Abstract Quantum walks are a powerful framework for the development of quantum algorithms, with lackadaisical quantum walks (LQWs) standing out as an efficient model for spatial search. In this work, we investigate how broken-link decoherence affects the performance of LQW-based search on a two-dimensional toroidal grid. We show through numerical simulations that while decoherence drives the loopless walk toward a uniform distribution and eliminates its search capability, the inclusion of self-loops significantly mitigates this effect. In particular, even under noise, the marked vertex remains identifiable with probability well above uniform, demonstrating that self-loops enhance the robustness of LQWs in realistic scenarios. These findings extend the known advantages of LQWs from the noiseless setting to noisy environments, consolidating self-loops as a valuable resource for designing resilient quantum search algorithms.
Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in one and two dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with [Formula: see text] success probability after [Formula: see text] time steps. In two dimensions, our numerical simulation up to [Formula: see text] for specific sets of target states suggests that the lackadaisical quantum walk can search one of the [Formula: see text] target states in [Formula: see text] time steps.
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
8×8 to
64×64, the transpiled gate count increases from
3.63×105 to
4.38×106 and the circuit depth from
2.13×105 to
2.56×106. 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.
The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.
Abstract Lackadaisical quantum walk (LQW) is a quantum analog of a classical lazy walk, where each vertex has a self-loop of weight l . For a regular N × N 2D grid LQW can find a single marked vertex with O (1) probability in O ( N log N ) steps using l = d / N , where d is the degree of the vertices of the grid [11]. For multiple marked vertices, however, l = d / N is not optimal as the success probability decreases with the increase of the number of marked vertices [12]. In this paper, we numerically study search by LQW for different types of 2D grids—triangular, rectangular and honeycomb—with multiple marked vertices. We show that in all cases the weight l = m ⋅ d / N , where m is the number of marked vertices, still leads to O (1) success probability.
The lackadaisical quantum walk, which is a quantum walk with a weighted self-loop at each vertex, has been shown to speed up dispersion on the line and improve spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, owing to each graph's vertex transitivity. In this paper, we propose lackadaisical quantum walks where the self-loops have different weights. We investigate spatial search on the complete bipartite graph, which can be irregular with
N1 and
N2 vertices in each partite set, and this naturally leads to self-loops in each partite set having different weights
l1 and
l2, respectively. We analytically prove that for large
N1 and
N2, if the
k marked vertices are confined to, say, the first partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its nonlackadaisical value when
l1=kN2/2N1 and ${N}_{2}>(3\ensuremath{-}2\sqrt{2}){N}_{1}$, regardless of
l2. When the initial state is stationary under the quantum walk, however, then the success probability is improved when
l1=kN2/2N1, now without a constraint on the ratio of
N1 and
N2, and again independent of
l2. Next, when marked vertices lie in both partite sets, then for either initial state there are many configurations for which the self-loops yield no improvement in quantum search, no matter what weights they take.
The lackadaisical quantum walk is a lazy version of a discrete-time, coined quantum walk, where each vertex has a weighted self-loop that permits the walker to stay put. They have been used to speed up spatial search on a variety of graphs, including periodic lattices, strongly regular graphs, Johnson graphs, and the hypercube. In these prior works, the weights of the self-loops preserved the symmetries of the graphs. In this paper, we show that the self-loops can break all the symmetries of vertex-transitive graphs while providing the same computational speedups. Only the weight of the self-loop at the marked vertex matters, and the remaining self-loop weights can be chosen randomly, as long as they are small compared to the degree of the graph.