Exponential algorithmic speedup by a quantum walk

Childs, Andrew M.; Cleve, Richard; Deotto, E.; Farhi, Edward; Gutmann, Sam; Spielman, Daniel A.

doi:10.1145/780542.780552

Cited by 592 publications

(444 citation statements)

References 28 publications

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“…This potential was recently demonstrated by Childs, Cleve, Deotto, Farhi, Gutmann and Spielman [8], who prove that there exists a black-box problem for which a quantum algorithm based on quantum walks gives an exponential speed-up over any classical randomized algorithm. The key to this algorithm is that a quantum walk is able to permeate a particular graph while any classical random walk (or any classical randomized algorithm, for that matter) cannot.…”

Section: Introductionmentioning

confidence: 90%

“…Note that in terms of implementation, this does not mean that the walk mixes in constant time; some number of operations that is polynomial in the degree of the graph and in some accuracy parameter is required to implement such a walk, assuming the ability to compute the neighbors of each vertex. See [2,8] for further details.…”

Section: Proposition 7 Any Continuous-time Quantum Walk On the Cayleymentioning

confidence: 99%

“…One of the topics that has recently received attention in the quantum computing community that highlights these differences is the the study of quantum computational variants of random walks, or quantum walks [1,3,5,6,8,9,11,15,18,19,23]. (A recent survey on quantum walks by Kempe [16] is an ideal starting point for background on quantum walks.)…”

Section: Introductionmentioning

confidence: 99%

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Continuous-Time Quantum Walks on the Symmetric Group

Gerhardt

Watrous

2003

Lecture Notes in Computer Science

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In this paper we study continuous-time quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference.

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Section: Introductionmentioning

confidence: 90%

Section: Proposition 7 Any Continuous-time Quantum Walk On the Cayleymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuous-Time Quantum Walks on the Symmetric Group

Gerhardt

Watrous

2003

Lecture Notes in Computer Science

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“…the averaged group velocity of the Bloch waves (see SI G). This expansion can be seen as a continuous quantum walk [20][21][22][23][24]. For comparison, classical (thermal) hopping of a particle (e.g.…”

Section: Non-interacting Casementioning

confidence: 99%

Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms

et al. 2012

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Transport properties are among the defining characteristics of many important phases in condensed matter physics. In the presence of strong correlations they are difficult to predict even for model systems like the Hubbard model. In real materials they are in general obscured by additional complications including impurities, lattice defects or multi-band effects. Ultracold atoms in contrast offer the possibility to study transport and out-of-equilibrium phenomena in a clean and well-controlled environment and can therefore act as a quantum simulator for condensed matter systems. Here we studied the expansion of an initially confined fermionic quantum gas in the lowest band of a homogeneous optical lattice. While we observe ballistic transport for non-interacting atoms, even small interactions render the expansion almost bimodal with a dramatically reduced expansion velocity. The dynamics is independent of the sign of the interaction, revealing a novel, dynamic symmetry of the Hubbard model.In solid state physics, transport properties are among the key observables, the most prominent example being the electrical conductivity, which e.g. allows to distinguish normal conductors from insulators or superconductors. Furthermore, many of today's most intriguing solid state phenomena manifest themselves in transport properties, examples including high-temperature superconductivity, giant magnetoresistance, quantum hall physics, topological insulators and disorder phenomena. Especially in strongly correlated systems, where the interactions between the conductance electrons are important, transport properties are difficult to calculate beyond the linear response regime. In general, predicting out-of-equilibrium fermionic dynamics represents an even harder problem than the prediction of static properties like the nature of the ground state. In real solids further complications arise due to the effects of e.g. impurities, lattice defects and phonons. These complications render an experimental investigation in a clean and well controlled ultracold atom system highly desirable. While the last years have seen dramatic progress in the control of quantum gases in optical lattices [1-3], a thorough understanding of the dynamics in these systems is still lacking. Genuine dynamical experiments can not only uncover new dynamic phenomena but are also essential to gain insight into the timescales needed to achieve equilibrium in the lattice [4,5] or to adiabatically load into the lattice [6,7].Using both bosonic and fermionic [8-10] atoms, it has become possible to simulate models of strongly interacting quantum particles, for which the Hubbard model [11] * ulrich.schneider@lmu.de is probably the most important example. A major advantage of these systems compared to real solids is the possibility to change all relevant parameters in real-time by e.g. varying laser intensities or magnetic fields. While first studies of dynamical properties of both bosonic and fermionic quantum gases [12][13][14] have already been performed, a remaining...

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“…Recently, both continuous-time [3] and discrete-time [4] QWs are found to be universal for quantum computation. A number of quantum algorithms based on QWs have already been proposed in [5][6][7][8][9][10]. In addition, QWs in graph [11], on a line with a moving boundary [12], with multiple coins [13] or decoherent coins [14] have been discussed also.…”

Section: Introductionmentioning

confidence: 99%