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arXiv:2405.07211

Equivariant QAOA and the Duel of the Mixers

Boris Tsvelikhovskiy, Ilya Safro, Yuri Alexeev

Published May 12, 2024• 0 citations
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Abstract

Constructing an optimal mixer for Quantum Approximate Optimization Algorithm (QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving combinatorial optimization problems. We present a systematic methodology for constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the inherent symmetries of classical optimization problem objectives. The key to our approach is to identify an operator that commutes with the action of the group of symmetries on the QAOA underlying Hilbert space and meets the essential technical criteria for effective mixer Hamiltonian functionality. We offer a construction method specifically tailored to the symmetric group $S_d$, prevalent in a variety of combinatorial optimization problems. By rigorously validating the required properties, providing a concrete formula and corresponding quantum circuit for implementation, we establish the viability of the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical mixer $B$ commutes only with a subgroup of $S_d$ of significantly smaller order than the group itself, enhancing the efficiency of the proposed approach. To evaluate the effectiveness of our methodology, we compare two QAOA variants utilizing different mixer Hamiltonians: conventional $B=\sum X_i$ and the newly proposed $H_M$ in edge coloring and graph partitioning problems across various graphs. We observe statistically significant differences in mean values, with the new variant consistently demonstrating superior performance across multiple independent simulations. Additionally, we analyze the phenomenon of poor performance in alternative warm-start QAOA variants, providing a conceptual explanation supported by recent literature findings.

DOI
10.48550/arxiv.2405.07211

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Constructing an optimal mixer for Quantum Approximate Optimization Algorithm (QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving combinatorial optimization problems. We present a systematic methodology for constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the inherent symmetries of classical optimization problem objectives. The key to our approach is to identify an operator that commutes with the action of the group of symmetries on the QAOA underlying Hilbert space and meets the essential technical criteria for effective mixer Hamiltonian functionality. We offer a construction method specifically tailored to the symmetric group SdS_dSd​, prevalent in a variety of combinatorial optimization problems. By rigorously validating the required properties, providing a concrete formula and corresponding quantum circuit for implementation, we establish the viability of the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical mixer BBB commutes only with a subgroup of SdS_dSd​ of significantly smaller order than the group itself, enhancing the efficiency of the proposed approach. To evaluate the effectiveness of our methodology, we compare two QAOA variants utilizing different mixer Hamiltonians: conventional B=∑XiB=\sum X_iB=∑Xi​ and the newly proposed HMH_MHM​ in edge coloring and graph partitioning problems across various graphs. We observe statistically significant differences in mean values, with the new variant consistently demonstrating superior performance across multiple independent simulations. Additionally, we analyze the phenomenon of poor performance in alternative warm-start QAOA variants, providing a conceptual explanation supported by recent literature findings.
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, where,where d = 2^\ell,toensurecompatibilitywithqudit−basedquantumarchitectures.Inparticular,weconstructQAOAmixersinvariantunderthefullsymmetricgroup, to ensure compatibility with qudit-based quantum architectures. In particular, we construct QAOA mixers invariant under the full symmetric group,toensurecompatibilitywithqudit−basedquantumarchitectures.Inparticular,weconstructQAOAmixersinvariantunderthefullsymmetricgroup S_{d}aswellasitscyclicsubgroupas well as its cyclic subgroupaswellasitscyclicsubgroup \mathbb {Z}_{d} \subset S_{d}.TheseconstructionsarenaturalinthattheyrespectthedecompositionoftheHilbertspaceintoisotypiccomponentsunderthesymmetrygroupaction.Notably,tothebestofourknowledge,theQAOAalgorithmbasedonthe. These constructions are natural in that they respect the decomposition of the Hilbert space into isotypic components under the symmetry group action. Notably, to the best of our knowledge, the QAOA algorithm based on the.TheseconstructionsarenaturalinthattheyrespectthedecompositionoftheHilbertspaceintoisotypiccomponentsunderthesymmetrygroupaction.Notably,tothebestofourknowledge,theQAOAalgorithmbasedonthe \mathbb {Z}_{d}−invariantmixerprovidesthefirstexampleofaQAOAprotocolwhosedynamics(uptofinalmeasurement)areconfinedentirelywithinanontrivialirreduciblerepresentationofasymmetrygroupoftheobjectivefunction.Althoughourworkdoesnotinvestigatethebenefitsofexploitingsuchsubspacesascomputationalresources,wethinkthattheveryrealizationofavariationalalgorithmwhoseevolutionisrestrictedtoanontrivialsymmetry−adaptedsubspaceisoffundamentalconceptualinterest.Weprovideclosed−formexpressionsforthesemixers,togetherwithexplicitquantumcircuitimplementations.Toempiricallyevaluateourapproach,wecompareQAOAvariantsemployingthestandardmixer-invariant mixer provides the first example of a QAOA protocol whose dynamics (up to final measurement) are confined entirely within a nontrivial irreducible representation of a symmetry group of the objective function. Although our work does not investigate the benefits of exploiting such subspaces as computational resources, we think that the very realization of a variational algorithm whose evolution is restricted to a nontrivial symmetry-adapted subspace is of fundamental conceptual interest. We provide closed-form expressions for these mixers, together with explicit quantum circuit implementations. To empirically evaluate our approach, we compare QAOA variants employing the standard mixer−invariantmixerprovidesthefirstexampleofaQAOAprotocolwhosedynamics(uptofinalmeasurement)areconfinedentirelywithinanontrivialirreduciblerepresentationofasymmetrygroupoftheobjectivefunction.Althoughourworkdoesnotinvestigatethebenefitsofexploitingsuchsubspacesascomputationalresources,wethinkthattheveryrealizationofavariationalalgorithmwhoseevolutionisrestrictedtoanontrivialsymmetry−adaptedsubspaceisoffundamentalconceptualinterest.Weprovideclosed−formexpressionsforthesemixers,togetherwithexplicitquantumcircuitimplementations.Toempiricallyevaluateourapproach,wecompareQAOAvariantsemployingthestandardmixerB = \sum X_{i}withthoseusingourproposedHamiltonianswith those using our proposed HamiltonianswiththoseusingourproposedHamiltoniansH_{M}andandandH_\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.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by 5.30%5.30\%5.30% on ER random graphs and 5.41%5.41\%5.41% on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only 15.30%15.30\%15.30% and 25.18%25.18\%25.18% of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.
Quantum optimization algorithms, such as the Quantum Approximate Optimization Algorithm, are emerging as promising heuristics for solving complex combinatorial problems. To improve performance, several extensions to the standard QAOA framework have been proposed in recent years. Two notable directions include: warm-starting techniques, which incorporate classical approximate solutions to guide the quantum evolution, and custom mixer Hamiltonians, such as XY mixers, which constrain the search to feasible subspaces aligned with the structure of the problem. In this work, we propose an approach that integrates these two strategies: a warm-start initialization with an XY mixer ansatz, enabling constraint-preserving quantum evolution biased toward high-quality classical solutions. The method begins by reformulating the combinatorial problem as a MaxCut instance, solved approximately using the Goemans-Williamson algorithm. The resulting binary solution is relaxed and used to construct a biased superposition over valid one-hot quantum states, maintaining compatibility with the XY mixer's constraints. We evaluate the approach on 5-city instances of the Traveling Salesperson Problem, a canonical optimization problem frequently encountered as a subroutine in real-world Vehicle Routing Problems. Our method is benchmarked against both the standard XY-mixer QAOA and a warm-start-only variant based on MaxCut relaxation. Results show that the proposed combination consistently outperforms both baselines in terms of the percentage and rank of optimal solutions, demonstrating the effectiveness of combining structured initializations with constraint-aware quantum evolution for optimization problems.
This paper investigates quantum heuristics based on Mixer Hamiltonians, which allow the search to be restricted to a specific subspace and enable warm-start strategies for solving the Traveling Salesman Problem (TSP). Approaches involving Mixer Hamiltonians can be integrated into the Quantum Approximate Optimization Algorithm (QAOA), where the Mixer acts as a mapping function that transforms qubit strings into feasible solution sets. We first introduce a swap-based mixer tailored to the TSP, which ensures that only qubit strings representing valid TSP solutions are explored during the QAOA process. Second, we propose a warm-start technique that initializes QAOA with a solution generated by any classical heuristic, thereby promoting faster convergence. These two contributions are combined into a Warm-Start QAOA framework with a Swap-Based Mixer, leveraging both structural and initialization advantages. Experimental results on a custom TSP instance involving five customers demonstrate the effectiveness of this approach, providing, for the first time, a viable integration of warm-start and swap-based mixers for the TSP within a quantum optimization framework.
Abstract The Quantum Approximate Optimization Algorithm (QAOA) stands as a hybrid classical‐quantum algorithm utilized for addressing combinatorial optimization challenges. Central to its effectiveness is the initial mixer, which is responsible for instigating the optimization process by generating the starting state. However, conventional QAOA implementations often assign equal probabilities to all solutions at the outset, potentially resulting in suboptimal performance when tackling complex combinatorial optimization problems. In this study, a novel enhancement is proposed to the QAOA, leveraging the Difference of Convex Algorithm (DCA). This method aims to refine QAOA's performance by facilitating the discovery of optimal parameters through a continuous warm‐start approach, as originally introduced by Egger et al. Through experimentation utilizing datasets from prior studies focusing on the weighted maximum cut problem, the efficacy of our proposed method is evaluated. Comparative analysis against existing methodologies reveals a significant improvement in the approximate ratio achieved by our approach.