Quantum walks and quantum search on graphene lattices

This thesis details research I have carried out in the field of quantum walks, which are the quantum analogue of classical random walks. Quantum walks have been shown to offer a significant speed-up compared to classical random walks for certain tasks and for this reason there has been considerable interest in their use in algorithmic settings, as well as in experimental demonstrations of such phenomena. One of the most interesting developments in quantum walk research is their application to spatial searches, where one searches for a particular site of some network or lattice structure. There has been much work done on the creation of discrete- and continuous-time quantum walk search algorithms on various lattice types. However, it has remained an issue that continuous-time searches on two-dimensional lattices have required the inclusion of additional memory in order to be effective, memory which takes the form of extra internal degrees of freedom for the walker. In this work, we describe how the need for extra degrees of freedom can be negated by utilising a graphene lattice, demonstrating that a continuous-time quantum search in the experimentally relevant regime of two-dimensions is possible. This is achieved through alternative methods of marking a particular site to previous searches, creating a quantum search protocol at the Dirac point in graphene. We demonstrate that this search mechanism can also be adapted to allow state transfer across the lattice. These two processes offer new methods for channelling information across lattices between specific sites and supports the possibility of graphene devices which operate at a single-atom level. Recent experiments on microwave analogues of graphene that adapt these ideas, which we will detail, demonstrate the feasibility of realising the quantum search and transfer mechanisms on graphene

Quantum walks and quantum search on graphene lattices

This thesis details research I have carried out in the field of quantum walks, which are the quantum analogue of classical random walks. Quantum walks have been shown to offer a significant speed-up compared to classical random walks for certain tasks and for this reason there has been considerable interest in their use in algorithmic settings, as well as in experimental demonstrations of such phenomena. One of the most interesting developments in quantum walk research is their application to spatial searches, where one searches for a particular site of some network or lattice structure. There has been much work done on the creation of discrete- and continuous-time quantum walk search algorithms on various lattice types. However, it has remained an issue that continuous-time searches on two-dimensional lattices have required the inclusion of additional memory in order to be effective, memory which takes the form of extra internal degrees of freedom for the walker. In this work, we describe how the need for extra degrees of freedom can be negated by utilising a graphene lattice, demonstrating that a continuous-time quantum search in the experimentally relevant regime of two-dimensions is possible. This is achieved through alternative methods of marking a particular site to previous searches, creating a quantum search protocol at the Dirac point in graphene. We demonstrate that this search mechanism can also be adapted to allow state transfer across the lattice. These two processes offer new methods for channelling information across lattices between specific sites and supports the possibility of graphene devices which operate at a single-atom level. Recent experiments on microwave analogues of graphene that adapt these ideas, which we will detail, demonstrate the feasibility of realising the quantum search and transfer mechanisms on graphene

Quantum search on graphene lattices

We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought- after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication

Microwave experiments simulating quantum search and directed transport in artificial graphene

A series of quantum search algorithms have been proposed recently providing an algebraic speedup compared to classical search algorithms from N to \sqrt{N}, where N is the number of items in the search space. In particular, devising searches on regular lattices has become popular in extending Grover’s original algorithm to spatial searching. Working in a tight-binding setup, it could be demonstrated, theoretically, that a search is possible in the physically relevant dimensions 2 and 3 if the lattice spectrum possesses Dirac points. We present here a proof of principle experiment implementing wave search algorithms and directed wave transport in a graphene lattice arrangement. The idea is based on bringing localized search states into resonance with an extended lattice state in an energy region of low spectral density—namely, at or near the Dirac point. The experiment is implemented using classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene. Furthermore, we investigate the scaling behavior experimentally using linear chains

Quantum walks and quantum search on graphene lattices

Quantum walks have been very useful in developing search algorithms in quantum information, in particular for devising of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we demonstrate that a continuous-time quantum walk search is possible in two dimensions by changing the search topology to a graphene lattice, utilizing the Dirac point in the energy spectrum. This is made possible by making a change to standard methods of marking a particular site in the lattice. Various ways of marking a site are shown to result in successful search protocols. We further establish that the search can be adapted to transfer probability amplitude across the lattice between specific lattice sites thus establishing a line of communication between these sites

The effectiveness of agricultural stewardship for improving water quality at the catchment scale: Experiences from an NVZ and ECSFDI watershed

Agriculture is estimated to be responsible for 70% of nitrate and 30-50% of phosphorus pollution, contributing to ecological and water treatment problems. Despite the fact that significant gaps remain in our understanding, it is known that agricultural stewardship can be highly effective in controlling water pollution at the plot and field scales. Knowledge at the catchment scale is, to a large extent, entirely lacking though and this is of paramount concern given that the catchment is the management unit used by regulatory authorities. The few studies that have examined the impact of agricultural stewardship at the catchment scale have found that Nitrate Vulnerable Zones (NVZs) in the UK have resulted in little improvement in water quality which concurs with the current catchment study. In addition to NVZs, there was little evidence to suggest that the England Catchment Sensitive Farming Delivery Initiative had impacted water quality and suggestions have been made for improvements, such as ensuring that stewardship measures are used in key pollution source areas and their implementation and impacts are monitored more closely. This will be essential if agricultural catchment management schemes are going to provide the benefits expected of them. Nevertheless, more intensive monitoring than that carried out by regulators showed a significant trend in decreasing winter nitrate peaks in some streams which is hypothesised to be due to recent reduced inorganic fertiliser application as a result of increasing prices. It was concluded that, collectively, these findings indicate that agricultural stewardship measures have the potential to improve water quality at the catchment scale but that voluntary schemes with insufficient financial reward or regulatory pressure are unlikely to be successful

The effectiveness of agricultural stewardship for improving water quality at the catchment scale: Experiences from an NVZ and ECSFDI watershed

Agriculture is estimated to be responsible for 70% of nitrate and 30–50% of phosphorus pollution, contributing to ecological and water treatment problems. Despite the fact that significant gaps remain in our understanding, it is known that agricultural stewardship can be highly effective in controlling water pollution at the plot and field scales. Knowledge at the catchment scale is, to a large extent, entirely lacking though and this is of paramount concern given that the catchment is the management unit used by regulatory authorities. The few studies that have examined the impact of agricultural stewardship at the catchment scale have found that Nitrate Vulnerable Zones (NVZs) in the UK have resulted in little improvement in water quality which concurs with the current catchment study. In addition to NVZs, there was little evidence to suggest that the England Catchment Sensitive Farming Delivery Initiative had impacted water quality and suggestions have been made for improvements, such as ensuring that stewardship measures are used in key pollution source areas and their implementation and impacts are monitored more closely. This will be essential if agricultural catchment management schemes are going to provide the benefits expected of them. Nevertheless, more intensive monitoring than that carried out by regulators showed a significant trend in decreasing winter nitrate peaks in some streams which is hypothesised to be due to recent reduced inorganic fertiliser application as a result of increasing prices. It was concluded that, collectively, these findings indicate that agricultural stewardship measures have the potential to improve water quality at the catchment scale but that voluntary schemes with insufficient financial reward or regulatory pressure are unlikely to be successful