Realizable Quantum Adiabatic Search

Grover's unstructured search algorithm is one of the best examples to date for the superiority of quantum algorithms over classical ones. Its applicability, however, has been questioned by many due to its oracular nature. We propose a mechanism to carry out a quantum adiabatic variant of Grover's search algorithm using a single bosonic particle placed in an optical lattice. By studying the scaling of the gap and relevant matrix element in various spatial dimensions, we show that a quantum speedup can already be gained in three dimensions. We argue that the suggested scheme is realizable with present-day experimental capabilities.Comment: 6 pages, 4 figure

Continuous-Time Quantum Algorithms for Unstructured Problems

We consider a family of unstructured problems, for which we propose a method for constructing analog, continuous-time quantum algorithms that are more efficient than their classical counterparts. In this family of problems, which we refer to as `scrambled output' problems, one has to find a minimum-cost configuration of a given integer-valued n-bit function whose output values have been scrambled in some arbitrary way. Special cases within this set of problems are Grover's search problem of finding a marked item in an unstructured database, certain random energy models, and the functions of the Deutsch-Josza problem. We consider a couple of examples in detail. In the first, we provide a deterministic analog quantum algorithm to solve the seminal problem of Deutsch and Josza, in which one has to determine whether an n-bit boolean function is constant (gives 0 on all inputs or 1 on all inputs) or balanced (returns 0 on half the input states and 1 on the other half). We also study one variant of the random energy model, and show that, as one might expect, its minimum energy configuration can be found quadratically faster with a quantum adiabatic algorithm than with classical algorithms.Comment: 8 pages, 4 figure

Period Finding with Adiabatic Quantum Computation

We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.Comment: 6 page