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arXiv:2209.04162

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

Li, Xinying, Yun Shang

Published Sep 9, 2022• 0 citations
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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 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 $\varepsilon$ 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.

DOI
10.48550/arxiv.2209.04162

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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((ε−1)\Heg)Θ((\varepsilon^{-1})\sqrt{\Heg})Θ((ε−1)\Heg​) to Θ(log⁡(ε−1)\Heg)Θ(\log(\varepsilon^{-1})\sqrt{\Heg})Θ(log(ε−1)\Heg​) and reduces the number of ancilla qubits required from Θ(log⁡(ε−1)+log⁡\Heg)Θ(\log(\varepsilon^{-1})+\log\sqrt{\Heg})Θ(log(ε−1)+log\Heg​) to Θ(log⁡log⁡(ε−1)+log⁡\Heg)Θ(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})Θ(loglog(ε−1)+log\Heg​), and the souffle problem is avoided while the success probability is improved, where ε\varepsilonε denotes the precision and \Heg\Heg\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.
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 NNN-vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps hhh 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.
Sampling from the stationary distribution is one of the fundamental tasks of Markov-chain-based algorithms and has important applications in machine learning, combinatorial optimization, and network science. For the quantum case, quantum sampling from Markov chains corresponds to preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. A different quantum sampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. We build a quantum sampling algorithm that not only accelerates nonregular graphs, but also keeps the speed up of existing quantum algorithms for regular graphs. In nonregular graphs, the invocation of the quantum fast-forwarding algorithm accelerates previous state-of-the-art quantum sampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms, we decrease the runtime by a factor of lognlognlogn, where nnn is the number of graph vertices. In regular graphs, our results match other quantum algorithms and the reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared with the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm achieves complete quadratic acceleration (without log factor) over the classical case without any limit. In addition, we construct a different reflection around the stationary state with fewer ancilla qubits and believe it may have an independent application.
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\epsilonϵ. More specifically, for an NNN-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps hhh satisfies h≥ln⁡(2ϵ)Nh \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}h≥ln(ϵ​2​)N​, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given ϵ∈(0,1]\epsilon\in (0,1]ϵ∈(0,1]. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.
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) PPP on a graph with a set of unknown marked vertices, one can define a related absorbing walk P′{P}^{'}P′ where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s)H(s)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.
Quantum-walk algorithms can quadratically reduce the dependence of spatial search on the classical hitting time HT\mathrm{HT}HT. Realising this speedup using standard quantum-walk algorithms, however, requires deep coherent circuits, limiting their applicability on near-term and early fault-tolerant quantum devices. We derive a parameterised family of hybrid quantum-classical spatial search algorithms that continuously interpolates between the classical and fully coherent quantum regimes, trading off circuit depth for total query complexity. For any γ∈[0,1]γ\in[0,1]γ∈[0,1], our algorithms use circuits of depth O~(HTγ/2)\widetilde{O}(\mathrm{HT}^{γ/2})O(HTγ/2) and find a marked vertex with constant success probability at total cost O~(HT1−γS+HT1−γ/2(U+C))\widetilde{O}(\mathrm{HT}^{1-γ}\mathsf{S} + \mathrm{HT}^{1-γ/2}(\mathsf{U} + \mathsf{C}))O(HT1−γS+HT1−γ/2(U+C)), where S\mathsf{S}S, U\mathsf{U}U, and C\mathsf{C}C denote the setup, update, and checking costs, respectively. We also derive an optimal continuous depth-query trade-off for fixed-point amplitude amplification, providing a potential building block for other hybrid designs.