Qunasys Inc Team Models Nonlinear Fluid Dynamics for Quantum Simulation

Scientists at QunaSys Inc, in collaboration with Tokyo Gas Co, have achieved a significant milestone in quantum computational fluid dynamics (CFD) by simulating fluid dynamics incorporating nonlinearity using a quantum circuit for the first time. The simulation leverages a Carleman-linearized Boltzmann method coupled with a Taylor-expansion-based ordinary differential equation (ODE) solver, representing a crucial step towards more realistic and industrially relevant quantum simulations of fluid behaviour. This construction provides a key baseline for reducing computational costs and developing future simulations encompassing higher dimensions and complex geometries, potentially revolutionising fields such as aerospace engineering, weather forecasting, and materials science.

Logarithmic scaling unlocks efficient quantum simulation of nonlinear fluids

Gate and qubit complexities now scale logarithmically with grid size and the degree of nonlinearity, representing a marked improvement over prior linear simulations. These earlier simulations were fundamentally limited in their ability to model the complex interactions inherent in real-world fluids, often relying on simplifying assumptions that compromised accuracy. This advancement stems from the successful implementation of second-order Carleman linearization of the Boltzmann equation, a mathematical technique used to transform nonlinear partial differential equations into a form more amenable to numerical solution. This enables the explicit quantum-circuit simulation of a one-dimensional fluid exhibiting nonlinear behaviour, a feat previously unattainable. The Boltzmann equation itself describes the statistical behaviour of a fluid system, considering the distribution of particles and their collisions.

Previously, quantum simulations were confined to simplified, linear fluid models, limiting their applicability to only the most basic scenarios. The construction provides a key baseline for reducing computational costs and enables future expansion to higher dimensions and more complex geometries, bringing industrially relevant quantum computational fluid dynamics closer to realisation. Second-order Carleman linearization was achieved, incorporating leading nonlinear effects, and results at a third-order truncation were presented for comparison, allowing for an assessment of the approximation’s accuracy and potential for refinement. The choice of second-order linearization represents a balance between computational complexity and the fidelity of the simulation, capturing the most significant nonlinear contributions without incurring excessive overhead.

The simulation employed a Taylor-expansion-based ODE solver, approximating the propagator, the mathematical operator that evolves the system in time, using truncated Taylor series up to the third order. This approach allows for a relatively efficient representation of the time evolution of the fluid, albeit with a limited degree of accuracy. The simulation utilised a quantum singular value transformation (QSVT) algorithm, a powerful technique for implementing linear transformations on quantum computers. Analysis revealed that the number of qubits needed scales logarithmically with grid size, nonlinearity from the Carleman linearization, and the utility of higher-order Taylor expansions, indicating efficient resource use. This logarithmic scaling is crucial for tackling increasingly complex simulations, as the number of qubits required grows much more slowly than the size of the problem. Specifically, the logarithmic relationship suggests that doubling the grid resolution or the degree of nonlinearity only requires a modest increase in qubit resources, making large-scale simulations more feasible.

Nonlinear fluid dynamics achieved via quantum state preparation

QunaSys Inc. and Tokyo Gas Co. Ltd. scientists have successfully demonstrated a quantum simulation incorporating nonlinear fluid behaviour, marking an important step towards modelling real-world phenomena. Scaling these simulations beyond simplified models has been a significant hurdle in the path to industrially useful quantum computational fluid dynamics, due to the exponential growth of computational resources required by traditional methods. The lattice Boltzmann method, upon which this work builds, is a popular technique for simulating fluid flows by modelling the behaviour of microscopic particles. However, the current construction focuses solely on preparing the final state of the fluid, with the complete time-dependent evolution remaining unaddressed. This means the simulation calculates the state of the fluid at a specific point in time, but does not simulate how the fluid reaches that state.

This work establishes a foundation for modelling the complete evolution of fluid dynamics, building upon lattice Boltzmann methods, a technique for simulating fluids that offers advantages in handling complex geometries and boundary conditions. It also opens questions regarding the simulation of time-dependent fluid flows and extension to more complex, three-dimensional scenarios. The team successfully demonstrated the first quantum-circuit simulation of a fluid incorporating nonlinear dynamics, moving beyond previous linear approximations. The favourable scaling of gate and qubit requirements with both the size of the simulated area and the strength of the nonlinearity suggests potential for future development, potentially enabling the simulation of larger and more complex fluid systems with limited quantum resources. Further research will focus on addressing the limitations of the current state preparation approach and exploring methods for simulating the full time-dependent behaviour of fluids, potentially through the development of quantum algorithms for solving the time evolution operator directly. The ultimate goal is to create a fully functional quantum CFD solver capable of tackling real-world engineering problems with unprecedented accuracy and efficiency.

Researchers successfully demonstrated a quantum-circuit simulation of fluid dynamics, incorporating nonlinear effects for the first time. This achievement represents a step towards utilising quantum computers to model complex fluid flows, a task currently limited by the resources required for traditional computation. The simulation prepared the final state of a one-dimensional fluid, with gate and qubit requirements scaling favourably with grid size and nonlinearity. The authors intend to extend this work by simulating the complete time-dependent evolution of fluids and exploring more complex, three-dimensional scenarios.