Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry

Abstract

The variational preparation of complex quantum states using the quantum approximate optimization algorithm (QAOA) is of fundamental interest, and becomes a promising application of quantum computers. Here, we systematically study the performance of QAOA for preparing ground states of target Hamiltonians near the critical points of their quantum phase transitions, and generating Greenberger-Horne-Zeilinger (GHZ) states. We reveal that the performance of QAOA is related to the translational invariance of the target Hamiltonian: Without the translational symmetry, for instance due to the open boundary condition (OBC) or randomness in the system, the QAOA becomes less efficient. We then propose a generalized QAOA assisted by the parameterized resource Hamiltonian (PRH-QAOA), to achieve a better performance. In addition, based on the PRH-QAOA, we design a low-depth quantum circuit beyond one-dimensional geometry, to generate GHZ states with perfect fidelity. The experimental realization of the proposed scheme for generating GHZ states on Rydberg-dressed atoms is discussed. Our work paves the way for performing QAOA on programmable quantum processors without translational symmetry, especially for recently developed two-dimensional quantum processors with OBC.

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

“…Conventional numerical methods for understanding these states are hindered by the exponential size of the Hilbert space, making it difficult to generate a theoretical understanding of experimental observations. QAOA or related generalizations [38,61,[67][68][69][70] offer a potential route to overcome conventional computing bottlenecks. Some successes along these lines have been observed in certain contexts, however, advances in methodology and quantum computing technologies are needed to extend these methods to complicated and larger-scale problems where quantum computational approaches may have a significant impact in understanding and developing materials for technological applications.…”

“…One approach to address these problems uses the quantum approximate optimization algorithm, which was originally designed to find approximate solutions to difficult combinatorial optimization problems [21] that are often expressed in terms of Ising Hamiltonians [56]. Empirical performance of QAOA has been characterized for a variety of combinatorial problems [36,[57][58][59][60] and this has also led to generalizations [61][62][63][64][65][66] that have been applied to preparing chemical ground states [67] as well as ground state preparation for onedimensional [38,68] and two-dimensional [69] quantum spin models in theory and experiment [70].…”

“…This is, importantly, a departure from the standard goal of QAOA in the context of approximate combinatorial optimization, where the goal is to find approximate solutions that are not necessarily the ground states. While QAOA is not expected to efficiently find exact ground states for generic NP-hard optimization problems, it may still prove useful for finding ground states of specific structured problems such as materials problems on a lattice similar to those we explore here [38,[68][69][70]]. We use numerical calculations to assess the theoretical performance of QAOA for ground-state preparation.…”

Red-QAOA: Efficient Variational Optimization through Circuit Reduction

“…Egger et al [58] introduced warm-starting quantum optimization using classical relaxations of optimization problems and showed effectiveness in portfolio optimization and MAXCUT problems. Other works [54,59,60] have also utilized domain-specific knowledge of QAOA to enhance execution fidelity, while some works [61,62] proposed efficient Hamiltonian transitions and expectation calculations.…”

Characterization of variational quantum algorithms using free fermions

“…Other factors not immediately linked to the expressibility of the circuit are known to affect the optimization, such as the boundary conditions used; these have been observed to greatly affect the success of optimization, and strategies have recently been pro-posed to address this issue [86]. Another set of such factors is related to the individual characteristics of states to be prepared, such as locality of correlations and entanglement [20,42,60,93,94,96].…”