Gate-level Implementation and Resource Analysis of Lackadaisical Quantum Walk Search

Abstract

Lackadaisical quantum walks (LQW) extend discrete-time quantum walks (DTQW) by introducing weighted self-loops, enabling improved spatial-search performance through controlled localization of the walker. Despite substantial theoretical progress, practical gate-level implementations suitable for quantum hardware remain largely unexplored, limiting evaluation under realistic architectural constraints, noise, and resource requirements. In this work, we present a gate-level implementation framework for lackadaisical quantum walk search. The proposed construction encodes the position and coin spaces into qubit registers, and realizes the walk dynamics through oracle, coin, and flip-flop shift operations. We validate the circuit by reproducing the expected search behavior for single and multiple marked vertices and by analyzing the effect of the self-loop weight on the success probability. We further evaluate the implementation under realistic noisy settings using superconducting hardware's noise models. Logical resource analysis shows that, for grids ranging from $8\times 8$ to $64\times 64$, the transpiled gate count increases from $3.63\times 10^5$ to $4.38\times 10^6$ and the circuit depth from $2.13\times 10^5$ to $2.56\times 10^6$. Finally, fault-tolerant resource estimates based on a surface-code model using the Microsoft Quantum Resource Estimator demonstrate the substantial space-time trade-off associated with magic-state production. The results establish a practical circuit-level pathway for implementing the LQW search and provide a basis for evaluating its performance.

27 pages, 15 figures