Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models

Abstract

The application of Quantum Computing (QC) to fluid dynamics simulation has developed into a dynamic research topic in recent years. With many flow problems of scientific and engineering interest requiring large computational resources, the potential of QC to speed-up simulations and facilitate more detailed modeling forms the main motivation for this growing research interest. Despite notable progress, many important challenges to creating quantum algorithms for fluid modeling remain. The key challenge of non-linearity of the governing equations in fluid modeling is investigated here in the context of lattice-based modeling of fluids. Quantum circuits for the D1Q3 (one-dimensional, three discrete velocities) Lattice Boltzmann model are detailed along with design trade-offs involving circuit width and depth. Then, the design is extended to a one-dimensional lattice model for the non-linear Burgers equation. To facilitate the evaluation of non-linear terms, the presented quantum circuits employ quantum computational basis encoding. The second part of this work introduces a novel, modular quantum-circuit implementation for non-linear terms in multi-dimensional lattice models. In particular, the evaluation of kinetic energy in two-dimensional models is detailed as the first step toward quantum circuits for the collision term of two- and three-dimensional Lattice Boltzmann methods. The quantum circuit analysis shows that with O(100) fault-tolerant qubits, meaningful proof-of-concept experiments could be performed in the near future.

Floating-Point Arithmetic with Consistent Rounding on a Quantum Computer

“…This enables the range of numbers that can accurately be presented to be tailored to the considered problem. This was demonstrated for quantum circuit implementation of one-dimensional lattice models for fluid dynamics [16,17]. For the previously considered examples with M ¼ 4, an asymmetric bias of EXP bias ¼ 5 means that the qubit strings |011|000i and |011|111i define the number 8=32 and 15=32, respectively.…”

“…An early work related to quantum circuits for floating-point arithmetic involved the complexity analysis of floating-point addition and multiplication by Häner and coworkers [8], which showed the significant challenges involved in terms of circuit complexity for IEEE-754 type precision. This motivated the present author to introduce floating-point formats for quantum algorithms with a reduced precision [12,16,17], such that the required number of qubits and the circuit complexity remain within limits of near-future quantum computers with an increased level of fault tolerance as compared to NISQ-era hardware.…”

Efficient and Fail-Safe Collisionless Quantum Boltzmann Method

“…After these early results by Yepez et al, the QCFD field became stagnant for about a decade until its recent resurgence, in particular, in the form of quantum Boltzmann methods. Most recent are the methods presented in [TS20; Bud20; Bud21; MVS22; SM22 ;Ste23], that all have their own strengths and weaknesses. Some papers include a streaming and specular reflection mechanism, but no collision methods yet [TS20; Bud20; SM22].…”

“…Due to the high costs of quantum state preparation and the chance of measurement errors this 'stop-and-go' strategy is hardly usable in practice. Other algorithms have managed to create a unitary collision operator, but have not yet been able to combine this with a streaming step into one start-to-end algorithm [MVS22;Ste23].…”

Quantum Carleman Lattice Boltzmann Simulation of Fluids

“…Similar to the case with classical computers [11], algorithms for the simulation of classical fluids on a quantum computer using lattice kinetic theory started with focus on lattice gas [13][14][15] to lattice Boltzmann [15][16][17]. However, we make a distinction between those that attempt to achieve a quantum analogue [18] versus those which use of a quantum computer is due to the advantage it affors when carrying out the arithmetic [19][20][21] In contrast, the Navier-Stokes fluid self-advection operator which is non-local and non-linear at once. The result is that information travels along space-time material lines defined by the flow velocity dx dt = u(x,t), while in lattice kinetic theory information moves along straight lines defined by discrete velocities v, namely dx dt = v, where v is a constant.…”