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Type II Quantum Computing Algorithm for Computational Fluid Dynamics

James Scoville

Published Oct 3, 2012• 0 citations
OpenAlex

Abstract

An algorithm is presented to simulate fluid dynamics on a three qubit type II quantum computer: a lattice of small quantum computers that communicate classical information. The algorithm presented is called a three qubit factorized quantum lattice gas algorithm. It is modeled after classical lattice gas algorithms which move virtual particles along an imaginary lattice and change the particles' momentums using collision rules when they meet at a lattice node. Instead of moving particles, the quantum algorithm presented here moves probabilities, which interact via a unitary collision operator. Probabilities are determined using ensemble measurement and are moved with classical communications channels. The lattice node spacing is defined to be a microscopic scale length. A mesoscopic governing equation for the lattice is derived for the most general three qubit collision operator which preserves particle number. In the continuum limit of the lattice, a governing macroscopic partial differential equation-the diffusion equation-is derived for a particular collision operator using a Chapman-Enskog expansion. A numerical simulation of the algorithm is carried out on a conventional desktop computer and compared to the analytic solution of the diffusion equation. The simulation agrees very well with the known solution.

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An algorithm is presented to simulate fluid dynamics on a three qubit type II quantum computer: a lattice of small quantum computers that communicate classical information. The algorithm presented is called a three qubit factorized quantum lattice gas algorithm. It is modeled after classical lattice gas algorithms which move virtual particles along an imaginary lattice and change the particles' momentums using collision rules when they meet at a lattice node. Instead of moving particles, the quantum algorithm presented here moves probabilities, which interact via a unitary collision operator. Probabilities are determined using ensemble measurement and are moved with classical communications channels. The lattice node spacing is defined to be a microscopic scale length. A mesoscopic governing equation for the lattice is derived for the most general three qubit collision operator which preserves particle number. In the continuum limit of the lattice, a governing macroscopic partial differential equation-the diffusion equation-is derived for a particular collision operator using a Chapman-Enskog expansion. A numerical simulation of the algorithm is carried out on a conventional desktop computer and compared to the analytic solution of the diffusion equation. The simulation agrees very well with the known solution.
Presented is a quantum lattice gas for Navier-Stokes fluid dynamics simulation. The quantum lattice-gas transport equation at the microscopic scale is presented as a generalization of the classical lattice-gas transport equation. A special type of quantum computer network is proposed that is suitable for implementing the quantum lattice gas. The quantum computer network undergoes a partial collapse of the wavefunction at every time step of the dynamical evolution. Each quantum computer in the network comprises only a few qubits, which are entangled for only a short time period. A Chapman-Enskog type analysis of the quantum computer network indicates that the total system of qubits behaves exactly like a viscous lattice-gas fluid at the macroscopic scale. Because of the quantum mechanical nature of the scattering process, superposition of outgoing collisional possibilities occurs. The quantum lattice gas obeys detail balance in its collisions and is therefore an unconditionally stable algorithm for fluid dynamics simulation.
Quantum-computing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specifically, the computational simulation of fluid dynamics). The quantum algorithm is called a quantum lattice gas. An analytical treatment of the microscopic quantum lattice-gas system is carried out to predict its behavior at the mesoscopic scale. At the mesoscopic scale, a lattice Boltzmann equation with a nonlocal collision term that depends on the entire system wave function, governs the dynamical system. Numerical results obtained from an exact simulation of a one-dimensional quantum lattice model are included to illustrate the formalism. A symbolic mathematical method is used to implement the quantum mechanical model on a conventional work- station The numerical simulation indicates that classical viscous damping is not present in the one-dimensional quantum lattice-gas system.
Presented is a factorized quantum lattice-gas algorithm to model the diffusion equation. It is a minimal model with two qubits per node of a one-dimensional lattice and it is suitable for implementation on a large array of small quantum computers interconnected by nearest-neighbor classical communication channels. The quantum lattice-gas system is described at the mesoscopic scale by a lattice-Boltzmann equation whose collision term is unconditionally stable and obeys the principle of detailed balance. An analytical treatment of the model is given to predict a macroscopic effective field theory. The numerical simulations are in excellent agreement with the analytical results. In particular, numerical simulations confirm the value of the analytically calculated diffusion constant. The algorithm is time-explicit with numerical convergence that is first-order accurate in time and second-order accurate in space.
A lattice-gas algorithm for the one-dimensional diffusion equation is realized using radio frequency pulses in a one-dimensional spin system. The model is a large array of quantum two-qubit nodes interconnected by the nearest-neighbor classical communication channels. We present a quantum protocol for implementation of the quantum collision operator and a method for initialization and reinitialization of quantum states. Numerical simulations of the quantum-classical dynamics are in good agreement with the analytic solution for the diffusion equation.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
Fluid flow simulations marshal our most powerful computational resources. In many cases, even this is not enough. Quantum computers provide an opportunity to speed up traditional algorithms for flow simulations. We show that lattice-based mesoscale numerical methods can be executed as efficient quantum algorithms due to their statistical features. This approach revises a quantum algorithm for lattice gas automata to reduce classical computations and state preparation at every time step. For this, the algorithm approximates the qubit relative phases and subtracts them at the end of each time step. Phases are evaluated using the iterative phase estimation algorithm and subtracted using single-qubit rotation phase gates. This method optimizes the quantum resource required and makes it more appropriate for near-term quantum hardware. We also demonstrate how the checkerboard deficiency that the D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is validated by simulating two canonical partial differential equations: the diffusion and Burgers' equations on different quantum simulators. We find good agreement between quantum simulations and classical solutions for the presented algorithm.