--- title: Lackadaisical Quantum Walks url: https://www.emergentmind.com/topics/lackadaisical-quantum-walks-lqws type: topic --- # Lackadaisical Quantum Walks 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" [2508.13462] 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 [2002.11227], [2508.13462]. ## 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 \[ C_v(\ell)=2|s_c(\ell)\rangle\langle s_c(\ell)|-I_{d+1}. \] Globally, \[ C(\ell)=\bigoplus_{v\in V} C_v(\ell), \qquad U=S\bigl(I_P\otimes C(\ell)\bigr), \] where the flip-flop shift satisfies \[ S|v,u\rangle=|u,v\rangle,\qquad S|v,\circlearrowleft\rangle=|v,\circlearrowleft\rangle. \] When \(\ell=0\), the loop sector disappears and one recovers the ordinary coined Grover walk [2002.11227]. A weighted-graph formulation makes the same construction applicable with real-valued loop weights. In that framework, the local reference state at a vertex \(v\) is \[ |s_v\rangle=\frac{1}{\sqrt{\sum_{t\sim v} w_{vt}}}\sum_{u\sim v}\sqrt{w_{vu}}\,|u\rangle, \] and the coin remains a reflection \(C_v=2|s_v\rangle\langle s_v|-I\). This formalism proves that many models with \(\ell\) identical integer self-loops are exactly reducible to a single self-loop of real-valued weight \(\ell\), which is the standard modern parameterization of LQWs [1703.10134]. 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 \[ P_{v\to u}= \begin{cases} 1/(d+\ell), & u\in\Gamma(v),\\[2mm] \ell/(d+\ell), & u=v. \end{cases} \] 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 [2002.11227], [2002.11234]. ## 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 \(1/2\) to \(1\), 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 [1502.04567]. A later weighted-graph treatment extended the complete-graph analysis to real-valued \(l\) and showed improved success probability when \(l<3+2\sqrt{2}\approx 5.828\) [1703.10134]. On the infinite line, lackadaisicality does not imply classical-style slowing. For the model with laziness parameter \(\tau\ge 0\), coin dimension \(\Delta=\tau+2\), and Grover coin \(G\in U(\Delta)\), Fourier analysis yields a nonzero localization probability at the origin, \[ P_\infty(0)=2\,\frac{\tau+4-2\sqrt{2\tau+4}}{\tau^2}, \] together with ballistic spreading. The peak velocities are \[ v_\pm(\tau)=\pm\sqrt{\frac{\tau}{\tau+2}}, \] and the variance obeys \(\mathrm{Var}[X_t]=c(\tau,\alpha,\beta)t^2+o(t^2)\), so the walk remains ballistic for arbitrary \(\tau\) [1612.03370]. A related one-dimensional literature, phrased in terms of lazy quantum walks with a three-state coin, reached a compatible conclusion: the \(n\)th moment scales as \(O(t^n)\), the support remains linear in \(t\), and the occupancy rate fluctuates around a positive constant as \(t\to\infty\), 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 [1412.6891]. ## 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 \(O(1/\log N)\) in \(O(\sqrt{N\log N})\) steps, so amplitude amplification is required and the overall runtime is \(O(\sqrt N\log N)\). Adding a self-loop of weight \(4/N\) changes the behavior qualitatively: the marked-vertex probability reaches a constant near \(1\) in \(O(\sqrt{N\log N})\) steps, so no amplitude amplification is needed and the overall runtime becomes \(O(\sqrt{N\log N})\). In Wong’s simulations on a \(64\times 64\) grid, \(\max_t p(t)\approx 0.9755\) at \(t\approx 170\), and the fitted first-peak time is \(t_*(N)\approx 0.922466\sqrt{N\ln N}\) with correlation coefficient \(0.999993\) [1706.06939]. This \(4/N\) prescription is part of a broader pattern. For a unique marked vertex on a \(d\)-regular vertex-transitive graph, Rhodes and Wong proposed that the optimal self-loop weight is \[ \ell_{\mathrm{opt}}=\frac{d}{N}. \] 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 \(P_{\max}\approx 1\) at \(\ell=d/N\), with quadratic speedup \(T=\Theta(\sqrt N)\) in the tested families [2002.11227]. Høyer and Yu subsequently proved the corresponding search theorem on any regular locally arc-transitive graph. Choosing \(\ell=d/N\), the LQW finds a unique marked vertex with constant success probability in \[ T=O\bigl(\sqrt{\mathrm{HT}(P,\{m\})}\bigr), \] where \(\mathrm{HT}(P,\{m\})\) is the classical hitting time. This analytic result subsumes prior conjectures and numerical findings for the torus, cycle, and Johnson graphs, and explains the \(d/N\) rule through the relation between coined LQWs and quantum interpolated walks [2002.11234]. The same degree-over-size prescription also appears in other two-dimensional lattices. On triangular and honeycomb grids, adding self-loops of weight \(6/N\) and \(3/N\), respectively, yields \(O(\sqrt{N\log N})\) running time, matching the rectangular-grid mechanism after replacing the degree \(d\) by \(6\) or \(3\) [2007.13564]. ## 4. Multiple marked vertices, exceptional configurations, and nonuniform weights For multiple marked vertices, the single-mark prescription \(\ell=d/N\) 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 \(m\) of marked vertices if one keeps \(\ell=d/N\). Scanning over weights instead reveals a distinct optimum near \[ \ell\approx \frac{md}{N}, \] and choosing exactly \(\ell=md/N\) restores \(p_M^{\max}=\Theta(1)\) with first-peak time \[ T=\Theta\!\left(\sqrt{\frac{N}{m}\log\frac{N}{m}}\right). \] For random marked sets on a \(100\times 100\) grid, the average \(p_M^{\max}\) remains at least \(0.45\) in the three geometries, with degrees \(d=6,4,3\) for triangular, rectangular, and honeycomb lattices [2104.09955]. A broader numerical generalization to \(d\)-dimensional periodic grids recommends \[ \ell_{\mathrm{opt}}=\frac{2dM}{N} \] for \(M\) marked vertices. In that formulation, the runtime scales as \(O(\sqrt{N\log(N/M)/M})\) for \(d=2\) and \(O(\sqrt{N/M})\) for \(d>2\). 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 \(P_{\mathrm{marked}}(t)\) with \(P_{\mathrm{marked}}(t-2)\) rather than with \(P_{\mathrm{marked}}(t-1)\) [2106.06274]. 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 \(1\times 2\) or \(2\times 1\) dominoes, there exist stationary states, and the success probability can remain \(O(1/N)\) for all \(t\), eliminating any speedup over classical exhaustive search [1808.00672]. For clustered marks, several context-specific formulas have been proposed. For a \(\sqrt{k}\times\sqrt{k}\) cluster with odd \(k\), one study derived \[ w^*=\frac{4}{N\bigl(k+\lfloor \sqrt{k}/2\rfloor\bigr)} \] and reported an increase of the marked-block probability by \(\sim 0.2\) relative to the non-lackadaisical case, together with a classical post-processing scheme that finds all \(k\) marks in \(O(\sqrt{k})\) average time after one quantum-walk run [1804.01446]. For the exceptional clustered case with \(k=n^2\) and \(n\) odd, a later study instead proposed \[ l_*=\frac{4}{N(k+1)} \] or, more robustly, the interval \[ \frac{4}{N(k+2)}\le l\le \frac{4}{Nk}, \] and reported near-unit success probability in \(O(\sqrt N)\) steps [2107.02049]. 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 \(K_{N_1,N_2}\), if \(k\) marked vertices are confined to the first partite set, then for large \(N_1,N_2\) the success probability is improved from its non-lackadaisical value when \[ l_1=\frac{kN_2}{2N_1} \] and \(N_2>(3-2\sqrt2)N_1\), regardless of \(l_2\), for the uniform initial state. For the stationary initial state under the walk, the same choice \(l_1=kN_2/(2N_1)\) improves the success probability without any constraint on \(N_2/N_1\), again independent of \(l_2\). 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 [1905.05887]. ## 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 \(p\). If an edge \((v,w)\) is broken, amplitude that would have flowed between \(v\) and \(w\) is redistributed unitarily among the remaining outgoing coin directions at \(v\). For search with marked vertex \(m\), the oracle is \[ R=I_C\otimes\bigl(I_P-2|m\rangle\langle m|\bigr), \] and the one-step noisy channel is \[ \mathcal E(\rho)= (1-p)\,U'\rho\,U'^\dagger + p\sum_e U'_e\,\rho\,U'_e{}^\dagger, \] with \(U'=R[S(C\otimes I)]\) and \(U'_e\) the corresponding marked walk on the graph where edge \(e\) is broken. The marked-vertex success probability is \[ P_m(t)=\mathrm{Tr}\bigl[(I_C\otimes |m\rangle\langle m|)\rho_t\bigr], \qquad \rho_t=\mathcal E(\rho_{t-1}). \] [2508.13462] The numerical findings are sharply different for loopless and lackadaisical walks. On a \(16\times 16\) torus with \(p=0.01\), the loopless search with \(\ell=0\) rapidly decays toward the uniform value \(1/N\), whereas choosing \(\ell\approx 4/N\) keeps \(P_m(t)\) peaked near its noiseless optimum \(\sim O(1)\). The long-time average \[ \overline P(x,y)=\frac1T\sum_{t=1}^T P_t(x,y) \] shows that for \(\ell=0\) and \(p>0\) the distribution becomes essentially flat, but for \(\ell>0\) the marked site remains visible by a constant factor above \(1/N\). Up to \(p\sim 1\%\), the same \(\ell\)-scaling remains near-optimal, and the LQW retains \(P_m\gg 1/N\), while the loopless walk degrades to \(P_m\approx 1/N\) 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 [2508.13462]. ## 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 \(p(t)\), first-peak time \(T=\Theta(\sqrt N)\), and success probability \(p_{\mathrm{succ}}=\Theta(1)\) as the homogeneous choice with \(\ell^*\approx d/N\) [2108.13856]. Directed LQWs depart even more strongly from the undirected intuition. On the directed line, there is a threshold at \(\ell=1\): for \(\ell<1\) the classical lazy walk dominates, whereas for \(\ell>1\) the quantum walk outperforms the classical one. In the large-\(\ell\) regime, the quantum-to-classical gain obeys \[ G(\ell,\alpha)\simeq \frac{\ell}{\alpha+1}, \] where \(\alpha\) parameterizes the initial coin state. A forward-biased start \(\alpha=0\) maximizes the speedup, and the same two \(\ell\)-scaling regimes appear on a directed binary tree. This is a setting in which large self-loop weight enhances rather than suppresses quantum transport [2211.06167]. Several search-oriented extensions modify the loop sector itself. On the hypercube, a multi-self-loop lackadaisical quantum walk with partial phase inversion attaches \(m\) self-loops of individual weight \(l'=l/m\) to each vertex and flips the phase of only \(s