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Quantum Hamiltonian simulation of linearised Euler equations in complex geometries

Vladyslav Bohun, Andrij Kuzmak, Maciej Koch-Janusz

TL;DR

The work advances quantum PDE solving by introducing explicit quantum circuits that encode complex obstacle boundary conditions within finite-difference representations, preserving Hermiticity and keeping Trotter error unchanged. It then applies these techniques to the conservative regime of the linearized Euler equations with a background flow, deriving per-direction circuit blocks and combining them with obstacle-aware operators to model acoustic wave propagation and scattering. The results, validated against classical FDM and exact matrix exponentials, demonstrate accurate pressure-field dynamics on modest qubit counts and reveal polynomial, rather than exponential, resource scaling in the quantum setting. The methods pave the way for realistic quantum CFD simulations on near- to mid-term devices, while highlighting ongoing challenges such as robust readout and extending to non-conservative regimes.

Abstract

Quantum computing promises exponential improvements in solving large systems of partial differential equations (PDE), which forms a bottleneck in high-resolution computational fluid dynamics (CFD) simulations, in, among others, aerospace applications and weather forecasting. One approach is via mapping classical CFD problems to a quantum Hamiltonian evolution, for which recently an explicit quantum circuit construction has been shown in simple cases, allowing proof-of-concept execution on quantum processors. Here we extended this method to more complex and practically relevant cases. We first demonstrate how boundary conditions corresponding to arbitrary complex-shaped obstacles can be introduced in the quantum representations of elementary difference operators used to implement the PDE. We provide explicit and efficient circuit constructions, and show they neither increase the Trotter error, nor asymptotic gate complexity with respect to the free space equation. Using these methods we then derive quantum circuits for simulating the linearized Euler equations in a presence of a background fluid flow and obstacles. We illustrate our results by simulating the obtained quantum circuits for a number of boundary conditions, and compare the errors of the quantum solution to classical finite difference methods.

Quantum Hamiltonian simulation of linearised Euler equations in complex geometries

TL;DR

The work advances quantum PDE solving by introducing explicit quantum circuits that encode complex obstacle boundary conditions within finite-difference representations, preserving Hermiticity and keeping Trotter error unchanged. It then applies these techniques to the conservative regime of the linearized Euler equations with a background flow, deriving per-direction circuit blocks and combining them with obstacle-aware operators to model acoustic wave propagation and scattering. The results, validated against classical FDM and exact matrix exponentials, demonstrate accurate pressure-field dynamics on modest qubit counts and reveal polynomial, rather than exponential, resource scaling in the quantum setting. The methods pave the way for realistic quantum CFD simulations on near- to mid-term devices, while highlighting ongoing challenges such as robust readout and extending to non-conservative regimes.

Abstract

Quantum computing promises exponential improvements in solving large systems of partial differential equations (PDE), which forms a bottleneck in high-resolution computational fluid dynamics (CFD) simulations, in, among others, aerospace applications and weather forecasting. One approach is via mapping classical CFD problems to a quantum Hamiltonian evolution, for which recently an explicit quantum circuit construction has been shown in simple cases, allowing proof-of-concept execution on quantum processors. Here we extended this method to more complex and practically relevant cases. We first demonstrate how boundary conditions corresponding to arbitrary complex-shaped obstacles can be introduced in the quantum representations of elementary difference operators used to implement the PDE. We provide explicit and efficient circuit constructions, and show they neither increase the Trotter error, nor asymptotic gate complexity with respect to the free space equation. Using these methods we then derive quantum circuits for simulating the linearized Euler equations in a presence of a background fluid flow and obstacles. We illustrate our results by simulating the obtained quantum circuits for a number of boundary conditions, and compare the errors of the quantum solution to classical finite difference methods.
Paper Structure (16 sections, 3 theorems, 63 equations, 15 figures)

This paper contains 16 sections, 3 theorems, 63 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Differential equations as trotterized evolutions
  3. Difference operators
  4. Induced evolution operators as quantum circuits
  5. Implementing finite difference operators with obstacles
  6. Linearized Euler equations with a mean flow
  7. Algorithm complexity
  8. Numerical simulation and experiments
  9. Conclusions
  10. Circuit derivation for the conservative system with central finite difference operators
  11. Circuit derivation for the conservative system with $D^{+}$ and $D^{-}$ difference operators
  12. Circuit derivation for the $D^{\pm}$ operator with periodic boundary conditions
  13. Proof of Lemma \ref{['lemma1']}
  14. Improved obstacle circuit implementation on a QPU
  15. Additional simulations with complex obstacles
  16. ...and 1 more sections

Key Result

Lemma 1

The evolution operator $\exp{(-iH\tau)}$ with the Hamiltonian $H$ given in Eq. eq:hamiltonian and a time increment $\tau$ can be approximated in the first-order Lie-Trotter-Suzuki decomposition by the unitary operator ${\rm V}(\tau)$ Eq. eq:evolution_op with an approximation error (in the sense of o

Figures (15)

  • Figure 1: A quantum time-evolution simulation of the 2D linearized Euler equations (LEE) describing acoustic wave propagation in the presence of a constant background fluid flow, and an impenetrable airfoil-like obstacle. Here we plot the deviation of pressure from the equilibrium. The results were obtained by simulating the quantum circuits derived in this work, using $n=9$ qubits and a Trotter step of $\tau=0.05$.
  • Figure 2: The quantum circuit for the ${\rm U}_j(\lambda)$ operator (see Eq. \ref{['eq:umatrix']}) acting on the $q_{1},\dots,q_{j}$ register.
  • Figure 3: (left) A rectangular object placed inside a binary cell of the discretized domain. The cell is defined by the bit-string prefix $01$ in $x$-direction, and by $011$ in $y$-direction [see the main text]. (right) The matrix representation of the $2lD^\pm$ operator (here for the $x$-direction of the obstacle shown in the left panel), where rows which correspond to the points with the $x$-coordinate inside the obstacle are shaded in blue.
  • Figure 4: (Left) The quantum circuit implementing the difference operator (derivative) on the $x$ axis with an impenetrable object inside the 2D environment (with Dirichlet boundary conditions on its edges) defined by the bitstrings $b_{x,1}b_{x,2} = 01$ on the $x$ coordinate and by $b_{y,1}b_{y,2}b_{y,3} = 011$ on the $y$ coordinate. Here the $D^\pm$ blocks implement the obstacle-free evolution and the prefix-controlled operators (see the main text) remove the matrix elements corresponding to the obstacles. (Right) The corresponding circuit for the difference operator on the $y$ axis. Note that the"little-endian" convention is used, that is the left-most bits in the bit-strings correspond to the qubits with the largest index.
  • Figure 5: The quantum circuit implementing the $Q_{\alpha}$ time evolution Eq. \ref{['eq:evolution_op2']} for the $\alpha=x,y$ spatial direction.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof