Quadratic speedup for finding marked vertices by quantum walks

Ambainis, Andris; Gilyén, András; Jeffery, Stacey; Kokainis, Martins

doi:10.48550/arxiv.1903.07493

Cited by 5 publications

(28 citation statements)

References 10 publications

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“…We also give a conceptual bridge between the recent result of Ref. [AGJK19] and the original approaches by Szegedy [Sze04] and Krovi, Magniez, Ozols and Roland [KMOR16]. The latter showed that combining quantum walks with phase estimation or time averaging allows one to quadratically improve the hitting time of a single marked element, when starting from the stationary distribution.…”

Section: Introductionmentioning

confidence: 59%

“…Combining interpolated walks with another technique called quantum fast-forwarding, introduced in [AS19], Ref. [AGJK19] more recently showed how to also find a marked element in the general case. We will refer to this final result as the hitting time framework.…”

Section: Different Frameworkmentioning

confidence: 99%

“…The downside is that algorithms in this framework only detect rather than actually find marked vertices. On the other hand, the improved hitting time framework of [AGJK19] shows how to actually find marked vertices in the hitting time framework, provided that the walk starts from a quantum sample of the stationary distribution. Both works hence provide complementary but incompatible improvements over the initial hitting time framework.…”

Section: Contributionsmentioning

confidence: 99%

“…The latter showed that combining quantum walks with phase estimation or time averaging allows one to quadratically improve the hitting time of a single marked element, when starting from the stationary distribution. Ambainis et al [AGJK19] used a more involved technique called quantum fast-forwarding [AS19] to improve these results to yield quadratic speedups on the hitting time of arbitrary sets. In this work we reprove the same result using only simple quantum walks, thereby proving a conjecture from [AGJK19].…”

Section: Introductionmentioning

confidence: 99%

“…Ambainis et al [AGJK19] used a more involved technique called quantum fast-forwarding [AS19] to improve these results to yield quadratic speedups on the hitting time of arbitrary sets. In this work we reprove the same result using only simple quantum walks, thereby proving a conjecture from [AGJK19].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Unified Framework of Quantum Walk Search

Apers¹,

Gilyén²,

Jeffery³

2019

Preprint

Self Cite

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The main results on quantum walk search are scattered over different, incomparable frameworks, most notably the hitting time framework, originally by Szegedy, the electric network framework by Belovs, and the MNRS framework by Magniez, Nayak, Roland and Santha. As a result, a number of pieces are currently missing. For instance, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. In recent work by Ambainis et al., this problem was resolved for the more restricted hitting time framework, in which quantum walks must start from the stationary distribution.We present a new quantum walk search framework that unifies and strengthens these frameworks. This leads to a number of new results. For instance, the new framework not only detects, but finds marked elements in the electric network setting. The new framework also allows one to interpolate between the hitting time framework, which minimizes the number of walk steps, and the MNRS framework, which minimizes the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. Whereas the original frameworks only rely on quantum walks and phase estimation, our new algorithm makes use of a technique called quantum fast-forwarding, similar to the recent results by Ambainis et al. As a final result we show how in certain cases we can simplify this more involved algorithm to merely applying the quantum walk operator some number of times. This answers an open question of Ambainis et al.

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Section: Introductionmentioning

confidence: 59%

Section: Different Frameworkmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Unified Framework of Quantum Walk Search

Apers¹,

Gilyén²,

Jeffery³

2019

Preprint

Self Cite

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Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

Harrow¹,

Wei²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule.We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing "qsamples" instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution.In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.

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On analog quantum algorithms for the mixing of Markov chains

Chakraborty,

Luh,

Roland

2019

Preprint

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The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. We also make novel inroads into a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. This notion of quantum mixing has been explored for a handful of specific graphs and the derived upper bound for this quantity has been faster than its classical counterpart for some graphs while being slower for others. In this article, using several results in random matrix theory, we prove an upper bound on the quantum mixing time of Erdös-Renyi random graphs: graphs of n nodes where each edge exists with probability p independently. For example for dense random graphs, where p is a constant, we show that the quantum mixing time is O(n 3/2+o(1) ). Consequently, this allows us to obtain an upper bound on the quantum mixing time for almost all graphs, i.e. the fraction of graphs for which this bound holds, goes to one in the asymptotic limit.

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Quadratic speedup for finding marked vertices by quantum walks

Cited by 5 publications

References 10 publications

A Unified Framework of Quantum Walk Search

A Unified Framework of Quantum Walk Search

Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

On analog quantum algorithms for the mixing of Markov chains

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