Vortex Detection from Quantum Data

TL;DR

This work addresses the challenge of extracting physically meaningful vortex observables from quantum data produced by quantum differential equation solvers. It introduces quantum vortex detection (QVD), a sliding-window, Fourier‑based readout framework with sequential and parallel variants that operate directly on quantum-encoded flow fields. Key contributions include a concrete circuit design with windowing, contour extraction, and Fourier analysis that can detect Lamb-Oseen vortices, plus a density-spectrum approach enabling high-accuracy classification of vortical versus non-vortical flows. The results demonstrate that quantum-native readout can outperform standard quantum neural networks for this task and point to broad potential for quantum data analysis in CFD-like problems and topological data analysis.

Abstract

Quantum solutions to differential equations represent quantum data -- states that contain relevant information about the system's behavior, yet are difficult to analyze. We propose a toolbox for reading out information from such data, where customized quantum circuits enable efficient extraction of flow properties. We concentrate on the process referred to as quantum vortex detection (QVD), where specialized operators are developed for pooling relevant features related to vorticity. Specifically, we propose approaches based on sliding windows and quantum Fourier analysis that provide a separation between patches of the flow field with vortex-type profiles. First, we show how contour-shaped windows can be applied, trained, and analyzed sequentially, providing a clear signal to flag the location of vortices in the flow. Second, we develop a parallel window extraction technique, such that signals from different contour positions are coherently processed to avoid looping over the entire solution mesh. We show that Fourier features can be extracted from the flow field, leading to classification of datasets with vortex-free solutions against those exhibiting Lamb-Oseen vortices. Our work exemplifies a successful case of efficiently extracting value from quantum data and points to the need for developing appropriate quantum data analysis tools that can be trained on them.

Figures (8)

• Figure 1: Workflow for vortex detection from quantum data. A nonlinear system is modeled using a quantum differential equation solver, producing states that contain solutions with vortices (turbulent flow) or without vortices (laminar flow). Vortex detection is applied with sliding window-based spectral analysis, compressing relevant features of the flow for consequent detection. The detection circuit is trained on few examples and only requires several tuning parameters.
• Figure 2: Quantum circuit used to perform feature extraction in QVD. Quantum data in the form of a multi-dimensional flow field is encoded into state $\ket{\psi_{\rm{f}}}$. A shift gate followed by a permutation gate are applied to select a window from this flow whose elements are encoded into the top register of state $\ket{\psi_{\rm{w}}}$. Another permutation gate is applied to select a circular contour from the window whose ordered elements are encoded into the top register of state $\ket{\psi_{\rm{c}}}$. The QFT is applied to calculate the power spectrum of the contour, resulting in state $|\psi_{\rm{ps}}\rangle$. The low-frequency band of this power spectrum is extracted from state $|\psi_{\rm{lfps}}\rangle$ by measuring the bottom register of the circuit. This circuit assumes that $n_{\rm{f}} \geq n_{\rm{w}} > n_{\rm{c}} > n_{\rm{lfps}}$.
• Figure 3: Examples of performing feature extraction in QVD.(a) A two-dimensional vorticity flow in the $xy$-plane. A sliding window is selected and the contour points within each window (indicated by white crosses) are extracted. Note that this example is course-grained for brevity. (b) Two $7$-qubit windows are shown with $5$-qubit circular contours extracted. The top window is representative of a featureless background and the bottom window is representative of a centrally-located vortex. (c) The vorticity amplitudes extracted from the two contours. (d) The power spectra of the background and vortex, where frequencies are indexed by window pixel $\psi_{i,j}$. The power spectra is shown for the encoded window ($\rm{QFT}\ket{\psi_{\rm{w}}}$ on $7$ qubits), contour ($|\psi_{\rm{ps}}\rangle$ on $5$ qubits) and low-frequency band ($|\psi_{\rm{lfps}}\rangle$ on $3$ qubits), as obtained with the quantum circuit from Fig. \ref{['fig:generic_quantum_circuit']}. Note that vorticity and power spectrum are measured in arbitrary units.
• Figure 4: Quantum circuit for parallelized feature processing in quantum vortex detection. Conditioned on the superposition state $|+\rangle^{\otimes n_a}$, operators corresponding to different shifts are applied (each corresponding to associated ancilla state $|x\rangle$), followed by permutations and QFT. Projecting out the bottom register, the low-frequency amplitudes are pushed to the ancilla register, which is to be sampled in the frequency basis.
• Figure 5: Quantum data-driven feature processing in QVD.(a) A quantum dataset containing vorticity flow field solutions $|\psi_{i,j}\rangle$ categorized into non-vortical and vortical classes. (b) The density spectra for each pair of vortical and non-vortical fields, where frequencies are indexed by single power spectrum values of contours $N_{\rm{c}}$ processed in parallel. These spectra are obtained with the quantum circuit from Fig. \ref{['fig:parallel_quantum_circuit']} and truncated to $7$ qubits. The top vortical density spectrum corresponds to a field with $4$ vortices and the bottom to one with $7$ vortices. (c) The representative density spectrum distribution of the non-vortical and vortical classes obtained from sampling the individual density spectra of all fields in the dataset. Note that density spectrum is measured in arbitrary units.