We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques towards predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.
We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques towards predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.
A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement forp=1on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers.
We investigate the Maximum Cut (MaxCut) problem on different graph classes with the quantum approximate optimization algorithm (QAOA) using symmetries. In particular, heuristics on the relationship between graph symmetries and the approximation ratio achieved by a QAOA simulation are considered. To do so, we first solve the MaxCut problem on well-known graphs, then we consider simple and controllable perturbations of the graph and find again the approximate MaxCut with the QAOA. Through an analysis of the spectrum of the graphs and their perturbations, as well as a careful study of the associated automorphism groups, we aim to extract valuable insight into how symmetry impacts the performance of QAOA. These insights can then be leveraged to heuristically reduce the quantum circuit complexity, the number of training steps, or the number of parameters involved, thus enhancing the efficiency and effectiveness of QAOA-based solutions.
Constructing effective mixer Hamiltonians is essential for enhancing the performance of the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems. In this work, we develop a systematic methodology for designing QAOA mixers that align with the symmetries of the classical objective function, with the goal of achieving values (mean, median, and minimum over multiple runs) that are closer to the true optimum. Our main idea is to design QAOA operators that are explicitly adapted to the action of symmetry groups on the Hilbert space. We focus on subgroups of the symmetric group$ S_{d},where d = 2^\ell,toensurecompatibilitywithqudit−basedquantumarchitectures.Inparticular,weconstructQAOAmixersinvariantunderthefullsymmetricgroup S_{d}aswellasitscyclicsubgroup \mathbb {Z}_{d} \subset S_{d}.TheseconstructionsarenaturalinthattheyrespectthedecompositionoftheHilbertspaceintoisotypiccomponentsunderthesymmetrygroupaction.Notably,tothebestofourknowledge,theQAOAalgorithmbasedonthe \mathbb {Z}_{d}−invariantmixerprovidesthefirstexampleofaQAOAprotocolwhosedynamics(uptofinalmeasurement)areconfinedentirelywithinanontrivialirreduciblerepresentationofasymmetrygroupoftheobjectivefunction.Althoughourworkdoesnotinvestigatethebenefitsofexploitingsuchsubspacesascomputationalresources,wethinkthattheveryrealizationofavariationalalgorithmwhoseevolutionisrestrictedtoanontrivialsymmetry−adaptedsubspaceisoffundamentalconceptualinterest.Weprovideclosed−formexpressionsforthesemixers,togetherwithexplicitquantumcircuitimplementations.Toempiricallyevaluateourapproach,wecompareQAOAvariantsemployingthestandardmixerB = \sum X_{i}withthoseusingourproposedHamiltoniansH_{M}andH_\chi$on edge coloring and graph partitioning problems. Across multiple graph instances, our symmetry-adapted mixers consistently yield objective values closer to the optimum, demonstrating statistically significant improvements over classical baselines.
The Quantum Approximate Optimization Algorithm (QAOA) is one of the promising near-term algorithms designed to find approximate solutions for combinatorial optimization problems. The algorithm prepares a parametrized state that is aimed to maximize the expectation value of the objective function of the problem. The circuit for QAOA consists of p layers, and it depends on 2p parameters, the determination of which can be carried out using a variational quantum eigensolver (VQE) subroutine. This is a variational approach for finding optimal parameters, where the quantum state is prepared with a quantum computer, and a classical computer is used for calculating the corresponding objective function value and performing an outer loop optimization routine in order to find optimal parameters. Earlier work on the algorithm has shown that QAOA is closely related to the Quantum Adiabatic Algorithm, which results in noticeable patterns in the optimal parameters. To this purpose, two methods, INTERP and FOURIER, were proposed to exploit this structure by Zhou et al. (2018). In this thesis, I extend upon these results by examining the effectiveness of one of their methods, the INTERP method, on the Max-Cut problem from graph theory. The performance of QAOA using this method was studied on several classes of graphs and benchmarked against the currently best-known classical algorithm Goemans-Williamson, which achieves an approximation ratio of ρ ≈ 0.878 on arbitrary graphs. The classes of graphs examined include cyclic, weighted and unweighted 3-regular, and Erdős–Rényi graphs with edge probabilities of 0.5 and 0.75 and graph sizes ranging from 4 to 16. Moreover, it was investigated if the INTERP method offers an advantage compared to random initialization of parameters and whether or not the method is polynomial in p. It was found that different graph classes lead to similar but different parameter patterns. Furthermore, the performance of QAOA using the INTERP method was dependent on the graph class considered. For the small graphs investigated it was found that QAOA outperforms Goemans-Williamson on ER-0.50, ER-0.75 and weighted 3-regular graphs from p=7 and beyond with respect to how well the expectation of the objective function approximates the optimal value, while for unweighted 3-regular graphs p=6 suffices. However, the calculation of the parameters is costly and requires a number of function evaluations that is on the order of the number of possible bipartitions of small graph with sizes n ≤ 16. Moreover, the numerical results suggested that indeed the method is polynomial in p advocating that the algorithm might be a strong alternative to Goemans-Williamson for large graph sizes, once hardware allows it.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.
Introducing additional tunable parameters to quantum circuits is a powerful way of improving per-formance without increasing hardware requirements. A recently introduced multiangle extension of the quantum approximate optimization algorithm (ma-QAOA) signifi-cantly improves the solution quality compared with QAOA by allowing the parameters for each term in the Hamilto-nian to vary independently. Prior results suggest, however, considerable redundancy in parameters, the removal of which would reduce the cost of parameter optimization. In this work we show numerically the connection between the problem symmetries and the parameter redundancy by demonstrating that symmetries can be used to reduce the number of parameters used by ma-QAOA without decreasing the solution quality. We study Max-Cut on all 7,565 connected, non-isomorphic 8-node graphs with a nontrivial symmetry group and show numerically that in 67.4% of these graphs, symmetry can be used to reduce the number of parameters with no decrease in the objective, with the average ratio of parameters reduced by 28.1%. Moreover, we show that in 35.9% of the graphs this reduction can be achieved by simply using the largest symmetry. For the graphs where reducing the number of parameters leads to a decrease in the objective, the largest symmetry can be used to reduce the parameter count by 37.1% at the cost of only a 6.1% decrease in the objective. We demonstrate the central role of symmetries by showing that a random parameter reduction strategy leads to much worse performance.