Applications Of Qudits In Quantum Computing And Their Physical Realizations

Quantum computation is a brand-new promising interdisciplinary science subject integrating quantum mechanics and computer science. Thanks to the character of the superposition and the entanglement of the quantum state, quantum computing have the ability greatly exceeding their classical counterpart, for example, factoring the large number in polynomial time, quadratic speedup for the problem of searching in the unsorted database.The basic elements consisting of quantum computer are called qubits, two-dimensional quantum systems, which are the counterparts of bits consisting of the classical computer.There exist a large number of stable two-state physical systems in the classical world, so encoding with the two-state bits is selected naturally for the classical computer. However, lots of the quantum systems whose energy eigenstates are discrete have multiple stable states, for example the Zeeman’s levels of the ground states of the atoms and ions.Making full use of these states, encoding with high-dimensional quantum systems is a alternative. There are several advantages encoding with high-dimensional quantum systems: 1. the exponential increase of the available Hilbert space with the same amount of physical resources, 2. more efficient quantum logical gate, 3. enhanced security in quantum cryptography, 4. larger violations of nonlocality, and so on. The main works of our thesis are to study the applications of high-dimensional quantum systems in quantum computing, generalization of the quantum algorithms with qubits, and design schemes to realize the improved algorithms in real physical systems.The main works are as follows:1. We suggested a method to implement the Grover algorithm in an arbitrary dimensional Hilbert space using qudits whose dimensions may be different from each other. As far as we know, it is the first time to suggest a quantum computational proposal with different dimensional qudits which, besides the qudits themselves, could be also an advantage for making full use of the physical resources in the case of quantum information processing with physical systems having different level configurations, such as hybrid quantum computations. Furthermore, with the modified algorithm we can find the target state among the items to be examined much faster, thus leading to less dissipations and errors in the physical systems during the quantum information processing. To crystallize the idea, we have designed two corresponding physical realizing schemes:(a) We have proposed a scheme for the implementation of a six-item Grover search in cavity QED using a qutrit and a qubit encoded in the Zeeman’s level structure and shown that a large enough success rate and fidelity could be reached with current techniques of cavity QED.(b) Employing single-photon two-qudit states, we have designed a six-dimensional Grover algorithm with linear optics. Through twice improvement and simplification of the experimental scheme, it can be accomplished with our current laboratory conditions.2. Encoding with the freedoms of the single photon’s path and time, we have designed an experimental scheme for demonstrating the Shor’s algorithm using linear optics, factoring the number 21 in the case that the period is 3. The scheme shows how the qudits simplify the construction of the quantum logical circuits and the application of the time encoding in quantum computing. Though there are three Mach-Zehnder interferometers in the scheme, actually, only one of them is needed to stabilize. So the scheme is feasible to realize in the experiment. Finally, We have generalized the Bernstein-Vazirani algorithm and Simon algorithm in the qudit systems.3. We suggested a method to implement the partial search for the arbitrary size database partitioned into any number of blocks using multi-level systems. With this method fewer iteration steps are needed and carriers of the information are used more economically compared with the partial search with qubits. To crystalize the idea, we have proposed a scheme for the implementation of a twelve-item partial search of the database separated into three blocks using a qutrit and two qubit with SQUIDs in cavity QED. Through the appropriate modulation of the microwave pulses’ intensities, our scheme could overcome the non-identity of the cavity-SQUID coupling strengths. In addition, since the cavity field is only required to be virtually excited, the system is insensitive to the cavity decay which releases the experimental requirement. Numerical simulation under the influence of the cavity and SQUID decays shows that the scheme could be achieved with high fidelity under current state-of-the-art technology.4. We studied the dynamics of the multipartite systems off-resonantly interacting with electromagnetic fields, focusing on the large detuning condition for the effective Hamiltonian in this case. Two-level configuration and three-level lambda configuration under the conditions of two-photon resonance and nonresonace are analyzed theoretically and numerically in detail, and the exact solutions of the temporal evolution of the coupled systems are derived. Since the coupling strength is enhanced due to the many-atom interference effects, we suggest a more rigorous large detuning condition for the effective Hamiltonian in this case. It is significant to apply the rigorous large detuning condition in analyzing the multipartite systems interacting with fields, since the original large detuning condition may lead to the invalidity of the effective Hamiltonian and the errors of the parameters associated with the detuning, especially for the analyses without full numerical simulations. Furthermore, for the situation of many cavity photons involved in the interaction, the formulism of the large detuning condition is given, which clearly reflect the many-atom and many-photon interference effects. Diverse quantum systems, such as atom systems, superconducting devices, trapped ions, and so on, interacting with electromagnetic fields are important coupling model for quantum computing, wherein the large detuning condition is usually employed, thus the study about this issue is the basis of the choose of the appropriate parameters for the scheme design.