Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena

TL;DR

This work develops a hybrid classical-quantum variational algorithm to solve linear and nonlinear transport PDEs arising in thermo-fluid dynamics, including heat, Burgers', and wave equations, under engineering boundary conditions. It encodes discretized band matrices into quantum-nonparametric units (QNPUs) and evaluates a Ritz-Galerkin residual via Hadamard-test circuits, achieving polylogarithmic gate complexity in the number of qubits $n$ and enabling scalable VQA-based CFD. Extensive one-dimensional tests across transient heat conduction with spatially varying diffusivity, steady advection-diffusion, inviscid nonlinear convection (unidirectional and bidirectional), Burgers' equation, and the linear wave equation demonstrate strong agreement with classical finite-difference solutions, with some deviations at higher $n$ attributed to optimization challenges and ansatz expressiveness. The results indicate that VQA-based CFD can leverage near-term quantum hardware to address transport problems with nonlinearities, while highlighting open challenges in implicit upwind schemes, boundary-condition handling, and extensions to higher dimensions or non-uniform grids.

Abstract

This article proposes a Variational Quantum Algorithm to solve linear and nonlinear thermofluid dynamic transport equations. The hybrid classical-quantum framework is applied to problems governed by the heat, wave, and Burgers' equation in combination with different engineering boundary conditions. Topics covered include the encoding of band matrices, as in the consideration of non-constant material properties and upwind-biased first- and higher-order approximations, widely used in engineering Computational Fluid Dynamics, by the use of a mask function. Verification examples demonstrate high predictive agreement with classical methods. Furthermore, the scalability analysis shows a polylog scaling of the number of quantum gates with the number of qubits. Remaining challenges refer to the implicit construction of upwind schemes and the identification of an appropriate parameterization strategy of the quantum ansatz.

Figures (14)

• Figure 1: Sketch of the hybrid classical-QC optimization in the VQA. The bilinear and linear circuit on the right-hand side show, the Hadamard gate $\pmb{H}$, the ansatz gate $\pmb{U(\hbox{$\bm{\lambda}$}\xspace_c)}$, the QNPU module, and the measurement gauge on the ancilla. The abbreviation CP marks a control port while IP & OP denote input/output ports, respectively.
• Figure 2: Employed brick-layer ansatz as proposed in Ref. BravoPrieto2023. Each $\pmb{R_\text{Y}=exp{(-\lambda_ j\, i\, \textsc{Y}/2)}}$ gate is parameterized by a unique control parameter $\pmb{\lambda_j}$ of the control vector $\pmb{\hbox{$\bm{\lambda}$}\xspace_c}$ and $\pmb{i}$ is the imaginary unit applied to the Y-gate.
• Figure 3: QNPU schematic for the potential contribution $\pmb{P}$, e.g., for $\pmb{n=4}$ qubits. The ansatz gate for encoding $\pmb{p}$ is marked with $\pmb{P(\bar{\hbox{$\bm{\lambda}$}\xspace}_c)}$, where $\pmb{\bar{\hbox{$\bm{\lambda}$}\xspace}_c}$ are the trained parameters. The control ports are labeled CP, the input ports are labeled IP, and the output ports are labeled OP.
• Figure 4: QNPU schematic for the adder (shift) circuits, e.g., for $\pmb{n=4}$ qubits. The control ports are labeled CP, the input ports are labeled IP, and the output ports are labeled OP.
• Figure 5: QNPU schematic for the adder-potential $\pmb{{A}_p}$ for the example of $\pmb{n=4}$ qubits. The ansatz gate for the potential $\pmb{p}$ is indicated by $\pmb{P(\bar{\hbox{$\bm{\lambda}$}\xspace}_c)}$. The abbreviations CP, IP, OP mark the control, input, and output ports, respectively.