Quantum CFD

Quantum CFD#

Quantum algorithms for fluid dynamics simulation. Lattice Boltzmann methods, quantum linear solvers for Navier-Stokes, and turbulence modeling.

Why Quantum for CFD?#

Computational fluid dynamics requires solving systems of partial differential equations (PDEs) on fine spatial grids. After discretization, these become large sparse linear systems:

\[\mathbf{A}\mathbf{x} = \mathbf{b}\]

where \(\mathbf{A}\) is an \(N \times N\) sparse matrix, \(N\) can be \(10^6\) to \(10^9\) for practical 3D simulations.

Classical solvers (GMRES, conjugate gradient) scale as \(O(N \cdot \kappa \cdot s)\) where \(\kappa\) is the condition number and \(s\) is the sparsity. The HHL quantum algorithm promises \(O(\log N \cdot \kappa^2 \cdot s^2)\), an exponential improvement in \(N\) under specific conditions.

Quantum Lattice Boltzmann Method#

The lattice Boltzmann method (LBM) simulates fluid dynamics by tracking particle distribution functions on a lattice rather than solving the Navier-Stokes equations directly. Quantum LBM encodes distribution functions in qubit amplitudes:

\[f_i(\mathbf{x}, t) \rightarrow \alpha_i |i\rangle\]

A quantum circuit performs the collision and streaming steps:

Qiskit

from qiskit import QuantumCircuit
import numpy as np

def quantum_lbm_step(n_velocities: int = 9):
    """One step of D2Q9 quantum lattice Boltzmann."""
    n_qubits = int(np.ceil(np.log2(n_velocities)))
    qc = QuantumCircuit(n_qubits)

    # Encode distribution function
    # (amplitude encoding of f_i values)
    qc.initialize(distribution_amplitudes, range(n_qubits))

    # Collision operator (relaxation toward equilibrium)
    # Implemented as a unitary rotation
    qc.ry(omega * delta_t, 0)  # BGK collision

    # Streaming (permutation of velocity directions)
    # Implemented as SWAP network
    qc.swap(0, 1)

    return qc

PennyLane

import pennylane as qml
import numpy as np

n_qubits = 4  # Encodes D2Q9 lattice directions
dev = qml.device("default.qubit", wires=n_qubits)

@qml.qnode(dev)
def lbm_collision_step(distribution, omega):
    """Quantum collision operator for LBM."""
    # Encode current distribution
    qml.AmplitudeEmbedding(distribution, wires=range(n_qubits), normalize=True)

    # BGK collision: relax toward equilibrium
    for i in range(n_qubits):
        qml.RY(omega, wires=i)

    return qml.state()

Variational Quantum Linear Solver (VQLS)#

For near-term quantum hardware, VQLS provides an alternative to HHL that works with shallow circuits:

\[\min_{\boldsymbol{\theta}} \quad C(\boldsymbol{\theta}) = 1 - \frac{|\langle \mathbf{b} | \mathbf{A} | \psi(\boldsymbol{\theta}) \rangle|^2}{\langle \psi(\boldsymbol{\theta}) | \mathbf{A}^\dagger \mathbf{A} | \psi(\boldsymbol{\theta}) \rangle}\]

The cost function \(C\) is zero when \(|\psi(\boldsymbol{\theta})\rangle\) is proportional to \(\mathbf{A}^{-1}|\mathbf{b}\rangle\).

Current Limitations#

Challenge	Impact
Data loading	Encoding classical data into quantum states costs \(O(N)\), erasing the \(O(\log N)\) advantage
Readout	Extracting the full solution requires \(O(N)\) measurements
Matrix sparsity	HHL/VQLS require efficient Hamiltonian simulation of \(\mathbf{A}\)
Condition number	Both algorithms scale with \(\kappa\); CFD matrices are often ill-conditioned
Problem size	Current QPUs handle \(N \sim 10^2\); practical CFD needs \(N \sim 10^6\)+