---
title: Lackadaisical Quantum Walk
url: https://www.emergentmind.com/topics/lackadaisical-quantum-walk
type: topic
---

# Lackadaisical Quantum Walk

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A lackadaisical quantum walk (LQW) is a discrete-time coined quantum walk in which each vertex is endowed with a self-loop, allowing the walker to remain at its current vertex with nonzero amplitude. In the standard weighted formulation, the self-loop is assigned a real-valued weight \(l\), so the coin space is enlarged by one “stay-put” basis state and the local Grover reflection is taken about a weighted uniform superposition. Introduced as a quantum analogue of the classical lazy random walk, the LQW has become a tunable framework for spatial search, dispersion, and related algorithmic tasks on complete graphs, grids, hypercubes, and broader classes of regular or vertex-transitive graphs. Its central feature is that the self-loop weight is not merely decorative: depending on the graph, oracle, coin, and marked-set geometry, it can improve success probability to \(1\), slow the walk, leave the dynamics unchanged, or preserve search performance under noise [1502.04567].

## 1. Formal definition and weighted-self-loop construction

In a regular coined walk of degree \(d\), the lackadaisical modification enlarges the coin space from dimension \(d\) to \(d+1\) by adjoining a self-loop state. A standard choice is the weighted coin state
\[
|s_c\rangle=\frac{1}{\sqrt{d+l}}\left(\sum_{i=0}^{d-1}\ket{i}+\sqrt{l}\ket{\circlearrowleft}\right),
\]
with Grover-type coin
\[
C=2|s_c\rangle\langle s_c|-I_{d+1}.
\]
The shift acts as usual on directional states and trivially on the self-loop, so \(S\ket{i,j,\circlearrowleft}=\ket{i,j,\circlearrowleft}\) on grids and analogous identities hold on other graphs [2002.11227].

For search, the walk is typically modified by a phase oracle on the marked set. On regular grids and other symmetric graphs, a standard search step is written as
\[
U' = U\,(Q\otimes I_{d+1}),
\]
where \(U=S(I\otimes C)\) and \(Q\) flips the phase of marked vertices [2104.09955]. The initial state is usually the uniform superposition over vertices tensored with the weighted coin state \(|s_c\rangle\).

A foundational refinement is the weighted-graph formulation. A coined quantum walk on a weighted graph replaces the local uniform state by a weighted superposition of outgoing directions, and \(l\) identical self-loops can be merged into a single self-loop of real-valued weight \(l\). This generalization removes the integrality restriction from early formulations and makes lackadaisical walks naturally compatible with weighted-graph and Szegedy-walk viewpoints [1703.10134].

## 2. Complete-graph search and the original lackadaisical effect

The original complete-graph analysis established that the effect of self-loops is sharply coin-dependent. For search for a marked vertex on the complete graph of \(N\) vertices, the discrete-time walk with the phase flip coin exhibits the characteristic lackadaisical enhancement: with no self-loop (\(l=0\)), the success probability is \(1/2\); with one self-loop (\(l=1\)), the success probability becomes \(1\); and with additional self-loops the success probability decreases again. The runtime remains Grover-like, \(O(\sqrt{N})\), so the self-loop changes the constant factor and peak success probability rather than the \(\sqrt{N}\) scaling [1502.04567].

With the Ambainis–Kempe–Rivosh coin, the situation is different. Adding self-loops simply slows down the search, and the best performance is obtained with the smallest possible \(l\). The asymptotic comparison is also coin-dependent: with the phase flip coin, the walk is faster than classical when \(l\) scales less than \(N\), whereas with the AKR coin the requirement is that \(l\) scale less than \(N^2\) [1502.04567].

The complete-graph setting also isolates a contrast with continuous-time search. In continuous time, adding self-loops makes no difference at all: the walk still rotates from the uniform initial state to the marked state in time
\[
t=\frac{\pi\sqrt{N}}{2},
\]
with energy gap
\[
\Delta E=\frac{2}{\sqrt{N}}.
\]
Thus, on the complete graph, discrete-time and continuous-time lackadaisical modifications are not interchangeable [1502.04567].

The weighted formulation sharpened the complete-graph picture further. On the weighted complete graph, the loopless success probability \(1/2\) is improved whenever
\[
l < 3 + 2\sqrt{2} \approx 5.828,
\]
and in the regime \(l\ge \frac13\), \(l=o(N)\), the peak success probability is
\[
p_* \approx \frac{4l}{(l+1)^2}.
\]
This made the self-loop weight a continuous search parameter rather than merely a count of integer loops [1703.10134].

## 3. Two-dimensional grids and lattice families

On the periodic square grid, the standard loopless coined walk reaches success probability \(O(1/\log N)\) after \(O(\sqrt{N\log N})\) steps, so amplitude amplification is required and the overall runtime is \(O(\sqrt{N}\log N)\). Adding a self-loop of weight
\[
l=\frac{4}{N}
\]
changes this regime: the success probability becomes a constant near \(1\) in \(O(\sqrt{N\log N})\) steps, eliminating amplitude amplification and yielding an \(O(\sqrt{\log N})\) improvement over the typical loopless algorithm [1706.06939].

The same mechanism extends beyond the square lattice. On triangular and honeycomb 2D grids, adding a self-loop of weight \(6/N\) and \(3/N\), respectively, results in \(O(\sqrt{N\log N})\) running time with constant success probability. The numerators match the vertex degrees of the corresponding lattices, suggesting a degree-over-\(N\) rule for single-marked-vertex search on regular 2D geometries [2007.13564].

This degree-over-\(N\) pattern was elevated from numerical regularity to a broader structural principle. For vertex-transitive graphs with a unique marked vertex, the optimal self-loop weight was conjectured to be
\[
\ell=\frac{d}{N},
\]
where \(d\) is the degree of the loopless graph. The evidence included complete graphs, cycles, tori, complete bipartite graphs, strongly regular graphs, Johnson graphs, cubic lattices, and hypercubes [2002.11227]. A later analytic result proved that on any regular locally arc-transitive graph, choosing \(\ell=d/N\) yields constant-success search in \(O(\sqrt{HT})\) steps, where \(HT\) is the classical hitting time, thereby rigorously establishing a quadratic speedup for a broad graph class [2002.11234].

## 4. Multiple marked vertices, geometry, and exceptional configurations

The single-mark prescription does not generally survive the transition to multiple marked vertices. On a \(\sqrt{N}\times\sqrt{N}\) grid, lackadaisical search retains exceptional configurations analogous to those of the loopless walk. For two adjacent marked vertices, a stationary state exists and the probability of finding a marked vertex satisfies
\[
p_M=O\!\left(\frac{1}{N}\right),
\]
which is the same order as classical exhaustive search. More generally, blocks tileable by \(1\times 2\) and \(2\times 1\) sub-blocks admit similar stationary-state constructions, so self-loops do not eliminate the exceptional-configuration phenomenon [1808.00672].

Away from such obstructions, the optimal loop weight scales with the number and arrangement of marked vertices. On triangular, rectangular, and honeycomb 2D grids with multiple marked vertices, numerical study showed that
\[
l = m\cdot \frac{d}{N}
\]
gives \(O(1)\) success probability, with runtime
\[
O\!\left(\sqrt{\frac{N}{m}\log\frac{N}{m}}\right).
\]
The same work reported that the single-mark choice \(l=d/N\) degrades as the number of marked vertices \(m\) increases, whereas \(l=md/N\) tracks near-optimal behavior across the tested lattices [2104.09955].

For clustered marked states, geometry becomes decisive. When \(k\) marked states form a \(\sqrt{k}\times\sqrt{k}\) cluster in a 2D grid, the tuned choice
\[
l=\frac{4}{N\left(k+\left\lfloor\frac{\sqrt{k}}{2}\right\rfloor\right)}
\]
with \(k\) odd increases the success probability by \(\sim 0.2\), producing probabilities close to \(0.8\) for \(\sqrt{N}\le 14\) and above \(0.95\) for larger \(N\) in the reported simulations [1804.01446]. A related later study of the exceptional odd-square-cluster regime proposed a narrow optimal band
\[
\frac{4}{N(k+2)} \le l \le \frac{4}{Nk},
\]
with only one value in that interval effectively optimal for a given configuration and first-peak success probability close to \(1\) for \(k=9\) across grid sizes from \(10\times 10\) to \(100\times 100\) [2107.02049].

The dependence on marked-set density and spacing was also made explicit in higher-dimensional grid studies. For multiple solutions on \(d\)-dimensional grids, a proposed optimal adjustment is
\[
l=\frac{2dm}{N},
\]
and the reported success probability is inversely related to solution density and directly related to the relative spacing between marked vertices [2106.06274].

## 5. General graph-theoretic theory and heterogeneous or symmetry-broken models

The most general analytic theory presently available connects lackadaisical walks to quantum interpolated walks. On a \(d\)-regular locally arc-transitive graph with one marked vertex, the self-loop choice
\[
\ell=\frac{d}{N}
\]
makes the lackadaisical walk equivalent, under an isometry, to a quantum interpolated walk with interpolation parameter \(s=1-\ell/d\). This correspondence yields constant success probability in
\[
O\!\left(\sqrt{HT(P(G),\{m\})}\right)
\]
steps and explains why the same self-loop rule had previously appeared across torus, cycle, Johnson, and related graph families [2002.11234].

The homogeneous-loop assumption is not essential. On vertex-transitive graphs, symmetry can be broken at all unmarked vertices without changing the asymptotic speedup: only the marked vertex’s self-loop weight matters, and the remaining self-loop weights may be chosen randomly provided they are small compared to the graph degree [2108.13856]. This result separates the algorithmic role of the marked vertex from the symmetry of the background graph.

Irregular graphs require genuinely nonhomogeneous weights. On the complete bipartite graph with partite sets of sizes \(N_1\) and \(N_2\), if the \(k\) marked vertices lie in the first partite set, then with the typical initial uniform state the success probability is improved from its non-lackadaisical value when
\[
l_1=\frac{kN_2}{2N_1}
\]
and
\[
N_2 > (3 - 2\sqrt{2}) N_1,
\]
regardless of \(l_2\). With the stationary initial state, the same \(l_1\) improves the success probability without a constraint on the ratio \(N_1/N_2\), again independently of \(l_2\). When marked vertices lie in both partite sets, there are many configurations for which self-loops yield no improvement, no matter what weights they take [1905.05887].

Directed lackadaisical walks reveal a different phenomenon. On a directed line and a directed binary tree, two regimes occur: for \(l<1\), the classical walker dominates; for \(l>1\), the quantum walker dominates. In the quantum-dominated regime, the mean position exhibits two scaling laws,
\[
\langle x\rangle \propto l^{-1/2}
\quad\text{and}\quad
\langle x\rangle \propto l^{-1},
\]
with crossover controlled by \(t^*=t/\sqrt{l}\). This shows that large self-loop strength need not destroy quantum speedup [2211.06167].

## 6. Line dynamics, hypercube search, noise, and later extensions

The term “lackadaisical” can be misleading outside search settings. On the line, with \(\tau\) self-loops per vertex and Grover coin of dimension \(\tau+2\), the walker exhibits localization at the origin with asymptotic probability
\[
\hat P_0=2\cdot\frac{\tau+4-2\sqrt{2\tau+4}}{\tau^2},
\]
independent of the initial left/right amplitudes \(\alpha,\beta\) as long as the initial coin state lies in the left/right subspace. At the same time, the traveling peaks move with velocities
\[
v_R=\sqrt{\frac{\tau}{\tau+2}},\qquad
v_L=-\sqrt{\frac{\tau}{\tau+2}},
\]
and the walk spreads ballistically for arbitrary \(\tau\), with variance scaling as \(t^2\). Hence the lackadaisical quantum walker is “not lazy at all” in the classical sense [1612.03370].

On the hypercube, multiple marked vertices introduce a distinct obstacle: even without self-loops, neighbor vertices are amplified and approximately \(1/2\) of the system energy is concentrated in them. The proposed self-loop value for multiple marked vertices is
\[
l=\frac{n}{N}\cdot k,
\]
which yields success probability close to \(1\) for non-adjacent marked vertices. Adjacent marked vertices remain harder, and in that regime the success probability is directly proportional to the density of marked vertices in the neighborhood [2108.09399]. A more elaborate extension, the multi-self-loop lackadaisical quantum walk with partial phase inversion, distributes a fixed total loop weight \(l\) across \(m\) self-loops with \(l'=l/m\) and flips only a chosen subset of self-loop phases. On the hypercube this variant reported success probabilities close to \(1\) in
\[
O(\sqrt{(n+m)\cdot N})
\]
for \(1\) to \(12\) marked vertices in the tested regime [2305.01121].

Robustness under decoherence has also become part of the subject. Under broken-link decoherence on a 2D toroidal grid, the loopless walk is driven toward a uniform distribution and loses its search capability, whereas the inclusion of self-loops preserves marked-vertex probability significantly above the uniform baseline
\[
P_{\text{uniform}}=\frac{1}{N}.
\]
This indicates that the lackadaisical modification can function as a robustness mechanism, not only as a noiseless optimization [2508.13462].

A recent implementation-oriented extension is the lackadaisical alternating quantum walk on a \(2^n\times 2^n\) lattice. By embedding the self-loop as a stay-put branch in an alternating-walk architecture, it achieves circuit depth \(\mathcal{O}(n^2+nt)\), compared with \(\mathcal{O}(n^2 t)\) for the referenced controlled alternating quantum walk baseline, and was proposed as a NISQ-friendly entropy source for chaos-based symmetric-key generation [2604.15030].

The accumulated literature therefore presents the lackadaisical quantum walk not as a single algorithmic trick but as a family of self-loop-tuned coined walks. Its main invariant lesson is that self-loops are a genuine control parameter. They can enhance search on complete graphs and regular lattices, admit rigorous \(O(\sqrt{HT})\) theory on regular locally arc-transitive graphs, fail on exceptional marked configurations, require retuning for multiple marked vertices, and in some settings even accelerate ballistic transport rather than suppress it.

Source: https://www.emergentmind.com/topics/lackadaisical-quantum-walk