Quantum Approximate Optimization Algorithm with Fixed Number of Parameters

TL;DR

The paper addresses the growth of variational parameter spaces in QAOA by introducing Fixed-Parameter-Count QAOA (FPC-QAOA), which fixes the number of trainable parameters regardless of system size or circuit depth. It decouples schedule optimization from circuit digitization, using three monotone schedule functions parameterized by a fixed count and reconstructed via cubic Hermite interpolation, enabling arbitrarily deep digitized evolutions under $NISQ$ constraints. Across random MaxCut benchmarks, connectivity topologies, and the Tail-Assignment Problem, FPC-QAOA achieves comparable or superior solution quality with significantly reduced classical optimization effort, a trend that persists on IBM Kingston hardware. This work demonstrates a practical, scalable approach to variational quantum optimization on near-term devices, with potential for broader applicability and robustness to hardware noise.

Abstract

We introduce a novel quantum optimization paradigm: the Fixed-Parameter-Count Quantum Approximate Optimization Algorithm (FPC-QAOA). It is a scalable variational framework that maintains a constant number of trainable parameters regardless of the number of qubits, Hamiltonian complexity, or circuit depth. By separating schedule function optimization from circuit digitization, FPC-QAOA enables accurate schedule approximations with minimal parameters while supporting arbitrarily deep digitized adiabatic evolutions, constrained only by NISQ hardware capabilities. This separation allows depth to scale without expanding the classical search space, mitigating overparameterization and optimization challenges typical of deep QAOA circuits, such as barren plateaus-like behaviors. We benchmark FPC-QAOA on random MaxCut instances and the Tail Assignment Problem, achieving performance comparable to or better than standard QAOA with nearly constant classical effort and significantly fewer quantum circuit evaluations. Experiments on the IBM Kingston superconducting processor with up to 50 qubits confirm robustness and hardware efficiency under realistic noise. These results position FPC-QAOA as a practical and scalable paradigm for variational quantum optimization on near-term quantum devices.

Figures (9)

• Figure 1: Conceptual overview of the Fixed-Parameter-Count Quantum Approximate Optimization Algorithm (FPC-QAOA). Three trainable, monotone schedule functions $F_1$, $F_2$, and $F_3$ are reconstructed via cubic Hermite interpolation from a fixed set of internal control points. At each Trotter step $j$, the schedules are sampled at time $\tau_j$ to generate angle parameters associated with the initial Hamiltonian $H_i$ ($R_x$ rotations), the problem Hamiltonian $H_p$ ($R_z$ and $R_{zz}$ evolutions), and the auxiliary Hamiltonian $H_{\mathrm{aux}}$ (local $R_z$ rotations). The total number of trainable parameters is independent of the number of Trotter steps $N$, allowing increasingly accurate digitalization without increasing the dimensionality of the classical optimization problem.
• Figure 2: (a) Enhancement ratio $\eta$ over 100 random MaxCut instances (up to $n=20$ nodes/qubits) for FPC-QAOA with three trainable parameters (one per schedule). Medians exceed one, indicating consistent improvement over standard QAOA. (b) Average number of classical optimization iterations versus Trotter depth. While QAOA requires an increasing number of iterations as depth grows, FPC-QAOA remains nearly constant, consistent with a fixed-size parameterization across depths.
• Figure 3: Enhancement ratio $\eta$ versus system size ($n=10$--$20$ nodes) at fixed Trotter depth ($N=3$). Colors indicate the total number of trainable parameters used by FPC-QAOA (3, 6, 9) and the corresponding QAOA baseline (6 parameters at three layers). FPC-QAOA maintains $\eta\gtrsim 1$ across sizes, with gains that tend to increase as the problem size grows.
• Figure 4: Graph topologies considered in the connectivity study: (a) cyclic ($C_n$) topology with periodic boundary conditions, (b) star ($S_n$) topology, and (c) wheel ($W_n$) topology combining cycle and star.
• Figure 5: Enhancement ratio $\eta$ for $C_n$ (top), $S_n$ (middle), and $W_n$ (bottom). Each panel aggregates 100 random instances at $n\in\{10,15,20\}$ with local and pairwise weights sampled uniformly in $[-1,1]$. FPC-QAOA uses three trainable parameters, whereas the QAOA baseline uses $2N$ parameters at the same depth. Medians above one indicate consistent gains, with smaller---but still positive---improvements on star graphs.