Quantum walks, quantum gates, and quantum computers

Abstract

The physics of quantum walks on graphs is formulated in Hamiltonian language, both for simple quantum walks and for composite walks, where extra discrete degrees of freedom live at each node of the graph. It is shown how to map between quantum walk Hamiltonians and Hamiltonians for qubit systems and quantum circuits; this is done for both a single-and multi-excitation coding, and for more general mappings. Specific examples of spin chains, as well as static and dynamic systems of qubits, are mapped to quantum walks, and walks on hyperlattices and hypercubes are mapped to various gate systems. We also show how to map a quantum circuit performing the quantum Fourier transform, the key element of Shor's algorithm, to a quantum walk system doing the same. The results herein are an essential preliminary to a Hamiltonian formulation of quantum walks in which coupling to a dynamic quantum environment is included.

Physical Implementation of Quantum Walks

“…The work carried out by Douglas and Wang (2009) and Loke and Wang (2011) can be used to efficiently implement such an oracle -using O(log(n)) elementary gates for a graph of order n -given a highly symmetric graph such as those considered in Hines and Stamp (2007), Reitzner et al (2009), Douglas and and Loke and Wang (2011). The work carried out by Douglas and Wang (2009) and Loke and Wang (2011) can be used to efficiently implement such an oracle -using O(log(n)) elementary gates for a graph of order n -given a highly symmetric graph such as those considered in Hines and Stamp (2007), Reitzner et al (2009), Douglas and and Loke and Wang (2011).…”

Quantum stochastic walks: A generalization of classical random walks and quantum walks

“…For the purposes of experimental implementation, the vertices of the graph in a walk can be implemented using a qubit per vertex (an inefficient or unary mapping) or by employing a quantum state per vertex (the binary or efficient mapping). The choice of mapping impacts the simulation efficiency and their robustness under decoherence [22,23,24]. The previous proposed approaches for exploring decoherence in quantum walks have added environmental-effects to a QW based on computational or physical models such as pure dephasing [17] but have not considered walks where the environmental effects are constructed axiomatically from the underlying graph.…”

Spin-based quantum computers made by chemistry: hows and whys

“…This Hamiltonian must be generalized to describe "composite quantum walks" (where discrete internal "spin" variables at each node can also couple to the walker) if we are to fully describe a QIPS [9]. The great advantage of this representation, at least for physicists and chemists, is that we have a good intuitive understanding of how particles move around such graphs.…”

“…One also clearly sees where things can get complicated, if the ǫ µ and T µν are strongly disordered or if their timedependence is non-trivial. How this works in practice can be found in the literature [8,9].…”