Quantum Computational Fluid Dynamics

Project Information

QCFD

Grant agreement ID: 101080085

DOI

10.3030/101080085

EC signature date 19 August 2022

Start date 1 November 2022

End date 30 April 2027

Funded under

• Digital, Industry and Space

Total cost

€ 4 935 146,25

EU contribution

€ 4 935 143,25

Coordinated by

UNIVERSITY OF HAMBURG
Germany

Periodic Reporting for period 2 - QCFD (Quantum Computational Fluid Dynamics)

Reporting period: 2024-05-01 to 2025-10-31

Summary of the context and overall objectives of the project

CFD is a mature field of research that facilitates scientific findings in physical and life sciences including climate research and technological progress in the energy, chemical and transportation industries. For diagnostic purposes, i.e. for analyzing existing flow configurations, powerful numerical CFD tools are widely available. Approximate physical models and scale-reducing simplifications and constraints are extensively used for these computations. The approximations are not broadly applicable which severely limits the reliability of CFD methods for predictive and design purposes. Quantum CFD (QCFD) promises to fundamentally change this paradigm by stripping away the need for limits, simplification and specialization, through the direct resolution of all relevant physical scales and phenomena. It promises to do so for a small fraction of the computational cost required by classical methods. The impact of such technology on the field of CFD and product design, in general, would be nothing short of revolutionary. It would massively strengthen numerical simulations and data sciences as the third pillar of today’s scientific enterprise next to theory and physical experiments.

The overarching goal of the QCFD is to rise to this challenge by developing a versatile quantum algorithmic framework for efficiently solving a wide range of CFD problems without making the usual compromises on accuracy. We develop QCFD algorithms, test and benchmark them on classical emulators and on quantum hardware developed in European Quantum Technology Flagship Projects. Our approach demonstrates the feasibility and potential advantages of QCFD compared to standard CFD methods and identifies the requirements on the quantum hardware. Our project tackles increasingly complex flow examples and utilizes them to iteratively benchmark, verify and improve the employed quantum algorithms. The resulting software framework shall enable users to run QCFD algorithms on suitable quantum flagship hardware when this becomes available.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The second project phase has been dedicated to increasing the level of complexity of flows amenable to QCFD algorithms and comparing their scaling and resources requirements to standard CFD methods. We made significant progress in developing tensor network methods for QCFD, translating them to quantum algorithms, carrying out gate-level simulations of QCFD algorithms and performing extensive systematic runs on quantum hardware and develop verification and benchmarking methods.

We have been analyzing complex boundary conditions, curvilinear coordinates, compressible flows including shock waves, the evolution of high-dimensional probability distribution functions, shear reactive flows, the Magnus effect for a rotating cylinder, Rayleigh-Benard convection, and flow through an S-bend pipe. We have improved and generalized algorithms for efficiently translating tensor network simulations into quantum circuits to run on quantum hardware. We have derived a quantum Nyquist-Shannon theorem that explicitly shows that the number of qubits required for accurate amplitude encoding of flow fields scales logarithmically with the Reynolds number. We showed that the tripartite mutual information regularizers serve as a powerful tool for detecting and preventing information scrambling and the emergence of barren plateaus in the optimization landscape. For comparing resource requirements between conventional and QCFD approaches we have introduced figures of merit quantifying memory and compute time requirements.

These findings further motivate the development of platform-optimized QCFD algorithms for near-term flagship quantum computers. Quantum algorithms for basic QCFD problems like the 1D Schrödinger equation for superfluid flows and Burgers equation were implemented and extensively tested. We mapped out the cost of evolving the Burgers equation in time to obtain a visible shockwave, in terms of the number of measurement shots, SWAP gates required for the specific quantum topology, entangling gate fidelities, and resilience to noise. A variety of optimizer methods, gradient improvement, variance reduction, noise mitigation, and circuit simplification studies were undertaken. In collaboration with hardware platform developers leading sources of errors have been suppressed and the quality of phase readout has been improved. The simulation of the Burgers equation at finite time has been successfully concluded on the AQT ion-based device with high fidelity. Importantly we have further developed validation and verification techniques for quantum simulations to be able to assess the quality and accuracy of solutions obtained in QCFD simulations.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Fig. 1a) shows the bond dimensions required for representing boundaries as a matrix product operator and the number of variables required to parametrize flow fields. The bond dimensions of matrix product operators (MPOs) remain small and can be implemented efficiently. The number of required parameters is reduced compared to finite difference methods. Fig. 1b) shows time required to calculate a time step. For sufficiently large grids the tensor network method outperforms the finite difference method. Fig. 1c) shows results for the flow fields and a comparison between finite difference results and tensor network results. This demonstrates the high accuracy of tensor network results. Fig. 1d) shows oscillations of the lift force in time for a stationary and a rotating cylinder. The obtained accuracy is better than 1% in most cases.

Fig. 2 shows the general setup used to translate tensor networks into quantum algorithms (platform independent). Different parts of the cost function are translated to quantum circuits via tensor quantum programming (e.g. Ulin, Unon-lin). The overall cost function is composed out of these terms by coupling their actions to auxiliary qubits. Measuring a single ancilla qubit reads out the cost function with minimal overhead.

Fig. 3 shows simulations of the Burgers equation on quantum hardware, obtained on AQT's trapped-ion IBEX-Q1 processor. Panels (a, c) show the cost function behavior different times and (b, d) the fluid velocity profiles demonstrating high fidelities.

Fig 4 illustrates the newly developed framework for bounded-error quantum simulations. The process is divided into two steps. The first on Hamiltonian and Lindbladian Learning was already developed in period 1. As shown in Fig. 4a) and 4b) this information is now used for model predictions of measured observables. Fig. 4c) show how to compare them with experimentally obtained results.

Flow around a cylinder: tensor networks compared to finite difference methods

Burgers equation in 1D: results from the AQT IBEX-Q1 processor

Quantum tensor programming: schematic process for transpiling tensor networks to quantum circuits

Bounded error quantum simulation: error bounds