Applying Grover-mixer Quantum Alternating Ansatz Algorithm to Higher-order Quadratic Unconstrained Optimization Problems

TL;DR

The paper investigates applying GM-QAOA, with a Grover diffusion mixer, to higher-order HUBO/PUBO optimization problems and demonstrates its advantages over XM-QAOA, especially as problem locality increases. It develops a Gaussian-energy, disorder-averaged analytical framework to model layer-wise GM-QAOA dynamics and introduces a resource-efficient parameterization GM-QAOA(a) that significantly reduces quantum evaluation overhead while delivering near-fully optimized performance. Numerical comparisons across Max-Cut on random hypergraphs and SK spin glasses show GM-QAOA’s monotonic improvement with depth and robustness to higher-order interactions, with a crossover depth where GM-QAOA surpasses XM-QAOA that grows with problem size. The work also provides a method for classically pre-optimizing angles using EVT-based estimates of $E_{ m min}$, enabling practical implementation on near-term hardware and suggesting pathways for qudit-based realizations and fixed-point scaling strategies.

Abstract

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.

Figures (6)

• Figure 1: (a) General structure of the QAOA circuit. (b) Two implementations of the mixing operator considered in this work: the standard transverse-field mixer (leading to XM-QAOA) and the Grover-type mixer (leading to GM-QAOA). Standard notation is used for the Hadamard gate, rotations about the Bloch $x$ axis, the Pauli-$X$ (inversion) gate, and the multi-controlled phase gate.
• Figure 2: Performance comparison between GM-QAOA and XM-QAOA for the Max-Cut problem on random hypergraphs (left panels) and the (SK) spin glass model (right panels). The plots show the ground-state success probability $P(E_{\min})$ as a function of the circuit depth (layer number) for different system sizes $n \in \{6,10,14\}$ and interaction orders $D=2$ (top row) and $D=4$ (bottom row). Red stars indicate the minimal circuit depth (critical point) at which GM-QAOA first surpasses the corresponding XM-QAOA performance for each instance family. Each data point represents an average over 100 randomly generated problem instances.
• Figure 3: Changing the critical depth, defined as the minimum circuit depth at which GM-QAOA outperforms XM-QAOA, and the corresponding success probability as functions of the problem parameters. Error bars denote the standard deviation over 100 random problem instances.
• Figure 4: Comparison of analytically optimized parameters $\beta_k$ (left panel) and $\gamma_k$ (right panel) as functions of the layer index $k$ for GM-QAOA(a), contrasted with the constant-angle variant GM-QAOA(c) introduced in Ref. zhukov2025grover. The data are averaged over 100 random SK instances with HUBO order $D=2$ (corresponding to the Ising case) for each system size $n$.
• Figure 5: Behavior of the success probability $P(E_{\rm min})$ as a function of the number of layers for four methods: layer-wise optimized GM-QAOA, XM-QAOA, and two analytical approaches—one with parameter optimization [GM-QAOA(a)] and one without [GM-QAOA(c)] -- for various problem sizes $n$ and HUBO orders $D$.