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arXiv:2610.23560

Electrostatic Limits on Quantum Spatial Search

Weikang Li

quant-phcs.CCSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

We prove that a quantum walker cannot search a two-dimensional lattice in Grover's $O(\sqrt N)$ time, resolving the long-standing open question posed by Aaronson and Ambainis in 2003. A walker that hops between neighbouring sites of an $L\times L$ lattice, $N=L^2$, needs $Ω(\sqrt{N\log N})$ local steps at least to find a marked vertex, even with unlimited memory and free queries. Hence, Tulsi's algorithm is asymptotically optimal. Our proof rests on a surprising correspondence with classical electrostatics. We treat the marked vertex as a point charge and weight the overlap between the walker's states with and without the mark by the potential drop, which vanishes where queries act. Only local steps change this weighted overlap, at a rate limited by the field energy, so the lower bound is set by the self-potential of the charge. In the plane this self-potential grows logarithmically with system size, like the Coulomb potential, leading to the extra root-logarithmic factor. Several marked vertices turn the charge into a conductor, and the lower bound depends on their arrangement through its capacitance.

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We prove that a quantum walker cannot search a two-dimensional lattice in Grover's O(N)O(\sqrt N)O(N​) time, resolving the long-standing open question posed by Aaronson and Ambainis in 2003. A walker that hops between neighbouring sites of an L×LL\times LL×L lattice, N=L2N=L^2N=L2, needs Ω(Nlog⁡N)Ω(\sqrt{N\log N})Ω(NlogN​) local steps at least to find a marked vertex, even with unlimited memory and free queries. Hence, Tulsi's algorithm is asymptotically optimal. Our proof rests on a surprising correspondence with classical electrostatics. We treat the marked vertex as a point charge and weight the overlap between the walker's states with and without the mark by the potential drop, which vanishes where queries act. Only local steps change this weighted overlap, at a rate limited by the field energy, so the lower bound is set by the self-potential of the charge. In the plane this self-potential grows logarithmically with system size, like the Coulomb potential, leading to the extra root-logarithmic factor. Several marked vertices turn the charge into a conductor, and the lower bound depends on their arrangement through its capacitance.
The spatial search problem consists of minimising the number of steps required to find a given site in a network, with the restriction that only an oracle query or a translation to a neighbouring site is allowed at each step. We propose a quantum algorithm for the spatial search problem on a triangular lattice with N sites and torus-like boundary conditions. The proposed algorithm is a special case of the general framework for abstract search proposed by Ambainis, Kempe and Rivosh (AKR) in Ambainis et al. (2005) and Tulsi in Tulsi (2008) applied to a triangular network. The AKR–Tulsi formalism was employed to show that the time complexity of the quantum search on the triangular lattice is O(Nlog⁡N)O(\sqrt{N \log N})O(NlogN​) .
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
A randomly walking quantum particle searches in Grover's Θ(N)Θ(\sqrt{N})Θ(N​) iterations for a marked vertex on the complete graph of NNN vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with NNN. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(N)Θ(\sqrt{N})Θ(N​) runtime.
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage \BHg{with} respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \cite{Childs_2014}. The numerical simulations showed that the walker finds the marked vertex in $O(N^{1/4} \log^{3/4} N) $ steps, with probability O(1/log⁡N)O(1/\log N)O(1/logN), for an overall complexity of O(N1/4log⁡7/4N)O(N^{1/4}\log^{7/4}N)O(N1/4log7/4N). We also proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm.
The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked vertex on a two-dimensional lattice. Our walk is based on the construction proposed by Falk, which tessellates the lattice with squares of size $2\ifmmode\times\else\texttimes\fi{}2$. Our walk uses minimal memory, O(NlogN)O(\sqrt{NlogN})O(NlogN​) applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a self-loop to the marked vertex---a technique we adapt from interpolated walks. We prove that with our explicit choice of self-loop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one.
A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.