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Geometric phases associated with the quantal observable space

Zeqian Chen

Published Oct 11, 2017• 0 citations
OpenAlex

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 observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. This scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang \cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space beyond the state space.

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Geometric phase that manifests itself in number of optic and nuclear experiments is shown to be a useful tool for realization of quantum computations in so called holonomic quantum computer model (HQCM). This model is considered as an externally driven quantum system with adiabatic evolution law and finite number of the energy levels. The corresponding evolution operators represent quantum gates of HQCM. The explicit expression for the gates is derived both for one-qubit and for multi-qubit quantum gates as Abelian and non-Abelian geometric phases provided the energy levels to be time-independent or in other words for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed for a Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupole NMR model (two-qubit gates). Generic quantum gates for these two models are derived. The possibility of construction of the universal quantum gates in both cases is shown.

arXiv:0707.0044•Jun 30, 2007•0 citations
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 observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. This scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang \cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space beyond the state space.
The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.
Geometric quantum phases enable the construction geometric quantum gates, whose action only depend on the evolution path in state space. Geometric gates may offer increased robustness to control parameter noise. In this report, the conditions under which non-adiabatic geometric and holonomic quantum gates preserve their geometric character under sequential composition are analysed. Grover’s search algorithm requiring non-commuting operations is used as a representative example. The basic concepts of qubits, quantum gates and quantum circuits are explained. The concept of geometric phases is introduced and derivations of Abelian and non-Abelian non-adiabatic geometric phases are provided (Aharonov-Anandan and Anandan phase respectively). Construction of geometric and holonomic universal quantum gate sets with the aforementioned geometric phases are briefly described.Explicit calculation of two non-commuting Abelian and non-adiabatic geometric time evolutions show that the successively combined operation is not geometric. A proof of successive combinations of non-Abelian and non-adiabatic geometric time evolutions preserving their non-adiabatic Abelian and geometric properties is given. The report concludes that it is in general not possible to perform quantum circuits completely geometrically with a geometric universal quantum gate set. However, performing quantum computations with gates from a holonomic universal gate set is geometric and holonomic throughout the computation.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek–Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed. The conception of quantum computer (QC) and quantum calculation developed in 80-th [1]–[2] were found to be fruitful both for computer science and mathematics as well as for physics [3]. Although a device being able to perform quantum calculations is far away from practical realization, there is a number of theoretical proposals of such a construct [4]–[16]. Recently holonomic realization of QC was proposed [17],[18]. It is based on the notion of non-Abelian Berry phase[19].To perform quantum computations in this approach one needs a parametrically driven quantum system described by the Hamiltonian H(λ) with adiabatically evolving parameters λ(t) = (λ1(t),..., λN(t)). Adiabaticity means that Ω ≪ ωmin where Ω is the characteristic frequency in the Fourier spectrum of λ(t) and ωmin is the minimal transition frequency in the spectrum
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.
Geometric phase that manifests itself in number of optic and nuclear experiments is shown to be a useful tool for realization of quantum computations in so called holonomic quantum computer model (HQCM). This model is considered as an externally driven quantum system with adiabatic evolution law and finite number of the energy levels. The corresponding evolution operators represent quantum gates of HQCM. The explicit expression for the gates is derived both for one-qubit and for multi-qubit quantum gates as Abelian and non-Abelian geometric phases provided the energy levels to be time-independent or in other words for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed for a Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupole NMR model (two-qubit gates). Generic quantum gates for these two models are derived. The possibility of construction of the universal quantum gates in both cases is shown.