Optimal spatial searches with long-range tunneling

TL;DR

This work establishes when long-range, power-law tunneling $1/r^{\alpha}$ enables Grover-optimal spatial search on low-dimensional lattices. By analyzing the continuous-time quantum walk generated by $\hat{H}_{\alpha}=-\gamma_0 L_{\alpha}-|w\rangle\langle w|$ and showing a two-state reduction at the critical point, the authors derive how the search time scales with system size $N$ and how the effective dimensionality $d_s=\frac{2d}{\alpha-d}$ governs optimality. They identify a critical exponent $\alpha_c=\frac{3d}{2}$: for $\alpha<\alpha_c$ the spectral gap supports Grover-like scaling in $d\le 4$ via mapping to a short-range model in dimension $d_s>4$, while $\alpha>\alpha_c$ leads to suboptimal scaling. The results provide quantitative guidance for implementing efficient analog quantum searches on platforms with tunable long-range interactions (e.g., Rydberg atoms, trapped ions, optical cavities), linking fundamental criticality to practical quantum information processing. Overall, the paper reveals how long-range connectivity can compensate for low physical dimensionality by effectively elevating the spectral dimension, enabling resource-efficient quantum search protocols.

Abstract

A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time $T$ can exhibit Grover's optimal scaling with the lattice size $N$, namely, $T\sim \sqrt{N}$, on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of $d>4$ dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance $r$ as $1/r^α$ and $0<α<3d/2$. For $α<d$ the dynamics mimics the one of a globally connected graph. For $d<α<d+2$, the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension $d_s=2d/(α-d)$, indicating that the search is optimal for $d_s>4$. Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.

Figures (12)

• Figure 1: (a) Illustrative graphic of search on a cubic lattice (hypercube with $d=3$) with nearest-neighbor couplings ($\alpha\rightarrow\infty$). Target node is depicted in red. (b) Schematic of the power-law scaling of the connectivity of a single site $\vec{i}$ in a two-dimensional cubic lattice.
• Figure 2: Participation ratio $1/\sum_j\vert\langle j \vert\psi_0\rangle \vert^4$ for the ground state $\vert\psi_0\rangle$vs$\gamma_0$, with the critical value $\gamma_0=\gamma_c$ separating the extended from the localized ground state. Inset shows scalar products $\vert\mathcal{S}_{i=0,1}\vert^2$ and $\vert\mathcal{W}_{i=0,1}\vert^2$ and the energy gap $E_1-E_0$ as a function of parameter $\gamma_0$. We set $d=1$ and $\alpha=0.6$; qualitatively similar results hold for other dimensions and tunneling exponents.
• Figure 3: Search optimality diagram illustrating the connection between the spatial dimension $d\in\mathbb{Z}$, in which the hypercube is embedded, and the power-law tunneling exponent $\alpha$, and how it relates to the time complexity. The critical exponent $\alpha_c=3d/2$, represented by the white points, separates regimes in which we have optimal ($d_s>4$, light blue) and suboptimal ($d_s<4$, dark blue) quantum spatial search, distinguished by the critical spectral dimension $d_{s}=4$.
• Figure 4: Upper bound to the search fidelity: $\chi_\alpha$ as a function of the number of lattice sites $N$ and long-range tunneling exponent $\alpha$ for (a) $d=1$, (b) $d=2$, and (c) $d=3$. Vertical lines correspond to $\alpha=d$ (solid) and $\alpha=\alpha_c=3d/2$ (dashed). Contours are shown for $\chi_\alpha=0.999,\,0.99,\,0.9$. (d) Asymptotic behavior of $\chi_\alpha$, Eq. \ref{['eq:chi']}, in the limit $N\rightarrow\infty$ as a function of $\alpha/d$. In Appendix \ref{['app:fidelity']} we show that, for $0<\alpha<d$, $\chi_\alpha=1$, while for $d<\alpha<3d/2$ it decreases monotonically to zero as $\chi_\alpha=\sqrt{3-2\alpha/d}/(2-\alpha/d)$.
• Figure A.1: Spectral gap $\Delta_\alpha$ of a one-dimensional hypercube as a function of (a) the power-law exponent $\alpha\in[0,1)$, (b) the power-law exponent $\alpha\in(1,3)$, and (c) the system size $N$, i.e. the number of vertices in the hypercube graph. Data points correspond to exact numeric results, while solid curves represent the asymptotic results in Eq. \ref{['eq:gap_scaling']}, with the constants $\mathscr{C}^{(1)}_{i=1,2}$ defined in Table \ref{['tab:spectral_gap_rescaled']}, see Eqs. \ref{['eq:spectral_gap_D1_asymptotic_1']} and \ref{['eq:spectral_gap_D1_asymptotic_2']}.