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- 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
- 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 modifies 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 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)

- 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)

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)

- Scott Aaronson, Andris Ambainis
- FOCS
- 2003

Can Grover’s algorithm speed up search of a physical region—for example a 2-D grid of size √ n× √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional… (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
- ICALP
- 1996

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

- Andris Ambainis, Ashwin Nayak, Amnon Ta-Shma, Umesh V. Vazirani
- J. ACM
- 2002

We consider the possibility of encoding <i>m</i> classical bits into many fewer <i>n</i> quantum bits (qubits) so that an arbitrary bit from the original <i>m</i> bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that quantum encoding cannot save more than… (More)

- Andris Ambainis
- J. Comput. Syst. Sci.
- 2003

The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f . This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M). This… (More)