Abstract: This paper introduces the Quadratic Quantum Variational Monte Carlo (Q$^2$VMC) algorithm, an innovative algorithm in quantum chemistry that significantly enhances the efficiency and accuracy of solving the Schrödinger equation. Leveraging the foundational principles of Quantum Variational Monte Carlo (QVMC) and inspired by the discretized imaginary-time Schrödinger evolution, Q$^2$VMC employs a novel quadratic update mechanism that integrates seamlessly with neural network-based ansatzes. Our extensive experiments showcase Q$^2$VMC's superior performance, achieving faster convergence and higher accuracy in wavefunction optimization across various molecular systems, without additional computational cost. This study not only advances the field of computational quantum chemistry but also highlights the importance of discretized evolution in variational quantum algorithms, offering a scalable and robust framework for future quantum research.