ZW-calculus is a useful graphical language for pure qubit quantum computing. It is via the translation of the completeness of ZW-calculus that the first proof of completeness of ZX-calculus was obtained. A d-level generalisation of qubit ZW-calculus (anyonic qudit ZW-calculus) has been given in [Hadzihasanovic 2017] which is universal for pure qudit quantum computing. However, the interpretation of the W spider in this type of ZW-calculus has so-called q-binomial coefficients involved, thus makes computation quite complicated. In this paper, we give a new type of qudit ZW-calculus which has generators and rewriting rules similar to that of the qubit ZW-calculus. Especially, the Z spider is exactly the same as that of the qudit ZX-calculus as given in [Wang 2021], and the new W spider has much simpler interpretation as a linear map. Furthermore, we establish a translation between this qudit ZW-calculus and the qudit ZX-calculus which is universal as shown in [Wang 2021], therefore this qudit ZW-calculus is also universal for pure qudit quantum computing.
Quantum information has typically focused on using 2-level qubits to perform simulation and computation. We propose to expand the number of levels for computation using qudits, where \\(d>2\\). Doing so could be a viable option for making trapped ion systems scalable for quantum computation. Our group in particular will use Barium ions because of some energy features and convenient laser wavelengths. This thesis presents much of the necessary background needed to work with Barium as a qudit for quantum computation. Energy structure, branching ratios, and saturation intensities are derived and presented. In addition, a method for selecting different isotopes of Barium for trapping is discussed. A method for measuring out the state of a Barium qudit is presented, with error rates estimated to be under \
1 for up to 5-level qudits. Finally, various optics projects which were necessary for building up our first ion trap are outlined.
Quantum algorithms are known for providing more efficient solutions to certain computational tasks than any corresponding classical algorithm. Here we show that a single qudit is sufficient to implement an oracle based quantum algorithm, which can solve a black-box problem faster than any classical algorithm. For 2d permutation functions defined on a set of d elements, deciding whether a given permutation is even or odd, requires evaluation of the function for at least two elements. We demonstrate that a quantum circuit with a single qudit can determine the parity of the permutation with only one evaluation of the function. Our algorithm provides an example for quantum computation without entanglement since it makes use of the pure state of a qudit. We also present an experimental realization of the proposed quantum algorithm with a quadrupolar nuclear magnetic resonance using a single four-level quantum system, i.e., a ququart.