Session P02: Minisymposia: Quantum Algorithms for Fluid Flows

From CFD to QCFD

Over the past two decades, quantum computing (QC) with its surfeit of efforts in devising novel quantum algorithms has strived to establish its might over its classical counterparts. For QC to emerge as the indispensable tool for practical applications, the exigency is not just for novel protocols that process quantum information but also for extracting it wisely in classical formats. Here, starting with a brief introduction to QC, we evaluate potential methods of conducting fluid mechanics research using QC, which we call Quantum Computation of Fluid Dynamics (QCFD).  We shall attempt to implement and evaluate an end-to-end performance demonstration of some modified Harrow-Hassidim-Lloyd (HHL) type algorithms to study flows such as the flow in a pipe, 1D Burgers equation and aspects of the Navier-Stokes equations.   

Quantum-Ready and Quantum-Inspired Computing for Fluid Dynamics

Within the past decade, significant progress has been made in using quantum computing  (QC)  for solving classical problems.  In this talk, an overview is made of the ways by which QC has shown promise for fluid dynamics research.  This is via both quantum-ready and quantum-inspired algorithms.  The former deals with problems that either have the potentials to benefit from quantum speed-up on universal gate-based digital computers, or those that can be solved on quantum simulators.  The latter deals with new classical algorithms that have emerged from quantum physics.    

Efficient quantum algorithm for dissipative nonlinear differential equations

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic $n$-dimensional ordinary differential equations. Assuming $R < 1$, where $R$ is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity $T^2 q \mathrm{poly}(\log T, \log n, \log 1/\epsilon)/\epsilon$, where $T$ is the evolution time, $\epsilon$ is the allowed error, and $q$ measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in $T$. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a novel convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for $R \ge \sqrt{2}$. Finally, we discuss potential applications, showing that the $R < 1$ condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of $R$.   

Quantum compiler for classical dynamics

We present a framework for simulating a measure-preserving, ergodic dynamical system by a finite-dimensional quantum system amenable to implementation on a quantum computer. The framework is based on a quantum feature map for representing classical states by density operators (quantum states) on a reproducing kernel Hilbert space $\mathcal{H}$. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on $\mathcal{H}$ such that quantum mechanical expectation values are consistent with pointwise function evaluation. Meanwhile, quantum states and observables on $\mathcal{H}$ evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. To achieve an exponential quantum advantage, the state of the quantum system is projected onto a density operator on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. The finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit with an $O(n)$ number of gates and no interchannel communication. Furthermore, the circuit features a quantum Fourier transform stage with $O(n^2)$ gates, which makes predictions of observables possible by measurement in the standard computational basis. In this talk, we describe this ``quantum compiler'' framework, and illustrate it with applications to low-dimensional dynamical systems.   

Quantum Machine Learning for Computational Fluid Dynamics

Recently there has been increased interest within the computational physics community in the study of neural ODE solvers, which use neural networks to attempt to find simpler solutions for systems of linear or non-linear ordinary differential equations.  In this talk, I will discuss the role that quantum machine learning could play in this space and show the advantages as well as disadvantages of such algorithms.  Speciifcally, I will discuss various flavors of quantum neural networks and show how quantum computers can be used to train them given classical data.  I will then present a no-go result, showing that non-linear differential equations that support chaotic dynamics cannot be directly solved using the most popular approaches and suggest new approaches to linearize the dynamics that may be more profitable for neural quantum ODEs as well as quantum algorithms based on the linear-system solver of Harrow Hassidim and Lloyd.   

Quantum reservoirs for simple fluid flows

Reservoir computing models are one way of implementing recurrent neural networks that can process sequential data. Classical reservoir computing models have been used recently with success to predict the time evolution of nonlinear dynamical systems and the low-order statistics of turbulent flows including turbulent convection flows which are driven by buoyancy forces. Here, we extend the classical model to a gate-based quantum reservoir computing model. The reservoir, which is a large sparse random network in the classical case, consists in its quantum version of a number of single-qubit random rotation gates in combination with 2-qubit entanglement gates. The performance of the model is tested for low-dimensional nonlinear Galerkin models of fluid flows, such as the Lorenz 63 model that describes two-dimensional thermal convection right above the onset of convection. The predictions, their dependence on the (hyper-)parameters of the quantum reservoir (e.g., the number of qubits), and a comparison to the classical results are presented and critically discussed.