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arXiv:2008.02941 · PRX Quantum

Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz

Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Carrasquilla, Yong Baek Kim, Henry Yuen

Published Dec 8, 2020• 250 citations
arXivHTMLPDF

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 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."We also numerically observe that the optimization landscape of the HVA becomes almost trap free, i.e., there are no suboptimal minima, when the ansatz is overparametrized.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 the speed of convergence to a good solution.In contrast to 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 overparametrization 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 the 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.

DOI
10.1103/prxquantum.1.020319

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Abstract The variational quantum eigensolver is a promising tool for simulating ground states of quantum many-body systems on noisy quantum computers. Its effectiveness relies heavily on the ansatz, which must be both hardware-efficient for implementation on noisy hardware and problem-specific to avoid local minima and convergence problems. In this article, we explore entanglement-informed ansatz schemes that naturally emerge from specific models, aiming to balance accuracy with minimal use of two-qubit entangling gates. We investigate three models of quasi-1D Hamiltonians focusing on entanglement barriers and long-range interactions. We find that including the entanglement structure in the parameterized quantum circuit reduces the resources necessary to achieve a given accuracy. A better assessment is obtained by analyzing how the ansatz captures the entanglement spectrum. Our comprehensive analysis provides a new perspective on the design of ansätze based on the expected entanglement structure of the approximated state.
We propose an efficient circuit structure of variational quantum circuit Ans\"atze used for the variational quantum eigensolver (VQE) algorithm in calculating gapped topological phases on the currently feasible noisy intermediate-scale quantum computers. An efficient circuit Ansatz should include two layers: the initialization layer and the variational layer. In the initialization layer, a fixed-depth circuit state with a compatible entanglement structure to the target topological phase is constructed. The circuit state is further adjusted subsequently to capture the details of the local correlations, which is dictated with the Hamiltonian, in the parametrized variational layer. Based on this strategy, we design a circuit Ansatz to investigate the symmetry-protected topological Haldane phase in a nonexactly solvable alternating spin-12\frac{1}{2}21​ Heisenberg chain by VQE calculations. Main characterizations of the Haldane phase, including the long-ranged string order, the fourfold nearly degenerate ground states associated with four different localized edge-mode patterns for the system with open boundaries, and the twofold degeneracy of the entanglement spectrum, are all observed for the optimized shallow circuit state with only one-depth variational layer both in numerical simulations and on real quantum computers. We further demonstrate that the computational capacity (i.e., expressibility) of this quantum circuit Ansatz is determined not by the system size but only by the intrinsic correlation length of the system, thus implying that the scalable VQE calculation is possible.
The variational quantum eigensolver (VQE), which is a quantum–classical hybrid approach, has latent powers to leverage near-term quantum devices by effectively managing a limited number of qubits with finite coherent lifetimes. While it is generally argued that the quantum approximate optimization algorithm (QAOA), which is a special case of VQE with a variational Ansatz based on the adiabatic theorem, may enable practical applications of noisy quantum devices for classical combinatorial optimizations, the strategy to improve the performance of this algorithm by increasing the circuit depth conflicts with the limited coherence time of near-term quantum devices. Here, we introduce strategies involving the VQE to reduce the circuit resources required for solving combinatorial optimizations. Our concept of a parameterized quantum circuit allows the Ansatz preparation to be achieved by only single-qubit operation. We find that the variational Ansatz without controlled X-gates leads to quick convergence in a classical subroutine used to determine the variational parameters. In addition, the variational Ansatz with optimized parameters maintains performance over the problem sizes both on the numerical simulation and IBM 27-qubit processor “ibm_kawasaki.” Therefore, the variational Ansatz introduced in this study has several advantages considering the total calculation time and performance scaling over the problem sizes. We also show that the variational Ansatz consisting of a lower number of gate operations than that of QAOA can approximate the eigenstates of diagonal Hamiltonians with high accuracy. We illustrate our ideas with a maximum-cut problem and show that near-term quantum applications may be feasible using short-depth circuits.