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IEEE Transactions on Quantum Engineering

Equivariant Quantum Approximate Optimization Algorithm

Boris Tsvelikhovskiy, Ilya Safro, Yuri Alexeev

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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$, to ensure compatibility with qudit-based quantum architectures. In particular, we construct QAOA mixers invariant under the full symmetric group$ S_{d}$as well as its cyclic subgroup$ \mathbb {Z}_{d} \subset S_{d}$. These constructions are natural in that they respect the decomposition of the Hilbert space into isotypic components under the symmetry group action. Notably, to the best of our knowledge, the QAOA algorithm based on the$ \mathbb {Z}_{d}$-invariant mixer provides the first example of a QAOA protocol whose dynamics (up to final measurement) are confined entirely within a nontrivial irreducible representation of a symmetry group of the objective function. Although our work does not investigate the benefits of exploiting such subspaces as computational resources, we think that the very realization of a variational algorithm whose evolution is restricted to a nontrivial symmetry-adapted subspace is of fundamental conceptual interest. We provide closed-form expressions for these mixers, together with explicit quantum circuit implementations. To empirically evaluate our approach, we compare QAOA variants employing the standard mixer$B = \sum X_{i}$with those using our proposed Hamiltonians$H_{M}$and$H_\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.

DOI
10.1109/tqe.2026.3654930

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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 groupSd S_{d}Sd​, whered=2ℓ d = 2^\elld=2ℓ, to ensure compatibility with qudit-based quantum architectures. In particular, we construct QAOA mixers invariant under the full symmetric groupSd S_{d}Sd​as well as its cyclic subgroupZd⊂Sd \mathbb {Z}_{d} \subset S_{d}Zd​⊂Sd​. 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 theZd \mathbb {Z}_{d}Zd​-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 mixerB=∑XiB = \sum X_{i}B=∑Xi​with those using our proposed HamiltoniansHMH_{M}HM​andHχH_\chiHχ​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.
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.
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.
The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most of the literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints. We generalize the “XY”-mixer designed to preserve the subspace of “one-hot” states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of the number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for an “XY”-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.
The quantum approximate optimization algorithm (QAOA) is a near-term quantum algorithm aimed at solving combinatorial optimization problems. Since its introduction, various generalizations have emerged, spanning modifications to the initial state, phase unitaries, and mixer unitaries. In this work, we present an analytical study of broad families of QAOA variants. We begin by examining a family of QAOA with product mixers, which includes single-body mixers parametrized by multiple variational angles, and derive exact analytical expressions for the cost expectation on weighted problem graphs in the single-layer ansatz setting. We then analyze a family of QAOA that employs many-body Grover-type mixers, deriving analogous analytical expressions for weighted problem hypergraphs in the setting of arbitrarily many circuit ansatz layers. For both families, we allow individual phase angles for each node and edge (hyperedge) in the problem graph (hypergraph). Our results reveal that, in contrast to product mixers, the Grover mixer is sensitive to contributions from cycles of all lengths in the problem graph, exhibiting a form of non-locality. Our study advances the understanding of QAOA's behavior in general scenarios, providing a foundation for further theoretical exploration.
The performance of the Quantum Approximate Optimization Algorithm (QAOA) is closely tied to the structure of the dynamical Lie algebra (DLA) generated by its Hamiltonians, which determines both its expressivity and trainability. In this work, we show that classical symmetries can be systematically exploited as a design principle for QAOA. Focusing on the MaxCut problem with global bit-flip symmetry, we analyze reduced QAOA instances obtained by fixing a single variable and study how this choice affects the associated DLAs. We show that the structure of the DLAs can change dramatically depending on which variable is held fixed. In particular, we construct explicit examples where the dimension collapses from exponential to quadratic, uncovering phenomena that do not appear in the original formulation. Numerical experiments on asymmetric graphs indicate that such reductions often produce DLAs of much smaller dimension, suggesting improved trainability. We also prove that any graph can be embedded into a slightly larger one (requiring only quadratic overhead) such that the standard reduced DLA coincides with the free reduced DLA, in most cases implying exponential dimension and irreducibility on the Hilbert space for reduced QAOA instances. These results establish symmetry-aware reduction as a principled tool for designing expressive and potentially trainable QAOA circuits.
Abstract The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.