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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)
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)
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)
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 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)
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)
We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared(More)