Optimal Quantum Spatial Search with One-Dimensional Long-Range Interactions

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 Oð ffiffiffi n p Þ time for n entries. However, the capability of models found in nature is largely unexplored-e.g., in one dimension only nearest-neighbor Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with Oð ffiffiffi n p Þ run time and high fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as 1=r α with distance r. In particular, near unit fidelity is achieved for α ≈ 1 and, in the limit n → ∞, we find a continuous transition from a region where optimal spatial search does exist (α < 1.5) to where it does not (α > 1.5). Numerically, we show that spatial search is robust to dephasing noise and that, for reasonable chain lengths, α ≲ 1.2 should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.

Ion trap long-range XY model for quantum state transfer and optimal spatial search

“…Classically, this problem requires O(N) queries (or, equivalently, time), where N is the number of nodes in the graph-the number of spins in the spin chain. Optimal spatial search is solving this problem with a quantum algorithm [13] in O( √ N) time, and, using the walk Hamiltonian of equation (20), this is possible for α < 1.5 [24]. The quantum algorithm starts by initialising the quantum state of the system as an equal superposition of single-excitation states |s⟩ =…”

“…The protocol works in the regime where optimal spatial search is possible in the long-range interaction setting [24]. Although the fidelity decreases as the interaction strength increases, these protocols will allow quantum state transfer for α < 1.5.…”

“…This analysis assumes the Hamiltonian can be accurately described by a degenerate subspace of the three most significant states. The degeneracy of the subspace is a good approximation for well-chosen γ, however, it does therefore rely on the same spectral gap conditions as optimal spatial search [22,24].…”

“…A number of high dimensional graphs were subsequently found that permit optimal spatial search [14][15][16][17][18][19][20][21][22][23]. Recently, optimal spatial search in one dimension using long-range interactions was found to be possible [24]. However, does this translate to a physical implementation?…”

Spatial search by continuous-time quantum walks on renormalized Internet networks

“…Later research has shown that spatial search by CTQW is optimal for other graph topologies such as the star graph [14,15], graphs with broken links [14], fractal graphs [16], Erdős-Rényi graphs [17], and 1D graphs with long-range interactions [18]. However, general conditions for a graph to be efficiently searchable are not known.…”