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.
Diabatic quantum annealing (DQA) is an alternative algorithm to adiabatic quantum annealing that can be used to circumvent the exponential slowdown caused by small minima in the annealing energy spectrum. We present the locally suppressed transverse-field (LSTF) protocol, a heuristic method for making stoquastic optimization problems compatible with DQA. We show that, provided an optimization problem intrinsically has magnetic frustration due to inhomogeneous local fields, a target qubit in the problem can always be manipulated to create a double minimum in the energy gap between the ground and first excited states during the evolution of the algorithm. Such a double energy minimum can be exploited to induce diabatic transitions to the first excited state and back to the ground state. In addition to its relevance to classical and quantum algorithmic speedups, the LSTF protocol enables DQA proof-of-principle and physics experiments to be performed on existing hardware, provided independent controls exist for the transverse qubit magnetization fields. We discuss the implications on the coherence requirements of the quantum annealing hardware when using the LSTF protocol, considering specifically the cases of relaxation and dephasing. We show that the relaxation rate of a large system can be made to depend only on the target qubit, presenting opportunities for the characterization of the decohering environment in a quantum annealing processor.
The efficiency of Adiabatic Quantum Annealing is limited by the scaling with system size of the minimum gap that appears between the ground and first excited state in the annealing energy spectrum. In general the algorithm is unable to find the solution to an optimisation problem in polynomial time due to the presence of avoided level crossings at which the gap size closes exponentially with system size. One promising avenue being explored to produce more favourable gap scaling is the introduction of non-stoquastic XX-couplings in the form of a catalyst -of particular interest are catalysts which utilise accessible information about the optimisation problem in their construction. Here we show extreme sensitivity of the effect of an XX-catalyst to subtle changes in the encoding of the optimisation problem. We observe that catalysts designed to enhance the minimum gap at an avoided level crossing can, under certain conditions, result in closing gaps in the spectrum. To understand the origin of these closing gaps, we study how the evolution of the ground state vector is altered by the presence of the catalyst. We find that the negative components of the ground state vector are key to understanding the response of the gap spectrum. In particular the closing gaps correspond to changes in which vector components become negative over the course of the evolution. We also consider how and when these closing gaps could be utilised in diabatic quantum annealing protocols -a promising alternative to adiabatic quantum annealing in which transitions to higher energy levels are exploited to reduce the run time of the algorithm.
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