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arXiv:2312.00077

Adiabatic-Passage-Based Parameter Setting for Quantum Approximate Optimization Algorithm

Mingyou Wu, Hanwu Chen

Published Nov 30, 2023• 0 citations
arXivHTMLPDF

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) exhibits significant potential for tackling combinatorial optimization problems. Despite its promise for near-term quantum devices, a major challenge in applying QAOA lies in the cost of circuit runs associated with parameter optimization. Existing methods for parameter setting generally incur at least a superlinear cost concerning the depth p of QAOA. In this study, we propose a novel adiabatic-passage-based parameter setting method that remarkably reduces the optimization cost, specifically when applied to the 3-SAT problem, to a sublinear level. Beginning with an analysis of the random model of the specific problem, this method applies a problem-dependent preprocessing on the problem Hamiltonian analytically, effectively segregating the magnitude of parameters from the scale of the problem. Consequently, a problem-independent initialization is achieved without incurring any optimization cost or pre-computation. Furthermore, the parameter space is adjusted based on the continuity of the optimal adiabatic passage, resulting in a reduction in the disparity of parameters between adjacent layers of QAOA. By leveraging this continuity, the cost to find quasi-optimal parameters is significantly reduced to a sublinear level.

DOI
10.48550/arxiv.2312.00077

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The Quantum Approximate Optimization Algorithm (QAOA) exhibits significant potential for tackling combinatorial optimization problems. Despite its promise for near-term quantum devices, a major challenge in applying QAOA lies in the cost of circuit runs associated with parameter optimization. Existing methods for parameter setting generally incur at least a superlinear cost concerning the depth p of QAOA. In this study, we propose a novel adiabatic-passage-based parameter setting method that remarkably reduces the optimization cost, specifically when applied to the 3-SAT problem, to a sublinear level. Beginning with an analysis of the random model of the specific problem, this method applies a problem-dependent preprocessing on the problem Hamiltonian analytically, effectively segregating the magnitude of parameters from the scale of the problem. Consequently, a problem-independent initialization is achieved without incurring any optimization cost or pre-computation. Furthermore, the parameter space is adjusted based on the continuity of the optimal adiabatic passage, resulting in a reduction in the disparity of parameters between adjacent layers of QAOA. By leveraging this continuity, the cost to find quasi-optimal parameters is significantly reduced to a sublinear level.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems.Despite its promise for near-term quantum applications, not much is currently understood about the QAOA's performance beyond its lowestdepth variant.An essential but missing ingredient for understanding and deploying the QAOA is a constructive approach to carry out the outer-loop classical optimization.We provide an in-depth study of the performance of the QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit nonadiabatic operations.Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasioptimal p-level QAOA parameters in O½polyðpÞ time, whereas the standard strategy of random initialization requires 2 OðpÞ optimization runs to achieve similar performance.We then benchmark the QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps.The comparison reveals that the QAOA can learn via optimization to utilize nonadiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps.Finally, we provide a realistic resource analysis on the experimental implementation of the QAOA.When quantum fluctuations in measurements are accounted for, we illustrate that optimization is important only for problem sizes beyond numerical simulations but accessible on near-term devices.We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for demonstrating quantum advantage on near-term devices, yet the physical origins of its efficacy remain poorly understood. In this work, we study QAOA for random kkk-SAT problems within a universal-mixer kkk-local search framework, establishing a formal correspondence between adiabatic state transfer and the QAOA ansatz. This correspondence yields a rigorous performance guarantee for random instances with clause density m=O(n1+ε)m=O(n^{1+ε})m=O(n1+ε) and circuit depth Θ(n2)Θ(n^2)Θ(n2). We further investigate the NISQ regime with shallow circuits of depth p=O(n)p=O(n)p=O(n). Surprisingly, the optimal parameters do not become stochastic under depth compression, but instead remain confined to a structured low-dimensional region, which we identify as a smooth adiabatic manifold. Numerical evidence indicates that this manifold persists across different circuit depths and arises from the variational suppression of adiabatic leakage. Based on this structure, we propose the smooth adiabatic-manifold parameterization (SAMP) strategy, transforming parameter optimization from an unstructured high-dimensional search into a guided refinement process. Numerical experiments on random 3-SAT instances show that SAMP achieves sublinear optimization scaling with circuit depth while providing robust zero-cost initialization for deep circuits.
The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading approach for combinatorial optimization on near-term quantum devices, yet its scalability is limited by the difficulty of optimizing 2p2p2p variational parameters for a large number ppp of layers. Recent empirical studies indicate that optimal QAOA angles exhibit concentration and transferability across problem sizes. Leveraging this observation, we propose a schedule-learning framework that transfers spectral-gap-informed adiabatic control strategies from small-scale instances to larger systems. Our method extracts the spectral gap profile of small problems and constructs a continuous schedule governed by ∂ts=κgq(s)\partial_t s = κg^q(s)∂t​s=κgq(s), where g(s)g(s)g(s) is the instantaneous gap and (κ,q)(κ, q)(κ,q) are global hyperparameters. Discretizing this schedule yields closed-form expressions for all QAOA angles, reducing the classical optimization task from 2p2p2p parameters to only 222, independent of circuit depth. This drastic parameter compression mitigates classical optimization overhead and reduces sensitivity to barren plateau phenomena. Numerical simulations on random QUBO and 3-regular MaxCut instances demonstrate that the learnt schedules transfer effectively to larger systems while achieving competitive approximation ratios. Our results suggest that gap-informed schedule transfers provide a scalable and parameter-efficient strategy for QAOA.
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Quantum Approximate Optimization Algorithm (QAOA) provides a way to solve combinatorial optimization problems using quantum computers. QAOA circuits consist of time evolution operators by the cost Hamiltonian and of state mixing operators, and embedded variational parameter for each operator is tuned so that the expectation value of the cost function is minimized. The optimization of the variational parameters is taken place on classical devices while the cost function is measured in the sense of quantum. To facilitate the classical optimization, there are several previous works on making decision strategies for optimal/initial parameters and on extracting similarities among instances. In our current work, we consider simplified QAOA parameters that take linear forms along with the depth in the circuit. Such a simplification, which would be suggested from an analogy to quantum annealing, leads to a drastic reduction of the parameter space from 2p to 4 dimensions with the any number of QAOA layers p. In addition, cost landscapes in the reduced parameter space have some stability on differing instances. This fact suggests that an optimal parameter set for a given instance can be transferred to other instances. In this paper we present some numerical results that are obtained for instances of the random Ising model and of the max-cut problem. The transferability of linearized parameters is demonstrated for randomly generated source and destination instances, and its dependence on features of the instances are investigated.