Efficient synthesis of contact-rich manipulation trajectories is essential for downstream robot learning. Gradient-based planning offers a promising approach, but jointly optimizing motion and contact decisions remains challenging. Meanwhile, many existing planners run on CPUs with limited throughput. We present CQDC-Warp, a GPU-accelerated differentiable quasi-dynamic simulator for batched model predictive control (MPC). We use it to study contact-guided trajectory optimization, in which intended robot--object contacts are specified before planning and encouraged through soft costs, while the simulated dynamics determine contact geometry and frictional response without a prescribed contact-mode schedule. Across $17{,}408$ trials on the $17$-object Allegro in-hand rotation benchmark, our framework achieves $79.8\%$ success, compared with $73.7\%$ for CFree and $65.2\%$ for IMPACT, and reduces mean trajectory-generation time by factors of $27.0$ and $50.2$, respectively. On a single NVIDIA RTX 4090 with $2{,}048$ parallel worlds, forward and backward throughput reach $2.06\times10^5$ and $2.25\times10^5$ world-steps per second, respectively. Further experiments cover manipulation with LEAP and Shadow hands, environmental contact exploitation, bimanual manipulation, whole-body control, and real-world trajectory replay on a Sharpa hand.

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