Abstract
Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail. In this paper, we focus on a special family of quantum circuits called the Hamiltonian Variational Ansatz (HVA), which takes inspiration from the quantum approximation optimization algorithm and adiabatic quantum computation. Through the study of its entanglement spectrum and energy gradient statistics, we find that HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied "hardware-efficient ansatz." We also numerically observe that the optimization landscape of HVA becomes almost trap free when the ansatz is over-parameterized. We observe a size-dependent "computational phase transition" as the number of layers in the HVA circuit is increased where the optimization crosses over from a hard to an easy region in terms of the quality of the approximations and speed of convergence to a good solution. In contrast with the analogous transitions observed in the learning of random unitaries which occur at a number of layers that grows exponentially with the number of qubits, our Variational Quantum Eigensolver experiments suggest that the threshold to achieve the over-parameterization phenomenon scales at most polynomially in the number of qubits for the transverse field Ising and XXZ models. Lastly, as a demonstration of its entangling power and effectiveness, we show that HVA can find accurate approximations to the ground states of a modified Haldane-Shastry Hamiltonian on a ring, which has long-range interactions and has a power-law entanglement scaling.
Highlights
We focus on a special family of quantum circuits called the Hamiltonian variational ansatz (HVA), which its takes inspiration from the quantum approximate optimization algorithm and adiabatic quantum computation
Through the study of its entanglement spectrum and energy-gradient statistics, we find that the HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied “hardware-efficient ansatz.”
Through the study of two prototypical models in condensed-matter physics, namely the 1D transverse-field Ising model (TFIM) and the XXZ model, we find that entanglement entropy and entanglement spectrum can shed light on the initialization and optimization properties of the HVA in the context of the variational quantum eigensolvers (VQEs) algorithm
Summary
With the advent of noisy intermediate-scale quantum (NISQ) computers [1], near-term quantum algorithms, such as variational quantum eigensolvers (VQEs), may offer computational capabilities beyond those of the best. A natural first approach is the random-quantum-circuit ansatz [2,3,4], which is capable of expressing a wide variety of states This has been shown to be ineffective for gradient-based optimization strategies due to the barren-plateau phenomenon [5,6,7,8], which causes the optimization of randomly initialized circuits to get stuck on flat areas in the cost landscape, where gradients are exponentially small. For a 1D gapped local Hamiltonian, the entanglement entropy of the ground state obeys an area law, i.e., the entanglement entropy grows proportional to the boundary area of the system instead of the system size [12] This remarkable result allows us to combat the exponential scaling of the Hilbert space, since this area law provides evidence that the relevant physics of a system only takes place in a restricted part of the full state space. We include the computational details in Appendix A, some additional numerical results in Appendix B, and extra results on the dynamics of entanglement entropy in Appendix C
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