Quantum Navier-Stokes Solvers

Quantum Navier-Stokes Solvers#

Applying HHL and variational quantum linear solvers to discretized Navier-Stokes equations. Complexity analysis and current limitations.

The Navier-Stokes Equations#

The incompressible Navier-Stokes equations govern fluid motion:

\[\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u}\]

\[\nabla \cdot \mathbf{u} = 0\]

where \(\mathbf{u}\) is the velocity field, \(p\) is pressure, \(\rho\) is density, and \(\nu\) is kinematic viscosity.

After spatial discretization (finite difference, finite volume, or finite element), each timestep reduces to solving:

\[\mathbf{A}\mathbf{x}^{n+1} = \mathbf{b}(\mathbf{x}^n)\]

This is where quantum linear solvers enter.

HHL for Navier-Stokes#

The HHL algorithm solves \(\mathbf{A}\mathbf{x} = \mathbf{b}\) in \(O(\log N \cdot \kappa^2 \cdot s^2 / \epsilon)\) time, exponentially faster in \(N\) than classical methods.

Requirements for HHL#

Requirement	Navier-Stokes Context	Challenge
Sparse \(\mathbf{A}\)	FD/FV matrices are sparse	Satisfied
Low condition number \(\kappa\)	CFD matrices: \(\kappa \sim 10^3 - 10^6\)	Problematic; preconditioning needed
Efficient \(\|b\rangle\) preparation	Requires quantum state encoding	\(O(N)\) cost negates \(O(\log N)\) advantage
Useful output	Often need full velocity field, not just expectation values	Readout bottleneck

Linearization#

The nonlinear convective term \((\mathbf{u} \cdot \nabla)\mathbf{u}\) must be linearized for quantum linear solvers. Common approaches:

Picard iteration: Replace \(\mathbf{u}\) with previous iterate \(\mathbf{u}^{(k)}\)
Newton linearization: Linearize around the current solution
Operator splitting: Separate advection from diffusion

Each approach converts the nonlinear PDE into a sequence of linear systems.

Variational Approach (VQLS)#

Qiskit

from qiskit import QuantumCircuit
from qiskit.circuit import ParameterVector
import numpy as np

def vqls_ansatz(n_qubits, depth=3):
    """Hardware-efficient ansatz for VQLS."""
    params = ParameterVector('theta', n_qubits * depth * 2)
    qc = QuantumCircuit(n_qubits)

    idx = 0
    for d in range(depth):
        for i in range(n_qubits):
            qc.ry(params[idx], i)
            qc.rz(params[idx + 1], i)
            idx += 2
        for i in range(n_qubits - 1):
            qc.cx(i, i + 1)

    return qc, params

# For a 2D Laplacian (diffusion term):
# A = tridiagonal(-1, 2, -1) / dx^2
# Decompose A into sum of Pauli strings for Hamiltonian simulation

PennyLane

import pennylane as qml
import numpy as np

n_qubits = 4  # Encodes 2^4 = 16 grid points
dev = qml.device("default.qubit", wires=n_qubits)

@qml.qnode(dev)
def vqls_cost(params, A_coeffs, A_paulis, b_state):
    """VQLS cost function for Navier-Stokes linear system."""
    # Prepare ansatz |x(theta)>
    for i in range(n_qubits):
        qml.RY(params[2*i], wires=i)
        qml.RZ(params[2*i+1], wires=i)
    for i in range(n_qubits - 1):
        qml.CNOT(wires=[i, i+1])

    # Compute <x|A†A|x> and |<b|A|x>|^2
    return qml.expval(qml.Hermitian(A_matrix, wires=range(n_qubits)))

Complexity Comparison#

Method	Complexity	Practical for CFD?
Classical CG/GMRES	\(O(N \cdot \kappa \cdot s)\)	Yes (standard tool)
HHL (ideal)	\(O(\log N \cdot \kappa^2 \cdot s^2 / \epsilon)\)	Only if data loading is cheap
VQLS (NISQ)	\(O(\text{poly}(\kappa) \cdot \text{circuit depth})\)	Proof-of-concept stage
Quantum-inspired classical	\(O(N \cdot \text{polylog}(N))\)	Growing competitor

The Data Loading Problem

The exponential speedup of HHL applies to the linear solve step. However, encoding the classical right-hand side \(\mathbf{b}\) into a quantum state and extracting the classical solution \(\mathbf{x}\) both require \(O(N)\) operations. For end-to-end quantum advantage, the problem must either start and end in quantum form, or use quantum random access memory (QRAM), which does not yet exist at scale.

Research Directions#

Quantum-quantum pipeline: Couple quantum PDE solvers with quantum optimization (avoid classical I/O)
QRAM development: Efficient quantum data loading would unlock HHL’s full potential
Preconditioning: Reduce \(\kappa\) before quantum solve to improve scaling
Hybrid schemes: Use quantum for the most expensive substep (e.g., pressure Poisson equation) and classical for the rest

Quantum Navier-Stokes Solvers

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