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.
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.
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 soufflé 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 soufflé 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 $Θ((\varepsilon^{-1})\sqrt{\Heg})$ to $Θ(\log(\varepsilon^{-1})\sqrt{\Heg})$ and reduces the number of ancilla qubits required from $Θ(\log(\varepsilon^{-1})+\log\sqrt{\Heg})$ to $Θ(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})$, and the souffle problem is avoided while the success probability is improved, where ε 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.
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.
Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC'20) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al.
We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) P on a graph with a set of unknown marked vertices, one can define a related absorbing walk P′ where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s) from the interpolated Markov chain $P(s)=(1\ensuremath{-}s)P+{\mathit{sP}}^{'}$ and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that, for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speedup for state-transitive reversible Markov chains with a unique marked vertex.
n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter ϵ. More specifically, for an N-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps h satisfies h≥ln(ϵ2)N, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk P and a single marked state, the quantum walk corresponding to P can find the solution using a number of queries that is quadratically smaller than the classical hitting time of P. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.