Optimal quantum spatial search with one-dimensional long-range interactions

TL;DR

This paper demonstrates that one-dimensional spin chains with long-range interactions decaying as 1/r^α can perform optimal quantum spatial search in O(√n) time, revealing a phase transition at α≈1.5 and robustness to noise.

Contribution

It proves the possibility of optimal quantum spatial search in 1D long-range interacting systems, extending beyond nearest-neighbor models.

Findings

Optimal search achieved for α≈1 with high fidelity.

A phase transition at α≈1.5 determines search feasibility.

Spatial search remains robust under realistic noise conditions.

Abstract

Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in $\mathcal O(\sqrt n)$ time for $n$ entries. However the capability of models found in nature is largely unexplored - e.g., in one dimension only nearest-neighbour Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with $\mathcal O(\sqrt n)$ run time and large fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as $1/r^\alpha$ with distance $r$. In particular, near unit fidelity is achieved for $\alpha\approx 1$ and, in the limit $n\to\infty$, we find a continuous transition from a region where optimal spatial search does exist ($\alpha<1.5$) to where it does not ($\alpha>1.5$). Numerically, we show that spatial search is robust to dephasing noise and that, for realistic conditions, $\alpha \lesssim 1.2$ should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.