Improvement of quantum walk-based search algorithms in single marked vertex graphs

TL;DR

This paper introduces generalized interpolated quantum walks that enhance success probabilities in quantum search algorithms, avoid souffle9 problems, and improve efficiency when combined with quantum fast-forwarding, with applications in quantum state construction and adiabatic processes.

Contribution

The paper proposes generalized interpolated quantum walks that improve success probability and efficiency of quantum search algorithms, avoiding souffle9 issues and enabling applications in stationary state construction.

Findings

Success probability is improved and souffle9 problems are avoided.

Reduction in walk operator calls from a0((\u03b5^{-1})a0\u221a{ ext{H}}_G) to a0(a0 ext{log}(a0 ext{log}(a0 ext{H})))

Reduced ancilla qubits from a0(a0(a0 ext{log}(a0 ext{H}))) to a0(a0 ext{log} ext{log}(a0 ext{H}))

Abstract

Quantum walks are powerful tools for building quantum search algorithms or quantum sampling algorithms named the construction of quantum stationary state. However, the success probability of those algorithms are all far away from 1. Amplitude amplification is usually used to amplify success probability, but the souffl\'e problems follow. 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 practical application of the algorithm when the optimal number of steps is not known. In this work, we define generalized interpolated quantum walks, which can both improve the success probability of search algorithms and avoid the souffl\'e problems. Then we combine generalized interpolation quantum walks with quantum fast-forwarding. The combination both reduce the times of calling walk operator of searching algorithm from $\Theta((\varepsilon^{-1})\sqrt{\Heg})$ to $\Theta(\log(\varepsilon^{-1})\sqrt{\Heg})$ and reduces the number of ancilla qubits required from $\Theta(\log(\varepsilon^{-1})+\log\sqrt{\Heg})$ to $\Theta(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})$, and the souffle problem is avoided while the success probability is improved, where $\varepsilon$ denotes the precision and $\Heg$ denotes the classical hitting time. Besides, we show that our generalized interpolated quantum walks can be used to improve the construction of quantum states corresponding to stationary distributions as well. Finally, we give an application that can be used to construct a slowly evolving Markov chain sequence by applying generalized interpolated quantum walks, which is the necessary premise in adiabatic stationary state preparation.