RFOX (Rotated-Field Oscillatory eXchange) quantum algorithm: Towards Parameter-Free Quantum Optimizers

Abstract

We introduce RFOX (Rotated-Field Oscillatory eXchange), a parameter-free quantum algorithm for combinatorial optimization. RFOX combines an almost constant non-stoquastic $XX$ catalyst with a weak harmonic $ZX$ counter-diabatic term. Using the Floquet-Magnus expansion, we derive a closed-form effective Hamiltonian whose first-order term retains the full $XX$ driver, while the leading correction consists of a single qubit $Y$ field at high drive frequency. This structure ensures that the instantaneous spectral gap remains essentially flat, independent of both the interpolation parameter and the disorder strength, modulated only by a $δ$ parameter. This behavior stands in stark contrast to the unpredictable gap reductions, or even collapses, exhibited by the $X$ (stoquastic), $XX$, and $X+sXX$ (non-stoquastic) driver schedules. Extensive noiseless simulations on random-field Ising model (RFIM) instances with 7, 9, and 12 qubits, across three magnetic-field ranges, validate these spectral predictions: RFOX attains near-optimal, and in some cases exact, ground states using up to an order of magnitude fewer Trotter slices. Its performance advantage grows with increasing disorder, as conventional methods slow down near vanishing gaps, whereas RFOX maintains a constant runtime scaling of $T \propto Δ_{\min}^{-2}$. Hardware experiments on IBM Quantum processors (Eagle r3 and Heron r1, with 12, 15, and 20 physical qubits) reproduce similar performance rankings. These results suggest that fixed-gap, non-stoquastic drivers augmented with analytically derived counter-diabatic terms offer a promising, scalable, and tuning-free route toward quantum optimizers for combinatorial optimization problems.