Search Results for « quantum walks »

Quantum walks of correlated particles offer the possibility of studying large-scale quantum interference; simulating biological, chemical, and physical systems; and providing a route to universal quantum computation. We have demonstrated quantum walks of two identical photons in an array of 21 continuously evanescently coupled waveguides in a SiO(x)N(y) chip. We observed quantum correlations, violating a classical limit by 76 standard deviations, and found that the correlations depended critically on the input state of the quantum walk. These results present a powerful approach to achieving quantum walks with correlated particles to encode information in an exponentially larger state space.

In some of the earliest work on quantum computing, Feynman showed how to implement universal quantum computation with a time-independent Hamiltonian. I show that this remains possible even if the Hamiltonian is restricted to be the adjacency matrix of a low-degree graph. Thus quantum walk can be regarded as a universal computational primitive, with any quantum computation encoded in some graph. The main idea is to implement quantum gates by scattering processes.

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

We analyse the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the 'quantum coin toss' in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.

We propose a new method for designing quantum search algorithms for finding a “marked” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 32–41] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [Quantum walk algorithm for Element Distinctness, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 22–31] and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

Computing Power of Quantum Mechanics There is much interest in developing quantum computers in order to perform certain tasks much faster than, or that are intractable for, a classical computer. A general quantum computer, however, requires the fabrication and operation a number of quantum logic devices (see the Perspective by Franson ). Broome et al. (p. 794 , published online 20 December) and Spring et al. (p. 798 , published online 20 December) describe experiments in which single photons and quantum interference were used to perform a calculation (the permanent of a matrix) that is very difficult on a classical computer. Similar to random walks, quantum walks on a graph describe the movement of a walker on a set of predetermined paths; instead of flipping a coin to decide which way to go at each point, a quantum walker can take several paths at once. Childs et al. (p. 791 ) propose an architecture for a quantum computer, based on quantum walks of multiple interacting walkers. The system is capable of performing any quantum operation using a subset of its nodes, with the size of the subset scaling favorably with the complexity of the operation.

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of $N$ items laid out in $d$ spatial dimensions can be searched in time of order $\sqrt{N}$ for $d>2$, and in time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$ for $d=2$. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that $\sqrt{N}$ speedup can also be achieved on the hypercube. We show that full $\sqrt{N}$ speedup can be achieved on a $d$-dimensional periodic lattice for $d>4$. In $d=4$, the quantum walk search algorithm takes time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$, and in $d<4$, the algorithm does not provide substantial speedup.

We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.