The simulation of fluid dynamics is relevant in a vast field of applications, ranging from climate modelling over the medical context to design optimization of airplanes. The most accurate simulations can be achieved with direct numerical simulations (DNS), which resolve all relevant scales but often require the use of computationally prohibitively large meshes. Here, quantum computers are a promising candidate to encode and process largescale solutions more efficiently when encoding the classical data into the amplitudes of the quantum state. However, the quantum gates are linear and unitary operations, while fluid flows are governed by non-unitary and non-linear dynamics. We solve these challenges by combining a variational quantum algorithm (VQA) with a flexible operator encoding through tensor network operators. We will explain how “tensor-programmable”-VQAs allow to map all building blocks of computational fluid dynamics (CFD), i.e. discretized fluid fields, (potentially complex) operators and time stepping schemes into a hybrid quantum-classical routine. As the approach allows to correctly track the normalization of potentially unnormalized classical fields, it is able to describe the evolution of classical and quantum systems and has been applied to various partial differential equations, as the Euler and the Burgers’ equation. The combination of tensor-programmable quantum circuits and variational time-stepping routines offers several advantages: First, a broad range of non-unitary operators is directly accessible, making the incorporation of different boundary conditions, higher order differentiation schemes or non-cartesian meshes straightforward. Second, the tensor network representation of fields and operators provide us with defined upper bounds in the circuit depths. Third, in contrast to previous VQA implementations, measuring one expectation value provides a direct measure of the quality of solution without the need of a complete state readout and the comparison with classical data.