Andris Ambainis

Quantum walks on graphs

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 the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

One-dimensional quantum walks

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 Hadamard walk has position that is nearly uniformly distributed in the range <italic>[-t/\sqrt 2, t/\sqrt 2]</italic> after <italic>t</italic> steps, which is in sharp contrast to the classical random walk, which has distance <italic>O(\sqrt t)</italic> from the origin with high probability. With an absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is <italic>2/&pgr</italic>, and with an additional absorbing boundary at location <italic>n</italic>, the probability that the walk exits to the left actually increases, approaching <italic>1/\sqrt 2</italic> in the limit. In the classical case both values are 1.

Quantum lower bounds by quantum arguments

We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with on 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. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques.

Quantum Walk Algorithm for Element Distinctness

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 also give an O(N/sup k/(k+1)/) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.

QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS

Quantum walks are quantum counterparts of Markov chains. In this article, we give a brief overview of quantum walks, with emphasis on their algorithmic applications.

Quantum walk algorithm for element distinctness

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 also give an O(N/sup k/(k+1)/) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.

1-way quantum finite automata: strengths, weaknesses and generalizations

We study 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 (greater than 7/9), then any 1-way QFAs can be simulated by a 1-way reversible automaton. However, quantum automata giving the correct answer with smaller probabilities are more powerful than reversible automata. Second, we show that 1-way QFAs can be very space-efficient. We construct a 1-way QFA that is exponentially smaller than any equivalent classical (even randomized) finite automaton. We think that this construction may be useful for design of other space-efficient quantum algorithms. Third, we consider several generalizations of 1-way QFAs. Here, our goal is to find a model which is more powerful than 1-way QFAs keeping the quantum part as simple as possible.

Upper Bound on Communication Complexity of Private Information Retrieval

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

Two-way finite automata with quantum and classical states

We introduce 2-way finite automata with quantum and classical states (2qcfas). This is a variant on the 2-way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfas; 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 classical.We show two languages for which 2qcfas are better than classical 2-way automata. First, 2qcfas can recognize palindromes, a language that cannot be recognized by 2-way deterministic or probabilistic finite automata. Second, in polynomial time 2qcfas can recognize {anbn|n ∈ N}, a language that can be recognized classically by a 2-way probabilistic automaton but only in exponential time.

Private quantum channels

We investigate how a classical private key can be used by two players, connected by an insecure one-way quantum channel, to perform private communication of quantum information. In particular, we show that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient. This result may be viewed as the quantum analogue of the classical one-time pad encryption scheme.