The Quantum Approximate Optimization Algorithm (QAOA) is among 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 constitute a generalized class of combinatorial optimization tasks characterized by intrinsically multi-variable interactions. We present a comprehensive numerical study demonstrating that GM-QAOA, unlike XM-QAOA, exhibits monotonic performance improvement with circuit depth and achieves superior results for HUBO problems. 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 parameterized GM-QAOA nearly matches the performance of the fully optimized algorithm while being far less demanding, establishing it as a highly effective approach for complex optimization tasks. These findings highlight GM-QAOA's potential and provide a practical pathway for its implementation on current quantum hardware.
The Quantum Approximate Optimization Algorithm (QAOA) is among 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 constitute a generalized class of combinatorial optimization tasks characterized by intrinsically multi-variable interactions. We present a comprehensive numerical study demonstrating that GM-QAOA, unlike XM-QAOA, exhibits monotonic performance improvement with circuit depth and achieves superior results for HUBO problems. 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 parameterized GM-QAOA nearly matches the performance of the fully optimized algorithm while being far less demanding, establishing it as a highly effective approach for complex optimization tasks. These findings highlight GM-QAOA's potential and provide a practical pathway for its implementation on current quantum hardware.
We comparatively study, through large-scale numerical simulation, the performance across a large set of Quantum Alternating Operator Ansatz (QAOA) implementations for finding approximate and optimum solutions to unconstrained combinatorial optimization problems. Our survey includes over 100 different mixing unitaries, and we combine each mixer with both the standard phase separator unitary representing the objective function and a thresholded version. Our numerical tests for randomly chosen instances of the unconstrained optimization problems Max 2-SAT and Max 3-SAT reveal that the traditional transverse-field mixer with the standard phase separator performs best for problem sizes of 8 through 14 variables, while the recently introduced Grover mixer with thresholding wins at problems of size 6. This result (i) corrects earlier work suggesting that the Grover mixer is a superior mixer based only on results from problems of size 6, thus illustrating the need to push numerical simulation to larger problem sizes to more accurately predict performance; and (ii) it suggests that more complicated mixers and phase separators may not improve QAOA performance.
The Grover mixer operator is a variational version of Grover's diffusion operator, introduced as a mixing operator for the quantum alternating operator ansatz (QAOA) and used in a variant known as Grover mixer QAOA (GM-QAOA). An important property of the QAOA with Grover mixer is that its expectation value is invariant over any permutation of states. As a consequence, the algorithm is independent of the structure of the problem. If, on the one hand, this characteristic raises serious doubts about the capacity of the algorithm to overcome the bound of the unstructured search problem, on the other hand, it can pave the way to its analytical study. In this sense, a prior work introduced a statistical approach to analyze GM-QAOA that results in an analytical expression for the expectation value depending on the probability distribution associated with the problem Hamiltonian spectrum. Although the method provides surprising simplifications in calculations, the expression depends exponentially on the number of layers, which makes direct analytical treatment unfeasible. In this work, we extend the analysis to the more simple context of Grover mixer threshold QAOA, a variant that replaces the phase separation operator of GM-QAOA to encode a threshold function. As a result, we obtain an expression for the expectation value independent of the number of layers and, with it, we provide bounds for different performance metrics. Furthermore, we extend the analysis to a more general context of QAOA with Grover mixer, which we called Grover-based QAOA. In that framework, which allows the phase separation operator to encode any compilation of the cost function, we generalize all the bounds by using an argument by contradiction with the optimality of Grover's algorithm on the unstructured search problem. As a result, we get the main contribution of this work, an asymptotic lower bound on the quantile achieved by the expectation value that formalizes the notion that the Grover mixer, at most, reflects a quadratic Grover-style speed-up over classical brute force. We apply that bound on the max-cut problem to the particular class of complete bipartite graphs and argue that the number of rounds required to achieve guarantees for any approximation ratio must grow exponentially with the number of vertices or edges, a severe limitation on the performance of the algorithm.
The quantum alternating operator ansatz (QAOA) represents a branch of quantum algorithms for solving combinatorial optimization problems. A specific variant, the Grover-mixer (GM) QAOA, ensures uniform amplitude across states that share equivalent objective values. This property makes the algorithm independent of the problem structure, focusing instead on the distribution of objective values within the problem. In this work we provide an alternative proof for the upper bound on the probability of measuring a computational basis state from a GM QAOA circuit with a given depth, which is a critical factor in QAOA cost. Using this, we derive the upper bounds for the probability of sampling an optimal solution and for the approximation ratio of maximum optimization problems, both dependent on the objective value distribution. Through numerical analysis, we link the distribution to the problem size and build the regression models that relate the problem size, QAOA depth, and performance upper bound. Our results suggest that the GM QAOA provides a quadratic enhancement in sampling probability and requires circuit depth that scales exponentially with problem size to maintain consistent performance.
The Quantum Alternating Operator Ansatz (QAOA) is a promising quantum heuristic to combinatorial optimization, designed for the Noisy Intermediate-Scale Quantum (NISQ) era. A key quantum operator of the algorithm is the mixing, which introduces quantum interference. This work investigates analytically the performance of QAOA with the Grover mixer, a mixing operator inspired by Grover’s algorithm. A critical feature of QAOA with Grover mixer is that it depends only on the probability distribution of the solutions, ignoring the structure of the problem. Due to this, it was conjectured that QAOA with this mixer is bounded to the quadratic speed-up of Grover’s algorithm over the classical brute force on the unstructured search problem. On the other hand, this feature allows the development of a statistical approach, which we explore in this work. For a variant of QAOA called Grover mixing QAOA (GM-QAOA), the analysis leads to an expression for the expectation value whose number of terms depends exponentially on the number of layers of the QAOA. In contrast, for Grover mixer Threshold QAOA (GM-Th-QAOA), a simpler variant that has an intimate relationship with Grover’s algorithm, we obtain a constant-time expression, which allows us to get bounds for different performance metrics, such as the statistical quantities of standard score and quantile. Subsequently, we generalized QAOA with the Grover mixer in a framework we called Grover-based QAOA. By using a contradiction argument with the optimality of Grover’s algorithm, we extend all the bounds of GM-Th-QAOA to the general context of Grover-based QAOA. As a result, we get the main contribution of this work: the proof of the aforementioned conjecture on the performance of QAOA with Grover mixer. We apply our bounds to the class of complete bipartite graphs on the Max-Cut problem, demonstrating that we cannot obtain a polynomial-time algorithm with Grover-based QAOA for these instances, showing that the performance limitation of the Grover mixer can be very severe. Our results suggest that achieving competitive performance with QAOA requires a mixing operator capable of leveraging the structure of combinatorial optimization problems.
The Quantum Alternating Operator Ansatz (QAOA) represents a branch of quantum algorithms for solving combinatorial optimization problems. A specific variant, the Grover-Mixer Quantum Alternating Operator Ansatz (GM-QAOA), ensures uniform amplitude across states that share equivalent objective values. This property makes the algorithm independent of the problem structure, focusing instead on the distribution of objective values within the problem. In this work, we prove the probability upper bound for measuring a computational basis state from a GM-QAOA circuit with a given depth, which is a critical factor in QAOA cost. Using this, we derive the upper bounds for the probability of sampling an optimal solution, and for the approximation ratio of maximum optimization problems, both dependent on the objective value distribution. Through numerical analysis, we link the distribution to the problem size and build the regression models that relate the problem size, QAOA depth, and performance upper bound. Our results suggest that the GM-QAOA provides a quadratic enhancement in sampling probability and requires circuit depth that scales exponentially with problem size to maintain consistent performance.
Grover-Mixer QAOA (GM-QAOA) is a promising quantum framework for constrained combinatorial optimization. However, its practical implementation requires the efficient preparation of a uniform superposition over feasible states. In this work, we propose a two-stage variational approach. First, a parameterized quantum circuit is trained to prepare this superposition, using an entropy-based cost function that favors a uniform distribution over feasible bitstrings. The trained circuit is then embedded in GM-QAOA, where it provides both the initial state and, together with its inverse, yields the Grover mixer. The resulting method is evaluated on two representative benchmark problems: the Maximum Weight Independent Set (MWIS) problem and the Multidimensional Knapsack Problem (MKP). While the results demonstrate consistent improvements over standard QAOA for MWIS, only marginal gains are observed for MKP, with the advantage diminishing as the problem size increases. The reduced performance on MKP reflects the growing difficulty of training the variational mixer, likely due to increasingly challenging optimization landscapes. Overall, these findings highlight both the potential of variational Grover mixers and the trade-off between their performance gains and the additional resources required for their training.