Holonomic Quantum Computation

Quantum computation and quantum information hold the permit to solve some difficult problems more efficiently than classical computer. However, implementa-tion of quantum information processing (QIP) poses daunting challenges. On the one hand, quantum coherence has to be maintained throughout the whole compu-tational process for most of the QIP protocols in spite of the decoherence induced by the unavoidable coupling with environmental degrees of freedom. On the other hand, one has to achieve an unprecedented level of control to enact quantum gates within the required high accuracy.To accomplish these tasks, several theoretical schemes have been devised s-ince the early days of QIP. Broadly speaking, strategies developed to remove the unwanted interaction with the environment to date fall in two categories:active techniques like quantum error correcting codes, quantum zeno effect and dynamical decoupling; symmetry-aided passive ones like decoherence-free subspaces (DFS) and noiseless subsystems (NS). Strategies developed to suppress the control parameter fluctuations includes quantum computation by geometrical and topological ones.Geometric QIP exploits Abelian or non-Abelian geometric phases to implement quantum gates. The latter is usually referred as holonomic quantum computation (HQC), which is studied in this thesis. Following the first adiabatic HQC proposals, many others have been considered. The motivating idea is that the geometric nature of the proposed quantum gates do not dependent on the evolution rate and endows them with some degree of inherent robustness against control imprecisions. To avoid the main drawback of the slowness due to the adiabaticity constraint, an non-adiabatic HQC based on non-degenerate subspace is proposed, which has been demonstrated experimentally, such like in superconducting system, NMR system and NV center.In this thesis, we study how to combine the ideas of HQC and that of the information-stabilizing strategies to to take advantage of the appealing features of both as well as approach to speed up adiabatic HQC to an non-adiabatic regime. The main contents in this thesis are as follows:First, the combination of the passive information-stabilizing strategy and non-adiabatic HQC is studied. We have shown how to find DFS/NS in a quantum system consisting of a set of qubits, which is coupled to a general collective envi-ronment. Qubits in the system interact with their environment in such a way that the interaction has permutation symmetry between them, which enables the emer-gence of DFS/NS that provides the computational space of HQC. We have shown that a universal set of sigle- and two-qubit gates can be achieved by non-adiabatic and non-Abelian quantum holonomies acting entirely within a noiseless subsystem In our scheme, each noiseless qubit can be encoded using four physical qubits as a block and geometrically manipulated by Heisenberg-like interactions. For the single-qubit gates, different paths in the NS can be chosen provided that different kinds of manipulation Hamiltonians can be performed. Arbitrary single-qubite gate can be achieved by combing different paths. For the two-qubit gate, a CNOT gate by geometric means is presented by coupling two physical blocks, which employs eight qubits. Finally, by numerical simulations, we have provided evidence of the robustness of the proposed hybrid scheme against symmetry-breaking interactions with the environment.Second, we have studied the combination of the active strategy and non-adiabatic HQC to release the collective environment constrain. While collective assumption holds well for some system like atoms in linear lattice, it is more rel-evant to consider the general (linear) environment in which each qubit interacts with its environment dependently. The detrimental effect of this kind of environ-ment can be removed by dynamical decoupling (DD) approach periodically. In our case, we choose XY-4 pulse as the decoupling sequence, which makes the average Hamiltonian vanished. The encoding of the computational logical state relies on the group algebra of the dynamical sequence. We point out that the cases are different when the system comprises even or odd qubits. In the even qubits case, the unitary operators constituting the dynamical decoupling group commutate, which implies that the group is an Abelian group, while in the odd qubits case, these operators form a non-Abelian group that has different group algebra. We consider them both and show how to find the DFS/NS in each case. Further, by finding the basis in the DD based DFS/NS, the encoding process has been addressed and the logical manipulation of the encoding states is presented. A universal set of qubit gates are constructed in both cases.Third, we show how to speed up the adiabatic HQC to non-adiabatic regime. Non-degenerate adiabatic process can be conducted in arbitrary rate by transi-tionless quantum driving algorithm (TQDA). To apply this algorithm to adiabatic HQC, we generalize TQDA to degenerate subspace, where non-Abelian geometric phases (quantum holonomies) are acquired after a cyclic evolution. We show how this generalized TQDA can be applied to realize high speed adiabatic HQC. This algorithm offers a general approach to construct HQC in quantum systems whose state space contains degenerate subspaces. To achieve the holonomies, we encode our qubit in a subspace of the system that undergoes a cyclic path. With an addi-tional Hamiltonian, non-adiabatic evolution is achieved. In this way, TQDA based geometric phases of both Abelian and non-Abelian kinds are realized. However, the required additional Hamiltonian may spoil the experimental feasibility of the original scheme since it introduces new energy transitions between various levels as well as level detunings. While the detunings could cause control errors, the new transitions imply new symmetries in the system. We show that this problem can be resolved by choosing specific paths in the parameter space. Finally, we show how the proposed scheme can be attainable in a superconducting circuit architecture.