Hybrid Real-Imaginary Time Evolution for Low-Depth Hamiltonian Simulation in Quantum Optimization

Counterdiabatic (CD) driving is a powerful technique for accelerating adiabatic quantum computing. However, it becomes self-limiting in complex optimizations like the Sherrington-Kirkpatrick model: long evolution times $T$ needed to traverse crossings force the CD strength to scale as $1\mathrm{/}T$, causing it to vanish before convergence and wasting the quantum resources invested in its implementation. We break this trade-off with a Hybrid adaptive variational quantum dynamics simulation (HAVQDS). HAVQDS combines adaptive real-time evolution for circuit compression with imaginary-time steps that suppress excitations at no extra gate cost. For the SK model (6–14 qubits), HAVQDS achieves higher approximation ratios than adiabatic or CD approaches, while reducing CNOT counts by 1–2 orders of magnitude, enabling high-fidelity quantum optimization.