The ingenious use of quantum effects has led to a significant number of quantum machine learning algorithms that offer computational speedups [1, 2]. While awaiting the demonstration of these quantum algorithms on full-fledge quantum computers equipped with quantum error correction, quantum computing has transitioned from theoretical ideas to the noisy intermediate-scale quantum (NISQ) technology era [3]. Hybrid quantum-classical algorithms using short-depth circuits are particularly suitable for implementation on NISQ devices. Many notable experimental demonstrations of NISQ devices employ hybrid algorithms for data classification [4] and quantum chemistry [5]. An ongoing quest is to find interesting applications on quantum computers with increasingly lower noise profile but not reaching a low enough threshold to enable continuous quantum error correction. Here we propose a hybrid quantum-classical algorithm that utilizes dissipative quantum dynamics for temporal information processing on gate-model NISQ quantum processors. Our approach exploits dissipative quantum systems as universal approximators for nonlinear maps with short-term or fading memory, important in a broad class of real-world problems including spoken digit recognition [6], neural modeling [7] and machine learning tasks (e.g., speech processing and natural language processing) [8, 9]. This is a quantum analogue of the universal function approximation property neural networks enjoy [10], but for nonlinear mappings from sequential input to sequential output data [11-13].