Finding high-quality parameters is a central obstacle to using the quantum approximate optimization algorithm (QAOA). Previous work partially addresses this issue for QAOA on unweighted MaxCut problems by leveraging similarities in the objective landscape among different problem instances. However, we show that the more general weighted MaxCut problem has significantly modified objective landscapes, with a proliferation of poor local optima. Our main contribution is a simple rescaling scheme that overcomes these deleterious effects of weights. We show that for a given QAOA depth, a single “typical” vector of QAOA parameters can be successfully transferred to weighted MaxCut instances. This transfer leads to a median decrease in the approximation ratio of only 2.0 percentage points relative to a considerably more expensive direct optimization on a dataset of 34,701 instances with up to 20 nodes and multiple weight distributions. This decrease can be reduced to 1.2 percentage points at the cost of only 10 additional QAOA circuit evaluations with parameters sampled from a pretrained metadistribution, or the transferred parameters can be used as a starting point for a single local optimization run to obtain approximation ratios equivalent to those achieved by exhaustive optimization in 96.35% of our cases.
Finding high-quality parameters is a central obstacle to using the quantum approximate optimization algorithm (QAOA). Previous work partially addresses this issue for QAOA on unweighted MaxCut problems by leveraging similarities in the objective landscape among different problem instances. However, we show that the more general weighted MaxCut problem has significantly modified objective landscapes, with a proliferation of poor local optima. Our main contribution is a simple rescaling scheme that overcomes these deleterious effects of weights. We show that for a given QAOA depth, a single “typical” vector of QAOA parameters can be successfully transferred to weighted MaxCut instances. This transfer leads to a median decrease in the approximation ratio of only 2.0 percentage points relative to a considerably more expensive direct optimization on a dataset of 34,701 instances with up to 20 nodes and multiple weight distributions. This decrease can be reduced to 1.2 percentage points at the cost of only 10 additional QAOA circuit evaluations with parameters sampled from a pretrained metadistribution, or the transferred parameters can be used as a starting point for a single local optimization run to obtain approximation ratios equivalent to those achieved by exhaustive optimization in 96.35% of our cases.
Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth p=1 applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for p≥1 we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for p=1 the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only 1.1 percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.4x on average.
Quantum approximate optimization algorithm (QAOA) have promising applications in combinatorial optimization problems (COPs). We investigated the MaxCut problem in three different families of graphs using QAOA ansats with parameter transfer initialization followed by targeted single layer optimization. For 3 regular (3R), Erdos Renyi (ER), and Barabasi Albert (BA) graphs, the parameter transfer approach achieved mean approximation ratios of 0.9443 for targeted-single layer optimization as compared to 0.9551 of full optimization. It represents 98.88 percent optimal performance, with 8.06 times computational speedup in unweighted graphs. But, in weighted graph families, optimal performance is relatively low (less than 90 percent) for higher nodes graph, suggesting parameter transfer followed by targeted-single-layer optimization is not ideal for weighted graph families, however, we find that for some weighted families (weighted 3-regular) this approach works perfectly. In 8.92 percent test cases, targeted single layer optimization outperformed the full optimization, indicating that complex parameter landscape can trap full optimization in sub-optimal local minima. To mitigate this inconsistency, ridge (L2) regularization is used to smoothen the solution landscape, which helps the optimizer to find better optimum parameters during full optimization and reduces these inconsistent test cases from 8.92 percent to 3.81 percent. This work demonstrates that efficient parameter initialization and targeted-single-layer optimization can improve the efficiency of QAOA with minimal performance degradation.
The quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. A near-optimal solution to the combinatorial optimization problem is achieved by preparing a quantum state through the optimization of quantum circuit parameters. Optimal QAOA parameter concentration effects for special MaxCut problem instances have been observed, but a rigorous study of the subject is still lacking. In this work we show clustering of optimal QAOA parameters around specific values; consequently, successful transferability of parameters between different QAOA instances can be explained and predicted based on local properties of the graphs, including the type of subgraphs (lightcones) from which graphs are composed as well as the overall degree of nodes in the graph (parity). We apply this approach to several instances of random graphs with a varying number of nodes as well as parity and show that one can use optimal donor graph QAOA parameters as near-optimal parameters for larger acceptor graphs with comparable approximation ratios. This work presents a pathway to identifying classes of combinatorial optimization instances for which variational quantum algorithms such as QAOA can be substantially accelerated.
Abstract The quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. Optimal QAOA parameter concentration effects for special MaxCut problem instances have been observed, but a rigorous study of the subject is still lacking. Due to clustering of optimal QAOA parameters for MaxCut, successful parameter transferability between different MaxCut instances can be explained and predicted based on local properties of the graphs, including the type of subgraphs (lightcones) from which graphs are composed as well as the overall degree of nodes in the graph (parity). In this work, we apply five different graph embedding techniques to determine good donor candidates for parameter transferability, including parameter transferability between different classes of MaxCut instances. Using this technique, we effectively reduce the number of iterations required for parameter optimization, obtaining an approximate solution to the target problem with an order of magnitude speedup. This procedure also effectively removes the problem of encountering barren plateaus during the variational optimization of parameters. Additionally, our findings demonstrate that the transferred parameters maintain effectiveness when subjected to noise, supporting their use in real-world quantum applications. This work presents a framework for identifying classes of combinatorial optimization instances for which optimal donor candidates can be predicted such that QAOA can be substantially accelerated under both ideal and noisy conditions.
The Quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. In a typical QAOA setup, a set of quantum circuit parameters is optimized to prepare a quantum state used to find the optimal solution of a combinatorial optimization problem. Several empirical observations about optimal parameter concentration effects for special QAOA MaxCut problem instances have been made in recent literature, however, a rigorous study of the subject is still lacking. We show that convergence of the optimal QAOA parameters around specific values and, consequently, successful transferability of parameters between different QAOA instances can be explained and predicted based on the local properties of the graphs, specifically the types of subgraphs (lightcones) from which the graphs are composed. We apply this approach to random regular and general random graphs. For example, we demonstrate how optimized parameters calculated for a 6-node random graph can be successfully used without modification as nearly optimal parameters for a 64-node random graph, with less than 1% reduction in approximation ratio as a result. This work presents a pathway to identifying classes of combinatorial optimization instances for which such variational quantum algorithms as QAOA can be substantially accelerated.
The quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. A near-optimal solution to the combinatorial optimization problem is achieved by preparing a quantum state through the optimization of quantum circuit parameters. Optimal QAOA parameter concentration effects for special MaxCut problem instances have been observed, but a rigorous study of the subject is still lacking. In this work we show clustering of optimal QAOA parameters around specific values; consequently, successful transferability of parameters between different QAOA instances can be explained and predicted based on local properties of the graphs, including the type of subgraphs (lightcones) from which graphs are composed as well as the overall degree of nodes in the graph (parity). We apply this approach to several instances of random graphs with a varying number of nodes as well as parity and show that one can use optimal donor graph QAOA parameters as near-optimal parameters for larger acceptor graphs with comparable approximation ratios. This work presents a pathway to identifying classes of combinatorial optimization instances for which variational quantum algorithms such as QAOA can be substantially accelerated.