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We propose a new method for designing quantum search algorithms forfinding a "marked" element in the state space of a classical Markovchain. The algorithm is based on a quantum walk à la Szegedy [25] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantumwalk in order to implement an… (More)

—We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck [1] and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm based methods… (More)

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $$M$$ M consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $${{\mathrm{HT}}}(P,M)$$ HT ( P , M ) of any reversible random walk $$P$$ P… (More)

Understanding NP-complete problems is a central topic in computer science (NP stands for nondeterministic polynomial time). This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps… (More)

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Recently, Ganor et al. provided such a separation in the distributional setting for a specific input distribution µ. We show that in the non-distributional setting,… (More)

Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann [1951] and has many applications in classical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superposition of… (More)

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each… (More)

It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, post-selection or non-local boxes. For instance, in the case of projective measurements , it is possible to solve this problem with protocols using one bit of… (More)

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma—computing… (More)

Adiabatic quantum optimization has attracted a lot of attention because small scale simulations gave hope that it would allow to solve NP-complete problems efficiently. Later, negative results proved the existence of specifically designed hard instances where adiabatic optimization requires exponential time. In spite of this, there was still hope that this… (More)