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 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.
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.
Abstract We develop a framework for analyzing layered quantum algorithms such as quantum alternating operator ansätze (QAOA). In the context of combinatorial optimization, our framework relates quantum cost gradient operators, derived from the cost and mixing Hamiltonians, to classical cost difference functions that reflect cost function neighborhood structure. By considering QAOA circuits from the Heisenberg picture, we derive exact general expressions for expectation values as series expansions in the algorithm parameters, cost gradient operators, and cost difference functions. This enables novel interpretability and insight into QAOA behavior in various parameter regimes. For single-level QAOA1 we show the leading-order changes in the output probabilities and cost expectation value explicitly in terms of classical cost differences, for arbitrary cost functions. This demonstrates that, for sufficiently small positive parameters, probability flows from lower to higher cost states on average. By selecting signs of the parameters, we can control the direction of flow. We use these results to derive a classical random algorithm emulating QAOA1 in the small-parameter regime, i.e. that produces bitstring samples with the same probabilities as QAOA1 up to small error. For deeper QAOA p circuits we apply our framework to derive analogous and additional results in several settings. In particular we show QAOA always beats random guessing. We describe how our framework incorporates cost Hamiltonian locality for specific problem classes, including causal cone approaches, and applies to QAOA performance analysis with arbitrary parameters. We illuminate our results with a number of examples including applications to QUBO problems, MaxCut, and variants of MaxSAT. We illustrate the generalization of our framework to QAOA circuits using mixing unitaries beyond the transverse-field mixer through two examples of constrained optimization problems, Max Independent Set and Graph Coloring. We conclude by outlining some of the further applications we envision for the framework.
Quantum Approximate Optimization Algorithm, further generalized as Quantum Alternating Operator Ansatz (QAOA), is a family of algorithms for combinatorial optimization problems. It is a leading candidate to run on emerging universal quantum computers to gain insight into quantum heuristics. In constrained optimization, penalties are often introduced so that the ground state of the cost Hamiltonian encodes the solution (a standard practice in quantum annealing). An alternative is to choose a mixing Hamiltonian such that the constraint corresponds to a constant of motion and the quantum evolution stays in the feasible subspace. Better performance of the algorithm is speculated due to a much smaller search space. We consider problems with a constant Hamming weight as the constraint. We also compare different methods of generating the generalized W-state, which serves as a natural initial state for the Hamming-weight constraint. Using graph-coloring as an example, we compare the performance of using XY model as a mixer that preserves the Hamming weight with the performance of adding a penalty term in the cost Hamiltonian.
The quantum approximate optimization algorithm (QAOA) is among the leading candidates for achieving quantum advantage on near-term processors. While typically implemented with a transverse-field mixer (XM-QAOA), the Grover-mixer variant (GM-QAOA) offers a compelling alternative due to its global search capabilities. This work investigates the application of GM-QAOA to higher-order unconstrained binary optimization (HUBO) problems, also known as polynomial unconstrained binary optimization (PUBO), which form a general class of combinatorial optimization problems involving multivariable interactions. We present a comprehensive numerical study demonstrating that GM-QAOA, unlike XM-QAOA, exhibits monotonic improvement in performance with circuit depth and achieves superior results for HUBO problems within a layerwise optimization framework. An important component of our approach is an analytical framework for modeling GM-QAOA dynamics, which enables a classical approximation of the optimal parameters and helps reduce the optimization overhead. Our resource-efficient parametrized version of GM-QAOA nearly matches the performance of the version optimized using the layerwise approach while being significantly less demanding, making it a highly effective approach for complex optimization tasks. These findings highlight the potential of GM-QAOA and provide a practical pathway for its implementation on current quantum hardware.
Abstract The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work (Lloyd, arXiv:1812.11075 ) argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbour ZZ terms, this set-up is quantum computationally universal and provides a universal gate set, i.e. all unitaries can be reached up to arbitrary precision. In the present paper, we complement this work by giving a complete proof and the precise conditions under which such a one-dimensional QAOA might produce a universal gate set. We further generalize this type of gate-set universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.
The quantum approximate optimization algorithm (QAOA) has emerged as a promising candidate for demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) devices. While various QAOA parameterization schemes exist, ranging from the original single-angle approach to the more expressive multi-angle quantum approximate optimization algorithm (MA-QAOA) and automorphic-angle quantum approximate optimization algorithm (AA-QAOA), each presents distinct trade-offs between expressiveness and classical optimization complexity. In this work, we introduce the k-interaction-angle quantum approximate optimization algorithm (kA-QAOA), a parameterization scheme that groups cost function terms by their k-body interaction order, providing a natural middle ground between parameter efficiency and solution quality. This approach is particularly well-suited for combinatorial optimization problems defined on hypergraphs, where multi-body interactions naturally arise in applications such as Boolean satisfiability and resource allocation with multi-party constraints. We benchmark kA-QAOA against standard single-angle quantum approximate optimization algorithm (SA-QAOA), MA-QAOA, and AA-QAOA on two problem classes: 3-uniform cyclic sign-alternating hypergraphs and random coefficient hypergraphs. Our results demonstrate that kA-QAOA achieves approximation ratios comparable to MA-QAOA while requiring significantly fewer function evaluations, thereby reducing quantum resource consumption.