Beyond Stoquasticity: Structural Steering and Interference in Quantum Optimization

Abstract

We present a theoretical analysis of the DIC-DAC-DOA algorithm, a non-stoquastic quantum algorithm for solving the Maximum Independent Set (MIS) problem. The algorithm runs in polynomial time and achieves exponential speedup over both transverse-field quantum annealing (TFQA) and classical algorithms on a structured family of NP-hard MIS instances, under assumptions supported by analytical and numerical evidence. The core of this speedup lies in the ability of the evolving ground state to develop both positive and negative amplitudes, enabled by the non-stoquastic XX-driver. This sign structure permits quantum interference that produces negative amplitudes in the computational basis, allowing efficient evolution paths beyond the reach of stoquastic algorithms, whose ground states remain strictly non-negative. In our analysis, the efficiency of the algorithm is measured by the presence or absence of an anti-crossing, rather than by spectral gap estimation as in traditional approaches. The key idea is to infer it from the crossing behavior of bare energy levels of relevant subsystems associated with the degenerate local minima (LM) and the global minimum (GM). The cliques of the critical LM, responsible for the anti-crossing in TFQA, can be efficiently identified to form the XX-driver graph. The resulting speedup can be attributed to two mechanisms: in the first stage, energy-guided localization within the same-sign block steers the ground state smoothly into the GM-supporting region, while in the second stage, the opposite-sign blocks are invoked and sign-generating quantum interference drives the evolution along an opposite-sign path. Finally, we derive scalable reduced models that provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.