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Optimally mapping a parallel application to compute and communication resources is increasingly important as both system size and heterogeneity increase. A similar mapping problem exists in gate-based quantum computing where the objective is to map tasks to gates in a topology-aware fashion. This is an NP-complete graph isomorphism problem, and existing task assignment approaches are either heuristic or based on physical optimization algorithms, providing different speed and solution quality trade-offs. Ising machines such as quantum and digital annealers have recently become available and offer an alternative hardware solution to solve this type of optimization problems. In this paper, we propose an algorithm that allows solving the topology-aware assignment problem using Ising machines. We demonstrate the algorithm on two use cases, i.e. classical task scheduling and quantum circuit gate scheduling. TIGER---topology-aware task/gate assignment mapper tool---implements our proposed algorithms and automatically integrates them into the quantum software environment. To address the limitations of physical solver, we propose and implement a domain-specific partition strategy that allows solving larger-scale problems and a weight optimization algorithm that allows tuning Ising model parameters to achieve better restuls. We use D-Wave's quantum annealer to demonstrate our algorithm and evaluate the proposed tool flow in terms of performance, partition efficiency, and solution quality. Results show significant speed-up compared to classical solutions, better scalability, and higher solution quality when using TIGER together with the proposed partition method. It reduces the data movement cost by 68\% in average for quantum circuit assignment compared to the IBM QX optimizer.