Modeling Noise in Quantum Computing of Scalar Convection

TL;DR

This work analyzes how gate noise on NISQ devices distorts quantum simulations of 1D scalar convection using a quantum spectral method that preserves spectral magnitudes in the noiseless limit. It derives a transition matrix based on Hamming distance for independent Pauli noise and validates it with density-matrix simulations and experiments on a superconducting processor, demonstrating a hierarchical noise-induced spectral decay. Data-driven PDE learning reveals that the noisy dynamics correspond to artificial diffusion and nonlinear source terms, enabling a deterministic PDE description of quantum errors. These results provide a principled framework for noise-aware quantum algorithms in CFD and point toward potential subgrid modeling opportunities that leverage hardware noise structure.

Abstract

Quantum computing holds potential for accelerating the simulation of fluid dynamics. However, hardware noise in the noisy intermediate-scale quantum era significantly distorts simulation accuracy. Although error magnitudes are frequently quantified, the specific physical effects of quantum noise on flow simulation results remain largely uncharacterized. We investigate the influence of gate noise on the quantum simulation of one-dimensional scalar convection. By employing a quantum spectral algorithm where ideal time advancement affects only Fourier phases, we isolate and analyze noise-induced artifacts in spectral magnitudes. We derive a theoretical transition matrix based on Hamming distances between computational basis states to predict spectral decay, and then validate this model against density-matrix simulations and experiments on a superconducting quantum processor. Furthermore, using data-driven sparse regression, we demonstrate that quantum noise manifests in the effective partial differential equation primarily as artificial diffusion and nonlinear source terms. These findings suggest that quantum errors can be modeled as deterministic physical terms rather than purely stochastic perturbations.

Figures (7)

• Figure 1: Solution procedure for the scalar convection in Eq. \ref{['eq:convection']}. First, the initial condition is encoded into the amplitudes of a quantum state. Next, a QFT maps the state into spectral space. The Hamiltonian simulation in Eq. \ref{['eq:evo']} is then applied for each time step. Finally, an inverse QFT reconstructs a quantum state whose amplitudes represent the solution.
• Figure 2: Schematic of different methods for scalar convection: ideal and noisy density-matrix simulations on a classical computer, and experiment on a superconducting processor.
• Figure 3: Evolution of a scalar field undergoing convection in a quantum simulation. (a) Initial profile $u(x,0)=(\cos x + \sin 2x + 2\cos 2x + 3\cos 3x)/10$. (b, c) Snapshots at $t=3.0$ and $t=6.0$ comparing the ideal solution in solid blue lines and density-matrix simulations with $p=8.3\times10^{-4}$ in dotted orange lines and $p=1.6\times10^{-3}$ in dashed green lines.
• Figure 4: Visualization of the transition matrices for $n=1,3,5,7$ qubits. (a) Theoretical prediction based on Eq. \ref{['eq:Ml_n']} with $p=8.3\times 10^{-4}$. (b) Density-matrix simulation using $p=8.3\times 10^{-4}$ for the depolarizing channel. (c) Experimental reconstruction on superconducting quantum processor "Yudu".
• Figure 5: Transition probabilities $M_n^l[k,0]$ (from ground state $\ket{0}$ to $\ket{k}$, blue hatched bars) and $M_n^l[0,k]$ (transition from $\ket{k}$ to ground state $\ket{0}$, orange hatched bars) for a three-qubit system after $l=320$ convection layers on superconducting quantum processor "Yudu", compared with the symmetric noise model (green dotted bars).