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.
Parafermions are Zn generalisations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. We study the simplest case of the Z3 parafermion chain and investigate the form of the non-topological gate that arises through direct short-range interaction of the parafermion edge modes. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, with the strongest order of the interaction generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We also illustrate the accessibility of highly non-contextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology.
The toric code is a simple and exactly solvable example of topological order realizing Abelian anyons. However, it was shown to support nonlocal lattice defects, namely twists, which exhibit non-Abelian anyonic behavior [Phys. Rev. Lett. 105, 030403 (2010)]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli-X upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed for quantum information tasks.
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