Non-Abelian Berry connections for quantum computation

Abstract

In the holonomic approach to quantum computation information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible, for a specific model, to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multi-partite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured.Comment: 5 pages, no figures, revtex, minor changes, version accepted by Phys. Rev A (rapid comm.

Mathematical foundations of holonomic quantum computer

“…Zanardi and Rasetti in [6] and [7] proposed such an idea using non-abelian Berry phase (quantum holonomy). See also [8] and [9] as another geometric models.…”

Geometric quantum computation on solid-state qubits

“…Recently, it has been proposed that controllable quantum operations can be achieved by a novel geometric principle as well [2,3]. When a quantum system undergoes an adiabatic cyclic evolution, it acquires a nontrivial geometric operation called a holonomy.…”

“…Otherwise, it becomes a non-Abelian unitary operation, i.e., a non-trivial rotation in the eigenspace. It has been shown that universal quantum computation is possible by means of holonomies only [2,3]. Further, holonomic quantum computation schemes have intrinsic tolerance to certain types of computational errors [4,5].…”

Self-protected quantum algorithms based on quantum state tomography

“…For example, decoherence-free subspaces (DFS) and noiseless subsystems (NS) [21,22,23,24] are based on the symmetry of the system-bath interaction, so do not require active detection and correction of errors. Another passive technique, holonomic quantum computation, is robust against stochastic errors in the control process [25,26]. When conditions are appropriate, passive strategies can be applied during the design of quantum algorithms.…”