves as a benchmark for real quantum devices on the way towards a useful quantum advantage.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values which do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than plain VQE, for the same amount of effort.
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