Quantum walk search based edge detection of images

TL;DR

The paper tackles edge detection in digital images by leveraging a discrete-time quantum walk search with a lackadaisical coin to amplify edge pixel amplitudes. Edges are identified by marking pixels through a gradient-threshold and evolving the quantum walk with a modified coin $\mathcal{C}_G$, enabling high measurement success probabilities $p_s$ and, in principle, a quadratic speedup over classical methods. A practical demonstration uses a small Qiskit circuit on $2\times2$ blocks (with $t=2$ iterations) showing high $p_s$ on a simulator ($p_s \approx 0.53$) and moderate performance on a fake IBM device ($\bar{p}_s \approx 0.40$), while full 2D implementation remains beyond current NISQ capabilities. The results indicate the proposed DTQWS approach offers superior edge-detection robustness and potential speed advantages compared to Hadamard edge detection (HED) and QSobel, though further work on error mitigation and large-scale deployment is needed.

Abstract

Quantum walk has emerged as an essential tool for searching marked vertices on various graphs. Recent advances in the discrete-time quantum walk search algorithm have enabled it to effectively handle multiple marked vertices, expanding its range of applications further. In this article, we propose a novel application of this advanced quantum walk search algorithm for the edge detection of images\textemdash a critical task in digital image processing. Given the probabilistic nature of quantum computing, obtaining measurement result with a high success probability is essential alongside faster computation time. Our quantum walk search algorithm demonstrates a high success probability in detecting the image edges compared to the existing quantum edge detection methods and outperforms classical edge detection methods with a quadratically faster speed. A small Qiskit circuit implementation of our method using a one-dimensional quantum walk search has been executed in Qiskit's $qasm\_simulator$ and $ibm\_sydney(fake)$ device.

Figures (5)

• Figure 1: (a) A two-dimensional lattice of size $4\times 6$, (b) six $2\times 2$ blocks of the $4\times 6$ two-dimensional lattice.
• Figure 2: Qiskit circuit for quantum edge detection with quantum walk search on a cycle with four vertices. Four dimensional coin space, having four basis states: left and right directions and two self-loops with $\bf s= 0.1$, $\bf t=2$.
• Figure 3: (a) $\bf 50 \times 50$ sample image, and its edge detection from quantum walk search algorithm with (b) regular or lackadaisical quantum walk with Grover coin with $\bf s=0.01$ (c) SKW coin and (d) $\bf \mathcal{C}_G$ coin with $\bf s= 0.01$. (b) -(d) are obtained by numerical analysis.
• Figure 4:
• Figure 5: A sample (a) $\bf 330 \times 350$ image from BSDS500 database david is used to obtain its edge from Qiskit circuit for the $2\times 2$ blocks of the image with (b) $\bf qasm\_simulator$(no noise), $\bf \bar{p}_s \approx 0.53$ and (c) $\bf ibm\_sydney$(fake) backend, $\bf \bar{p}_s \approx 0.40$, $\bf s =0.1$, $\bf t=2$. Also edge obtained with the two-dimensional numerical QWS of the $\bf 330 \times 350$ image with (d) QWS with $\bf s= 0.0001$, $\bf t=791$, $\bf p_s \approx 0.98$, (e) HED, $\bf \bar{p}_h \approx 0.0031$, (f) QSobel, $\bf p_q \approx 0.0095$. Edge obtained with raw the data(without post-processing) obtained from numerical analysis of the entire image with (g) QWS with $\bf s= 0.0001$, $\bf t=791$, $\bf p_s \approx 0.98$, (h) HED, $\bf \bar{p}_h \approx 0.0031$, (i) QSobel, $\bf p_q \approx 0.0095$.