Efficient Treatment of Non-Linearities for Quantum Computational Fluid Dynamics

To run a CFD algorithm on a quantum computer, its main components must be translated into a quantum-compatible form, namely the discretized fluid fields, the relevant operators, and the time-marching scheme. For this purpose, we use a hybrid quantum-classical approach. The fluid fields at a given time step are first discretized in space and then encoded into the amplitudes of a quantum state. This encoding is realized through a variationally parameterized ansatz circuit, with the circuit parameters represented by real-valued numbers that can be stored efficiently on a classical computer. If the solution is not normalized, an additional parameter is introduced to capture the norm. 

To advance the system by one time step, we variationally determine the classical parameters defining the next state by minimizing a problem-specific cost function. For this purpose, we embed the problem into a modified Hadamard test, which allows the evaluation of a single expectation value to be sufficient for the procedure. This adapted version of the Hadamard test also provides a convergence metric that quantifies the discrepancy between the numerically exact next time step and the variationally obtained one, without requiring a comparison to a classical reference solution [5]. 

To map the differential operators, and the time-dependent non-linearity to the quantum circuit, we employ a tensor network mapping into unitary gates [5,6].  

We lay special focus on the cost of the non-linear term and have adapted its implementation compared to previously presented approaches [5,8,9] to increase the success probabilities and reduce the measurement requirements.

We demonstrate the correct functioning and the beneficial scaling of the new method following two complementary strategies.

First, we have implemented the method with the alternative non-linear treatment and tested the expected scaling and working concept successfully at the example of the one-dimensional non-linear Burgers' equation using a quantum simulator. We have extended the approach to solve the equation in two and three dimensions, and are currently working on the simulations, which we plan to present at the conference.

Second, we use analytical and numerical arguments, to compute the scaling behavior and the required resources for solving real flow scenarios at the example of three-dimensional turbulent flow fields. We can show, that using the new implementation of the non-linearity, the success probability is significantly increased and does not show the same steep decay for increasing system size. 

Our methodology is an inherently classical-quantum hybrid approach, which exploits the strengths of classical and quantum tensor network routines to reduce the scaling compared to purely classical or purely quantum tensor network implementations. The close connection to classical tensor network methods does not only help to find efficient representations of differential and non-linear operators but does also provide upper bounds for circuits depths and operator applications for many CFD relevant problems [5,6,8,10]. 

Furthermore, we present an analytical and numerical study of the measurement and parameter requirements for the quantum approach at the example of different three-dimensional turbulent flow fields. This allows to predict cost and scaling for CFD-relevant use cases, beyond current hardware and simulator capabilities.