Universal Computation by Quantum Walk

Abstract

In some of the earliest work on quantum mechanical computers, Feynman showed how to implement universal quantum computation by the dynamics of a time-independent Hamiltonian. I show that this remains possible even if the Hamiltonian is restricted to be a sparse matrix with all entries equal to 0 or 1, i.e., the adjacency matrix of a low-degree graph. Thus quantum walk can be regarded as a universal computational primitive, with any desired quantum computation encoded entirely in some underlying graph. The main idea of the construction is to implement quantum gates by scattering processes.

Error tolerance of the boson-sampling model for linear optics quantum computing

“…Quantum walks (QWs) [14][15][16][17] have recently obtained a lot of interest as an approach to quantum information processing. It was noted in Ref.…”

Average position in quantum walks with a U(2) coin

“…According to the time evolution, QWs can be devided into discrete-time and continuous-time [2] QWs. Recently, both continuous-time [3] and discrete-time [4] QWs are found to be universal for quantum computation. A number of quantum algorithms based on QWs have already been proposed in [5][6][7][8][9][10].…”

Thermodynamic formalism for dissipative quantum walks

“…The standard CTQWs (the unitary analog of classical random walks) describe the coherent motion of an excitation on the vertices of an undirected graph [7][8][9][10]. They are a powerful tool in quantum computation, providing a useful approach to algorithms' design, and an alternative framework for universal quantum computation [11]. More recently CTQWs have also been used to describe transport properties in complex networks [12,13], while dissipative CTQWs have found applications in the context of quantum biology [14][15][16][17][18][19], and quantum state transfer in superconducting qubits [20,21].…”