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We present an implementation of the improved staggered quark action lattice QCD computation designed for execution on a GPU cluster. The parallelization strategy is based on dividing the space-time lattice along the time dimension and distributing the sub-lattices among the GPU cluster nodes. We provide a mixed-precision floating-point GPU implementation of the multi-mass conjugate gradient solver. Our single GPU implementation of the conjugate gradient solver achieves a 9x performance improvement over the highly optimized code executed on a state-of-the-art eight-core CPU node. The overall application executes almost six times faster on a GPU-enabled cluster vs. a conventional multi-core cluster. The developed code is currently used for running production QCD calculations with electromagnetic corrections.