Observable-Geometric Phases and Quantum Computation

Abstract

This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observablegeometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. IntroductionThe geometric phase can be used in quantum computing by implementing each of quantum gates in terms of geometric phases only [12]. This scheme for computing is called geometric quantum computation. Geometric quantum computation involves adiabatic or nonadiabatic, as well as Abelian or non-Abelian characteristics of the underlying quantum evolution for the quantum state. It is shown in [14,15] that the non-adiabatic Abelian geometric phases can be used to realize a universal set of quantum gates and thus are sufficient for universal all-geometric quantum computation. Furthermore, the non-adiabatic non-Abelian geometric phases are shown in [11] to achieve universality as well. We refer to [10] for a up-date review and background.Here, we present an alternative approach to the geometric phase from the observable point of view. Precisely, we introduce the concept of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the geometry of the observable space evolving according to the Heisenberg equation. In contrast to that classical (Riemannian) geometry of the state space is suited to the usual geometric phases, the geometry suited for the observable-geometric phases is a quantum geometric structure over the observable space induced by a complete measurement. This quantum-geometric structure of the observable space will be described in details in Section 5.The paper is organized as follows. In Section 2, we will define the notion of the observable-geometric phase using the propagator of a time-dependent Hamiltonian, which is shown to be independent of of the choice of the Hamiltonian as long as the Heisenberg equations involving the Hamiltonian. Two examples are included in which qubit systems subject to an orientated magnetic field, a rotating magnetic field, and a rotating plus a constant magnetic field. Subsequently, in Section 3, we show that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. The quantum gate based on the observable-geometric phases can be interpreted as the Abelian geometric gates of Zhu and Wang [14,15]; indeed the two schemes lead to the same gates. However, the ZW gates are related with the classical geometry of the state space of the system, while ours are connected with the quantum-geometric structure over the observable space. We wil...

Geometric Phases and The Sagnac Effect: Foundational Aspects and Sensing Applications

“…On the application side, particular attention was given to gravitational and non-inertial measurements. However, geometric phases are also an important player in quantum information and computation, [205][206][207] chemical physics, [5,22,208,209] and many other areas.…”

A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases

“…1) Note that for every n ≥ 1, β n may be a complex number (see Example 5.2 below). This is different from the ones of a quantum system in the Hermitian case as defined in [11]. 2) If h(t)'s are all Hermitian, then ψ * n (t) = ψ n (t) and the observable-geometric phases β n 's are all real and coincide with the ones defined in [11].…”

“…As usual, these geometric phases are associated with the quantum state. Recently, the notion of the geometric phase for the observable (the so-called observable-geometric phase) was introduced in [11], which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. In this section, we will study the observable-geometric phase in the non-Hermitian setting.…”

“…Observable-geometric phases. This subsection is devoted to the study of observable-geometric phases for non-Hermitian quantum systems, which was introduced in [11] for a quantum system in the Hermitian case. We first define this notion in the non-Hermitian case using the evolution system mentioned in Section 2.4.…”

“…where ψ * n (t) = e −iαn(t) |ψ * n (t) for every n ≥ 1. Following [11], this leads to the notion of the observable-geometric phase in the non-Hermitian setting as follows.…”