A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.
A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.
A randomly walking quantum particle searches in Grover's Θ(N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(N) runtime.
Abstract An ideal quantum walk transitions from one vertex to another with perfect fidelity, but in physical systems, the particle may be hindered by potential energy barriers. Then the particle has some amplitude of tunneling through the barriers, and some amplitude of staying put. We investigate the algorithmic consequence of such barriers for the quantum walk formulation of Grover’s algorithm. We prove that the failure amplitude must scale as O ( 1 / N ) for search to retain its quantum O ( N ) runtime; otherwise, it searches in classical O ( N ) time. Thus searching larger ‘databases’ requires increasingly reliable hop operations or error correction. This condition holds for both discrete- and continuous-time quantum walks.
The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N , while the second requires that l scale less than N 2 . Finally, continuous-time quantum walks differ from both of these discrete-time examples—the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.
When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.
Abstract Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities are far below 1. Amplitude amplification is usually used to amplify success probability, but the soufflé problem follows. Only stop at the right step can we achieve a maximum success probability. Otherwise, as the number of steps increases, the success probability may decrease, which will cause troubles in the practical application of the algorithm when the optimal number of steps is unknown. In this work, we define generalized interpolated quantum walks instead of amplitude amplification, which can not only improve the success probability but also avoid the soufflé problem. We choose a special case of generalized interpolated quantum walks and construct a series of new search algorithms based on phase estimation and quantum fast-forwarding, respectively. Especially, by combining our interpolated quantum walks with quantum fast-forwarding, we both reduce the time complexity of the search algorithm from Θ ( ( ε − 1 ) H T ) to Θ ( log ( ε − 1 ) HT ) and reduce the number of ancilla qubits required from Θ ( log ( ε − 1 ) + log H T ) to Θ ( log log ( ε − 1 ) + log H T ) , where ɛ denotes the precision and H T denotes the classical hitting time. In addition, we show that our generalized interpolated quantum walks can improve the construction of quantum stationary states corresponding to reversible Markov chains. Finally, we give an application to construct a slowly evolving Markov chain sequence by applying generalized interpolated quantum walks, which is the necessary premise in adiabatic stationary state preparation.
There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the souffl\'e problem, which mainly means that the success probability of finding a marked vertex could shrink dramatically, even to zero, when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the souffl\'e problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the souffl\'e problem for quantum walk-based search algorithms by presenting a quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $\ensuremath{\epsilon}$, the quantum algorithm can find a marked vertex on an N-vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps h satisfies $h\ensuremath{\ge}ln(\frac{2}{\sqrt{\ensuremath{\epsilon}}})\sqrt{N}+1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.