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This paper investigates the application of Quantum Support Vector Machines (QSVMs) with an emphasis on the computational advancements enabled by NVIDIA's cuQuantum SDK, especially leveraging the cuTensorNet library. We present a simulation workflow that substantially diminishes computational overhead, as evidenced by our experiments, from exponential to quadratic cost. While state vector simulations become infeasible for qubit counts over 50, our evaluation demonstrates that cuTensorNet speeds up simulations to be completed within seconds on the NVIDIA A100 GPU, even for qubit counts approaching 784. By employing multi-GPU processing with Message Passing Interface (MPI), we document a marked decrease in computation times, effectively demonstrating the strong linear speedup of our approach for increasing data sizes. This enables QSVMs to operate efficiently on High-Performance Computing (HPC) systems, thereby opening a new window for researchers to explore complex quantum algorithms that have not yet been investigated. In accuracy assessments, our QSVM achieves up to 95\% on challenging classifications within the MNIST dataset for training sets larger than 100 instances, surpassing the capabilities of classical SVMs. These advancements position cuTensorNet within the cuQuantum SDK as a pivotal tool for scaling quantum machine learning simulations and potentially signpost the seamless integration of such computational strategies as pivotal within the Quantum-HPC ecosystem.