The paper introduces a deep learning-based calibration framework for the quadratic rough Heston model, which extends the rough Heston model to incorporate the Zumbach effect (feedback of past returns on future volatility). The model is made tractable via a multi-factor approximation that replaces the fractional kernel with a sum of exponentials, resulting in a high-dimensional Markovian system. This approximation allows efficient Monte Carlo simulation. To overcome the computational burden of calibration, the authors train deep neural networks using two strategies: a Parameters-to-Market (PtM) network that directly maps implied volatility surfaces to model parameters, and a Market-to-Parameters (MtP) network that approximates the pricing function and is then used within a traditional optimizer. Both methods are trained on a large synthetic dataset generated from the model. Numerical experiments show that the neural networks achieve pricing accuracy within Monte Carlo confidence intervals. When calibrated to market data, the model jointly fits SPX and VIX implied volatilities remarkably well, with VIX option prices falling inside bid-ask spreads and an excellent reproduction of the SPX at-the-money skew. Additionally, the trained networks enable instantaneous computation of hedging quantities (Greeks), facilitating practical hedging applications. The work demonstrates that combining rough volatility models with deep learning provides a powerful and efficient tool for derivatives pricing, calibration, and risk management.