Quantum Algorithm for Fluid Flow Simulations: Promise and Challenges

The Lattice Boltzmann Carleman quantum algorithm, developed by Claudio Sanavio and Sauro Succi from the Fondazione Istituto Italiano di Tecnologia Center for Life NanoNeuroscience, is a quantum computing algorithm for fluid flows. It uses the Carleman linearization of the Lattice Boltzmann method to simulate fluid flows at moderate Reynolds numbers. The algorithm’s quantum circuit has a fixed depth, regardless of the number of lattice sites, but its depth is currently beyond the capabilities of existing quantum computers. Despite this, the algorithm holds promise for the future of fluid flow simulations as quantum computing technology continues to advance.

What is the Lattice Boltzmann Carleman Quantum Algorithm?

The Lattice Boltzmann Carleman quantum algorithm is a quantum computing algorithm for fluid flows. It is based on the Carleman linearization of the Lattice Boltzmann (LB) method. The algorithm was developed by Claudio Sanavio and Sauro Succi from the Fondazione Istituto Italiano di Tecnologia Center for Life NanoNeuroscience at la Sapienza in Rome, Italy.

The algorithm was designed to address fluid flows at moderate Reynolds numbers, specifically for Kolmogorov-like flows. The Reynolds number is a measure of the nonlinearity of a system, defined as the ratio between the inertial and viscous forces. The algorithm shows that at least for moderate Reynolds numbers between 10 and 100, the Carleman LB procedure can be successfully truncated at second order.

The quantum circuit implementing the single timestep collision operator has a fixed depth regardless of the number of lattice sites. However, the depth is of the order of ten thousands quantum gates, meaning that quantum advantage over classical computing is not attainable today but could be achieved in the near or midterm future.

How Does Quantum Computing Impact Fluid Flow Simulations?

Quantum computing holds promise to provide dramatic speed up to the solution of a number of major scientific problems, including advanced industrial and societal applications. The so-called quantum advantage stems from the deepest and most counterintuitive features of quantum mechanics, in particular superposition and entanglement of quantum states.

This feature provides a natural way out to the infamous curse of dimensionality plaguing the simulation of most quantum many-body problems, both classical and quantum. Yet realizing such a mindboggling potential faces with a number of steep challenges, primarily fast decoherence and the quantum noise affecting the operation of real-life quantum computers.

There is a mounting interest in learning whether the potential of quantum computing can be put at use also for solving the most compelling problems in classical physics, as typically described by strongly nonlinear partial differential equations. In this respect, fluid turbulence stands out as a prominent candidate, both in terms of fundamental physics and also in view of its pervasive applications in both natural and industrial phenomena.

What is the Lattice Boltzmann Method?

The Lattice Boltzmann method is a computational fluid dynamics technique. It is used to simulate fluid flows, and is particularly useful for simulating complex and irregular boundaries. The method is based on the Boltzmann equation, a statistical description of the behavior of a fluid.

The Lattice Boltzmann method simplifies the Boltzmann equation by discretizing the velocity space into a finite number of velocities. This makes the method computationally efficient, and it has been used to simulate a wide range of fluid flows, from simple laminar flows to complex turbulent flows.

In the context of the Lattice Boltzmann Carleman quantum algorithm, the Lattice Boltzmann method is used as the basis for the quantum computing algorithm. The algorithm uses the Carleman linearization of the Lattice Boltzmann method to simulate fluid flows at moderate Reynolds numbers.

How Does the Carleman Linearization Work?

The Carleman linearization is a mathematical technique used to transform a nonlinear system of equations into a linear system. This is done by introducing an infinite number of new variables, which represent the moments of the original variables.

In the context of the Lattice Boltzmann Carleman quantum algorithm, the Carleman linearization is used to deal with the nonlinearity of the Navier-Stokes equations, which govern the physics of dissipative fluids. The algorithm introduces two ways to deal with the infinite number of variables and shows how to conveniently cut down the size of the system of equations.

The Carleman linearization is a key component of the algorithm, as it allows for the successful truncation of the Carleman LB procedure at second order for moderate Reynolds numbers. This is a very encouraging result, as it suggests that the algorithm could be used to simulate fluid flows at these Reynolds numbers.

What are the Future Prospects for the Lattice Boltzmann Carleman Quantum Algorithm?

The Lattice Boltzmann Carleman quantum algorithm represents a significant step forward in the use of quantum computing for fluid flow simulations. However, there are still many challenges to overcome before the algorithm can be used in practical applications.

One of the main challenges is the depth of the quantum circuit implementing the single timestep collision operator. The depth is of the order of ten thousands quantum gates, which is currently beyond the capabilities of existing quantum computers. However, as quantum computing technology continues to advance, it is expected that this challenge will be overcome in the near or midterm future.

Another challenge is the multistep version of the algorithm, which remains an open topic for future research. Despite these challenges, the Lattice Boltzmann Carleman quantum algorithm holds great promise for the future of fluid flow simulations, and it is expected that further research and development will lead to significant advancements in this field.