Non-Abelian Berry connections for quantum computation

The field of quantum information and computation (QC) [1] brings together ideas and techniques from very different areas ranging from fundamental quantum physics to solid-state engineering and computer science. QC synergetically benefits from all these contributions and conversely quite often offers fresh viewpoints on old subjects. Recently it has been suggested [2] that even tools related to gauge theories [3] might play a fruitful role in the arena of QC. Indeed in ref. [2] the possibility of realizing quantum information processing by using nonabelian Berry holonomies [4] induced by moving along suitable loops in a control space M has been analysed. The computational capability stems from the features of the global geometry of the bundle of eigenspaces associated with a family F of Hamiltonians parametrized by points of M. The geometry is described by a non-trivial gauge potential A or connection, with values in the algebra u(n) of anti-hermitian matrices (n is the dimension of the computational space). Since the unitary transformations realizing the computations are nothing but the holonomies associated with the connection A, this conceptual framework for QC is referred to as Holonomic Quantum Computation (HQC). In a sense HQC can be considered as the (continuous) differential-geometric counterpart of the (discrete) topological QC with anyons described in refs. [5,6]. In this paper we shall provide further analysis of this proposal. After concisely reviewing the conceptual basis of HQC, we shall show how, in a specific relevant model, one can explicitly determine the sequence of loops necessary for generating any given quantum gate. Then we shall introduce HQC models with a natural multi-partite structure and discuss how this bears on the question of complexity. Finally we shall discuss how in principle one can implement HQC by repeated pulse control of a system with degenerate spectrum. Let us begin by recalling the basic ideas of HQC [2]. Quantum information is encoded in a n-fold degenerate eigenspace C of a Hamiltonian H0, with eigenvalue ε0. Operator, H0, belongs to a family F = {Hλ}λ∈M, H0 = Hλ0 , in which no energy level crossings occur as λ ranges over M. In the following we shall satisfy this latter condition by assuming, for simplicity, that the Hamiltonians Hλ are isospectral (Hλ = U(λ)H0 U(λ)†). The λ’s represent the “control” parameters that one has to drive in order to manipulate the coding states |ψ〉 ∈ C. In general the points of M, from the physical point of view, can be thought of as describing external fields, such as electric or magnetic fields, or couplings between subsystems. Let C be a loop in the control manifold M, with base point λ0 , C: [0, 1] 7→ M, C(0) = C(1) = λ0. We assume that C is traveled along slowly with respect to the longest dynamical time scale involved: in this case the evolution is adiabatic i.e., no transitions among different energy levels are induced. If |ψ〉in ∈ C is the initial state, at the end of this control process one gets |ψ〉out = e ε0 T ΓA(C) |ψ〉in. The first factor here is just an overall dynamical phase and in the following it will be omitted; let us just mention that such a decoupling of the fast dynamical evolution opens new possibilities about coherent and error avoiding encoding [6]. The second contribution, the holonomy ΓA(C) ∈ U(n), has a purely geometric origin and its appearance accounts for the non-triviality (curvature) of the bundle of eigenspaces over M. By introducing the Wilczek-Zee connection [7]