Lackadaisical Quantum Walks

• Lackadaisical quantum walks are discrete-time coined quantum walks on graphs with added weighted self-loops, allowing the walker to either move along an edge or remain at the vertex.
• Adjusting the self-loop weight (e.g., ℓ = d/N) can dramatically improve spatial search performance by raising success probabilities from subconstant to near unity without extra amplitude amplification.
• LQWs also enhance robustness against decoherence and can be tailored for irregular graphs or directed walks, offering practical advances in quantum search algorithms and error mitigation.

Searching arXiv for recent and foundational papers on lackadaisical quantum walks to support the article. Searching arXiv for "lackadaisical quantum walk" and key related works. Looking up core arXiv records for LQWs and spatial search. Searching arXiv for "Noise-Resilient Spatial Search with Lackadaisical Quantum Walks" (Vieira et al., 19 Aug 2025) and related LQW search papers. A lackadaisical quantum walk (LQW) is a discrete-time coined quantum walk on a graph in which each vertex is augmented by a weighted self-loop, so that the walker can either move along an edge or remain at the current vertex. In the standard formulation, the self-loop is absorbed into a generalized Grover coin and a flip-flop shift, producing a walk whose dynamics interpolate between ordinary coined walks and stay-put behavior. LQWs became prominent because the self-loop weight can change both spectral structure and search performance: on several graph families it raises the success probability of spatial search from subconstant to constant, and recent work shows that the same mechanism can also mitigate broken-link decoherence on the two-dimensional torus (Rhodes et al., 2020, Vieira et al., 19 Aug 2025).

1. Formal model and operator structure

Let $G=(V,E)$ be a $d$-regular graph on $N=|V|$ vertices, and let $\ell\ge 0$ denote the weight of a self-loop added to every vertex. The canonical Hilbert space is

$$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$

where $\mathcal H_P$ is the position register and $\mathcal H_C$ is the coin register with $d$ moving directions plus one loop direction $|\circlearrowleft\rangle$. The normalized coin state is

$$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$

and the local coin is the generalized Grover reflection

$d$0

Globally,

$d$1

where the flip-flop shift satisfies

$d$2

When $d$3, the loop sector disappears and one recovers the ordinary coined Grover walk (Rhodes et al., 2020).

A weighted-graph formulation makes the same construction applicable with real-valued loop weights. In that framework, the local reference state at a vertex $d$4 is

$d$5

and the coin remains a reflection $d$6. This formalism proves that many models with $d$7 identical integer self-loops are exactly reducible to a single self-loop of real-valued weight $d$8, which is the standard modern parameterization of LQWs (Wong, 2017).

The same model admits two important equivalences. First, with the flip-flop shift it is equivalent, up to relabeling of registers, to Szegedy’s quantization of the interpolated classical Markov chain

$d$9

Second, on regular locally arc-transitive graphs, Høyer and Yu related the coined LQW directly to quantum interpolated walks, which is the basis of the general search-time analysis on those graph classes (Rhodes et al., 2020, Høyer et al., 2020).

2. Early analytical picture and one-dimensional dynamics

Historically, LQWs were introduced as a quantum analogue of lazy random walks in the context of Grover search on the complete graph. In that setting, with the phase-flip coin, adding a self-loop boosts the success probability from $N=|V|$0 to $N=|V|$1, whereas additional self-loops decrease the success probability; with the Ambainis–Kempe–Rivosh coin, self-loops simply slow the search. The same work also showed that continuous-time quantum walks behave differently: self-loops make no difference at all, because they contribute only a multiple of the identity to the Hamiltonian (Wong, 2015). A later weighted-graph treatment extended the complete-graph analysis to real-valued $N=|V|$2 and showed improved success probability when $N=|V|$3 (Wong, 2017).

On the infinite line, lackadaisicality does not imply classical-style slowing. For the model with laziness parameter $N=|V|$4, coin dimension $N=|V|$5, and Grover coin $N=|V|$6, Fourier analysis yields a nonzero localization probability at the origin,

$N=|V|$7

together with ballistic spreading. The peak velocities are

$N=|V|$8

and the variance obeys $N=|V|$9, so the walk remains ballistic for arbitrary $\ell\ge 0$0 (Wang et al., 2016).

A related one-dimensional literature, phrased in terms of lazy quantum walks with a three-state coin, reached a compatible conclusion: the $\ell\ge 0$1th moment scales as $\ell\ge 0$2, the support remains linear in $\ell\ge 0$3, and the occupancy rate fluctuates around a positive constant as $\ell\ge 0$4, in contrast to classical random walks where the occupancy rate tends to zero. This suggests that self-loops can increase coverage of the accessible region without destroying quantum-ballistic scaling (Li et al., 2014).

3. Spatial search with a single marked vertex

The central algorithmic role of LQWs is spatial search. On the two-dimensional periodic grid, the loopless coined walk finds a marked vertex with success probability $\ell\ge 0$5 in $\ell\ge 0$6 steps, so amplitude amplification is required and the overall runtime is $\ell\ge 0$7. Adding a self-loop of weight $\ell\ge 0$8 changes the behavior qualitatively: the marked-vertex probability reaches a constant near $\ell\ge 0$9 in $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$0 steps, so no amplitude amplification is needed and the overall runtime becomes $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$1. In Wong’s simulations on a $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$2 grid, $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$3 at $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$4, and the fitted first-peak time is $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$5 with correlation coefficient $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$6 (Wong, 2017).

This $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$7 prescription is part of a broader pattern. For a unique marked vertex on a $$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$8-regular vertex-transitive graph, Rhodes and Wong proposed that the optimal self-loop weight is

$$\mathcal H=\mathcal H_P\otimes\mathcal H_C \cong \mathbb C^N\otimes \mathbb C^{d+1},$$9

They supplied analytical or numerical evidence on the complete graph, two-dimensional torus, cycle, regular complete bipartite graph, higher-dimensional cubic lattices, strongly regular graphs, Johnson graphs, and the hypercube. Their reported maxima satisfy $\mathcal H_P$0 at $\mathcal H_P$1, with quadratic speedup $\mathcal H_P$2 in the tested families (Rhodes et al., 2020).

Høyer and Yu subsequently proved the corresponding search theorem on any regular locally arc-transitive graph. Choosing $\mathcal H_P$3, the LQW finds a unique marked vertex with constant success probability in

$\mathcal H_P$4

where $\mathcal H_P$5 is the classical hitting time. This analytic result subsumes prior conjectures and numerical findings for the torus, cycle, and Johnson graphs, and explains the $\mathcal H_P$6 rule through the relation between coined LQWs and quantum interpolated walks (Høyer et al., 2020).

The same degree-over-size prescription also appears in other two-dimensional lattices. On triangular and honeycomb grids, adding self-loops of weight $\mathcal H_P$7 and $\mathcal H_P$8, respectively, yields $\mathcal H_P$9 running time, matching the rectangular-grid mechanism after replacing the degree $\mathcal H_C$0 by $\mathcal H_C$1 or $\mathcal H_C$2 (Nahimovs, 2020).

4. Multiple marked vertices, exceptional configurations, and nonuniform weights

For multiple marked vertices, the single-mark prescription $\mathcal H_C$3 is generally no longer optimal. On triangular, rectangular, and honeycomb two-dimensional grids with periodic boundary conditions, a numerical study found that the peak success probability decreases rapidly with the number $\mathcal H_C$4 of marked vertices if one keeps $\mathcal H_C$5. Scanning over weights instead reveals a distinct optimum near

$\mathcal H_C$6

and choosing exactly $\mathcal H_C$7 restores $\mathcal H_C$8 with first-peak time

$\mathcal H_C$9

For random marked sets on a $d$0 grid, the average $d$1 remains at least $d$2 in the three geometries, with degrees $d$3 for triangular, rectangular, and honeycomb lattices (Nahimovs et al., 2021).

A broader numerical generalization to $d$4-dimensional periodic grids recommends

$d$5

for $d$6 marked vertices. In that formulation, the runtime scales as $d$7 for $d$8 and $d$9 for $|\circlearrowleft\rangle$0. The same work also argued that two steps of the discrete-time walk correspond to one Grover rotation in the invariant marked/unmarked subspace, which leads to an even-step stopping rule based on comparing $|\circlearrowleft\rangle$1 with $|\circlearrowleft\rangle$2 rather than with $|\circlearrowleft\rangle$3 (Carvalho et al., 2021).

Marked-state geometry can nevertheless dominate weight tuning. On the two-dimensional torus, LQWs inherit exceptional configurations from the loopless walk. If the marked vertices can be tiled by $|\circlearrowleft\rangle$4 or $|\circlearrowleft\rangle$5 dominoes, there exist stationary states, and the success probability can remain $|\circlearrowleft\rangle$6 for all $|\circlearrowleft\rangle$7, eliminating any speedup over classical exhaustive search (Nahimovs, 2018). For clustered marks, several context-specific formulas have been proposed. For a $|\circlearrowleft\rangle$8 cluster with odd $|\circlearrowleft\rangle$9, one study derived

$$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$0

and reported an increase of the marked-block probability by $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$1 relative to the non-lackadaisical case, together with a classical post-processing scheme that finds all $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$2 marks in $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$3 average time after one quantum-walk run (Saha et al., 2018). For the exceptional clustered case with $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$4 and $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$5 odd, a later study instead proposed

$$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$6

or, more robustly, the interval

$$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$7

and reported near-unit success probability in $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$8 steps (Saha et al., 2021). These results indicate that, beyond the homogeneous single-mark regime, the optimal loop prescription is highly sensitive to marking pattern.

LQWs with nonhomogeneous loop weights provide a complementary route when the graph itself is irregular. On the complete bipartite graph $$|s_c(\ell)\rangle=\frac{1}{\sqrt{d+\ell}}\left(\sum_{u\in\Gamma(v)}|u\rangle+\sqrt{\ell}\,|\circlearrowleft\rangle\right),$$9, if $d$00 marked vertices are confined to the first partite set, then for large $d$01 the success probability is improved from its non-lackadaisical value when

$d$02

and $d$03, regardless of $d$04, for the uniform initial state. For the stationary initial state under the walk, the same choice $d$05 improves the success probability without any constraint on $d$06, again independent of $d$07. When marked vertices lie in both partite sets, however, there are many configurations for which self-loops yield no improvement, no matter what weights they take (Rhodes et al., 2019).

5. Decoherence and noise-resilient search

Recent work has extended LQW search from the noiseless setting to dynamic percolation noise. On the two-dimensional torus, broken-link decoherence is modeled by independently breaking each undirected edge at every time step with probability $d$08. If an edge $d$09 is broken, amplitude that would have flowed between $d$10 and $d$11 is redistributed unitarily among the remaining outgoing coin directions at $d$12. For search with marked vertex $d$13, the oracle is

$d$14

and the one-step noisy channel is

$d$15

with $d$16 and $d$17 the corresponding marked walk on the graph where edge $d$18 is broken. The marked-vertex success probability is

$d$19

(Vieira et al., 19 Aug 2025)

The numerical findings are sharply different for loopless and lackadaisical walks. On a $d$20 torus with $d$21, the loopless search with $d$22 rapidly decays toward the uniform value $d$23, whereas choosing $d$24 keeps $d$25 peaked near its noiseless optimum $d$26. The long-time average

$d$27

shows that for $d$28 and $d$29 the distribution becomes essentially flat, but for $d$30 the marked site remains visible by a constant factor above $d$31. Up to $d$32, the same $d$33-scaling remains near-optimal, and the LQW retains $d$34, while the loopless walk degrades to $d$35 in a few steps. In that sense, self-loops act as a built-in error-mitigation mechanism against the uniformizing effect of broken-link decoherence (Vieira et al., 19 Aug 2025).

6. Symmetry breaking, directed walks, and implementation-oriented variants

Uniform self-loop weights are not necessary for the leading asymptotic behavior on vertex-transitive graphs. In a symmetry-breaking variant, only the self-loop weight at the marked vertex matters asymptotically; the remaining loop weights may be chosen randomly, provided they are small compared to the degree. For the complete graph and numerically for complete bipartite, Johnson, Paley, and hypercube examples, the broken-symmetry walk has the same leading $d$36, first-peak time $d$37, and success probability $d$38 as the homogeneous choice with $d$39 (Rapoza et al., 2021).

Directed LQWs depart even more strongly from the undirected intuition. On the directed line, there is a threshold at $d$40: for $d$41 the classical lazy walk dominates, whereas for $d$42 the quantum walk outperforms the classical one. In the large-$d$43 regime, the quantum-to-classical gain obeys

$d$44

where $d$45 parameterizes the initial coin state. A forward-biased start $d$46 maximizes the speedup, and the same two $d$47-scaling regimes appear on a directed binary tree. This is a setting in which large self-loop weight enhances rather than suppresses quantum transport (Naredi et al., 2022).

Several search-oriented extensions modify the loop sector itself. On the hypercube, a multi-self-loop lackadaisical quantum walk with partial phase inversion attaches $d$48 self-loops of individual weight $d$49 to each vertex and flips the phase of only $d$50 of them at the marked state. The reported runtime is $d$51, and for $d$52 on a $d$53-dimensional hypercube the maximum success probabilities can be pushed close to $d$54 for $d$55 to $d$56 marked vertices by appropriate choices of $d$57, $d$58, and $d$59 (Souza et al., 2023).

Implementation-oriented variants have also appeared. A lackadaisical alternating quantum walk (LAQW) on an $d$60 periodic lattice uses two coin qubits so that $d$61 and $d$62 encode motion while $d$63 and $d$64 encode self-loops. Its circuit depth scales as $d$65, compared with $d$66 for the controlled alternating quantum walk baseline. Under IBM’s FakeTorino noise model, the reported depth stays below $d$67 for $d$68, $d$69, versus $d$70 for the comparator, and the paper uses the resulting walk as a quantum entropy source for a chaos-based symmetric-key generation scheme (Gibson et al., 16 Apr 2026).

At the same time, several central prescriptions remain empirical. The exact spectral derivation of the $d$71 Grover-oracle walk on the two-dimensional grid was left open in the original rectangular-grid speedup paper, and the multi-mark results on triangular, rectangular, and honeycomb grids were explicitly presented as numerical observations rather than proofs (Wong, 2017, Nahimovs et al., 2021). A plausible implication is that the mature part of the theory is now the single-mark, regular-graph regime, whereas multiple-mark structure, symmetry breaking outside the perturbative regime, and noisy search beyond broken-link decoherence remain active analytic frontiers.