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- Andris Ambainis
- STOC
- 2000

We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modiies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs.… (More)

- Andris Ambainis, Rusins Freivalds
- FOCS
- 1998

We consider 1-way quantum finite automata (QFAs). First, we compare them with their classical counterparts. We show that, if an automaton is required to give the correct answer with a large probability (over 9/10), then the power of 1-way QFAs is equal to the power of 1-way reversible automata. However, quantum automata giving the correct answer with… (More)

- A. Ambainis, L. Schulman, A. Ta-Shma
- 2003

We introduce 2-way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2-way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be… (More)

We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the <italic>Hadamard walk</italic>. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the… (More)

- Andris Ambainis
- 45th Annual IEEE Symposium on Foundations of…
- 2004

We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N/sup 2/3/) query quantum algorithm. This improves the previous O(N/sup 3/4/) quantum algorithm of Buhrman et al. and matches the lower bound by Shi. We… (More)

- Andris Ambainis
- ICALP
- 1996

We construct a scheme for private information retrieval with k databases and communication complexity O(n 1=(2k?1)).

We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined… (More)

- Andris Ambainis, Julia Kempe, Alexander Rivosh
- SODA
- 2005

We show how to search <i>N</i> items arranged on a √<i>N</i> × √<i>N</i> grid in time <i>O</i>(√<i>N</i> log <i>N</i>), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom. since it has been shown recently… (More)

- Andris Ambainis
- Theory of Computing
- 2005

We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions its polynomial degree is the same for all M ≥ N. Therefore, if we have a quantum query lower bound for some (possibly quite large) range M which is shown using the polynomials method, we immediately… (More)