Unitary transformations for quantum computing

Abstract

The last two decades have seen an enormous increase in the computational power of digital computers. This was due to the rapid technical development in manufacturing processes and controlling semiconducting structures on submicron scale. Concurrently, the electric circuits have encountered the first signs of the realm of quantum mechanics. Those effects may induce noise and thus they are typically considered harmful. However, the manipulation of the coherent quantum states might turn out be the basis of powerful computers – quantum computers. There, the computation is encoded into the unitary time evolution of a quantum mechanical state vector. Eventually, quantum mechanics could enable one, for example, to read secret electronic messages which are encrypted by the widely employed RSA cryptosystem – a task which is extremely laborious for the current digital computers.

This thesis presents a theoretical study of the coherent manipulations of pure quantum states in a quantum register, that is, quantum algorithms. An implementation of a quantum algorithm involves the initialization of the input state and its manipulation with quantum gates followed by the measurements. The physical implementation of each gate requires that it is decomposed into low-level gates whose physical realizations are explicitly known. Here, the problem is examined from two directions. Firstly, the numerical optimization scheme for controlling time-evolution of a closed quantum system is discussed. This yields a method for implementing quantum gates acting on up to three quantum bits, qubits. The approach is independent of the physical realization of the quantum computer, but it is considered explicitly for a proposed inductively coupled Josephson charge qubit register. Secondly, the techniques of numerical matrix computation are utilized to find a general method for decomposing an arbitrary n-qubit gate into a sequence of elementary gates, which act on one or two qubits.

The results of this thesis help to improve the implementation of quantum algorithms. The quantum circuit construction developed in the thesis is the first one to achieve the asymptotically minimal complexity in the number of elementary gates. In context of acceleration of quantum algorithms we present a gate-level study of Shor's algorithm and show how to accelerate the algorithm by merging several elementary gates into multiqubit gates. Finally, the requirements set by the resulting gate array are compared to the properties of superconducting qubits. This allows us to discuss the feasibility of the Josephson charge qubit register, for instance, as hardware for breaking the RSA cryptosystem.