Analog quantum algorithms for the mixing of Markov chains

Chakraborty, Shantanav; Luh, Kyle; Roland, Jérémie

doi:10.1103/physreva.102.022423

Cited by 14 publications

(3 citation statements)

References 53 publications

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“…At variance, equation ( 5) describes reversible dynamics which rules out a unique long time limit, but one can consider the long-time average distribution. In general, the time-averaged probability distribution gets close to it after a time known as mixing time which has been recently upper bounded for generic graphs when M G is the adjacency matrix [163,164], revealing that quantum walks typically take longer to mix than classical walks. Furthermore, unlike the probabilities the amplitudes are subject to interference effects which can lead to ballistic instead of diffusive spread [27] as demonstrated in figure 6.…”

Section: Walkers and Search Algorithmsmentioning

confidence: 99%

Complex quantum networks: a topical review

Nokkala,

Piilo,

Bianconi

2024

J. Phys. A: Math. Theor.

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These are exciting times for quantum physics as new quantum technologies are expected to soon transform computing at an unprecedented level. Simultaneously network science is ﬂourishing proving an ideal mathematical and computational framework to capture the complexity of large interacting systems. Here we provide a comprehensive and timely review of the rising ﬁeld of complex quantum networks. On one side, this subject is key to harness the potential of complex networks in order to provide design principles to boost and enhance quantum algorithms and quantum technologies. On the other side this subject can provide a new generation of quantum algorithms to infer signiﬁcant complex network properties. The ﬁeld features fundamental research questions as diverse as designing networks to shape Hamiltonians and their corresponding phase diagram, taming the complexity of many-body quantum systems with network theory, revealing how quantum physics and quantum algorithms can predict novel network properties and phase transitions, and studying the interplay between architecture, topology and performance in quantum communication networks. Our review covers all of these multifaceted aspects in a self-contained presentation aimed both at network-curious quantum physicists and at quantum-curious network theorists. We provide a framework that uniﬁes the ﬁeld of quantum complex networks along four main research lines: network-generalized, quantum-applied, quantum-generalized and quantum-enhanced. Finally we draw attention to the connections between these research lines, which can lead to new opportunities and new discoveries at the interface between quantum physics and network science.

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Section: Walkers and Search Algorithmsmentioning

confidence: 99%

Complex quantum networks: a topical review

Nokkala,

Piilo,

Bianconi

2024

J. Phys. A: Math. Theor.

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“…In the literature, one finds papers analyzing the quantum mixing time on cycles [5], Cayley graphs [5], hypercubes [28,29], two-dimensional lattices [30], and complete graphs [31]. Upper bounds for the quantum mixing time were obtained in [32,33]. Apers et al [34] discuss the simulation of the quantum mixing time by classical Markov chains with added memory, and concludes that quantum walk speedups are not necessarily diagnostic of quantum effects.…”

Section: Limiting Distribution and Mixing Timementioning

confidence: 99%

Quantum Walk on the Generalized Birkhoff Polytope Graph

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et al. 2021

Entropy

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We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is O(m/ϵ), where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is O(m1.5/ϵ).

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“…Google's PageRank algorithm [18] can be reduced to prepare the stationary distribution. Currently, the qsampling (quantum sampling) algorithms which achieve the quadratic speedup compared with the classical case are only known for some limited stationary distributions [19,20] and a series of sparse graphs [14,21]. In [22] we provide a new discrete-time quantum walk-based sampling algorithm in cost Θ( √ HT log 1 ε ), which is suitable for any reversible Markov chain with no additional requirement on graphs or stationary distributions and achieves quadratic speedup in a series common graphs such as sparse graphs.…”

Section: Introductionmentioning

confidence: 99%

Improvement of quantum walks search algorithm in single-marked vertex graph

Li,

Shang

2023

J. Phys. A: Math. Theor.

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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 souffl'e 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 souffl'e 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 $\Theta((\varepsilon^{-1})\sqrt{\Heg})$ to $\Theta(\log(\varepsilon^{-1})\sqrt{\Heg})$ and reduce the number of ancilla qubits required from $\Theta(\log(\varepsilon^{-1})+\log\sqrt{\Heg})$ to $\Theta(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})$, where $\varepsilon$ denotes the precision and $\Heg$ 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.

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Analog quantum algorithms for the mixing of Markov chains

Cited by 14 publications

References 53 publications

Complex quantum networks: a topical review

Complex quantum networks: a topical review

Quantum Walk on the Generalized Birkhoff Polytope Graph

Improvement of quantum walks search algorithm in single-marked vertex graph

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