Efficient discrete-time simulations of continuous-time quantum query algorithms

Cleve, Richard; Gottesman, Daniel; Mosca, Michele; Somma, Rolando D.; Yonge-Mallo, David L.

doi:10.1145/1536414.1536471

Cited by 27 publications

(73 citation statements)

References 20 publications

(15 reference statements)

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“…Given the results of Childs et al [6] and Cleve et al [19], in which quantum walks are shown to traverse a family of 'glued trees' exponentially faster than any possible classical algorithm, relative to a quantum oracle, we decided to look into quantum walks along glued trees in the non-oracular setting. Note that the algorithm presented in [6] employs continuous-time quantum walks, while in [19] it was shown to also be implementable by discrete-time quantum walks.…”

Section: Efficient Quantum Circuit Implementationmentioning

confidence: 99%

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Efficient quantum circuit implementation of quantum walks

Douglas

Wang

2009

Phys. Rev. A

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Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the 'glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.

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Section: Efficient Quantum Circuit Implementationmentioning

confidence: 99%

“…Note that the algorithm presented in [6] employs continuous-time quantum walks, while in [19] it was shown to also be implementable by discrete-time quantum walks. Both require the use of a quantum oracle.…”

Section: Efficient Quantum Circuit Implementationmentioning

confidence: 99%

Efficient quantum circuit implementation of quantum walks

Douglas

Wang

2009

Phys. Rev. A

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“…The problem it faces is that the norm of A G can be super-constant, which slows the simulation. Fortunately, a relationship between the discrete-and continuous-time quantum query complexity models, from [14], allows for removing the norm dependency, at a sub-logarithmic cost to the query complexity. By instead reflecting about the eigenvalue-zero subspace of A G , we efficiently remove the dependence on higher-energy portions of A G .…”

Section: Reflection Structure Of the Algorithmmentioning

confidence: 99%

“…The log over log log factor in Eq. (1.2) comes from converting a certain continuous-time query algorithm into a discrete-query algorithm following [14]. This conversion somewhat obscures the algorithm's structure.…”

Section: Introductionmentioning

confidence: 99%

Reflections for quantum query algorithms

Reichardt¹

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms.Our proof goes via the general adversary bound, a semidefinite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a boundederror query complexity that matches the general adversary bound.Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f • (g, . . . , g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.

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“…In fact, using the equivalence between continuous-and discrete-time query models in Ref. [24] yields a lower bound for the cost O(L/[ log(L/ )]). Our formal proof below, which uses a version of the adversary method in the continuous-time setting, will avoid the logarithmic correction in the cost.…”

Section: Nondegenerate Eigenpathmentioning

confidence: 99%

Necessary condition for the quantum adiabatic approximation

Boixo

Somma

2010

Phys. Rev. A

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A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.

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Efficient discrete-time simulations of continuous-time quantum query algorithms

Cited by 27 publications

References 20 publications

Efficient quantum circuit implementation of quantum walks

Efficient quantum circuit implementation of quantum walks

Reflections for quantum query algorithms

Necessary condition for the quantum adiabatic approximation

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