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arXiv:1410.5057

Signature of Non-Abelian to Abelian Transition in Spin Systems Through Geometric Phase

D. De Munshi, Manas Mukherjee

Published Oct 19, 2014• 0 citations
arXivHTMLPDF

Abstract

Abelian and Non-Abelian evolution of a quantum system manifests differently in the geometric phase acquired by the system under such evolutions. In this work we develop and study, using dressed state techniques, an experimentally realizable spin system which allows us to transit smoothly from non-Abelian to Abelian evolutions by changing externally controllable parameters. The study provides insights into the underlying physical phenomenon governing such a transition allowing us greater control on phase generation in a quantum system. The robustness of geometric phase against fluctuations of the external parameters of the system has also been studied in this work. The noise analysis has direct consequence on the present search for fault tolerant quantum gates using geometric phase.

DOI
10.48550/arxiv.1410.5057

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Abelian and Non-Abelian evolution of a quantum system manifests differently in the geometric phase acquired by the system under such evolutions. In this work we develop and study, using dressed state techniques, an experimentally realizable spin system which allows us to transit smoothly from non-Abelian to Abelian evolutions by changing externally controllable parameters. The study provides insights into the underlying physical phenomenon governing such a transition allowing us greater control on phase generation in a quantum system. The robustness of geometric phase against fluctuations of the external parameters of the system has also been studied in this work. The noise analysis has direct consequence on the present search for fault tolerant quantum gates using geometric phase.
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.
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V(\mathbit{\ensuremath{\lambda}}\left(t\right))$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter $\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.
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We have developed an adiabatic Abelian geometric quantum computation strategy based on the non-degenerate energy eigenstates in (but not limited to) superconducting phase-qubit systems. The fidelity of the designed quantum gate was evaluated in the presence of simulated thermal fluctuation in superconducting phase qubits circuit and was found to be rather robust against the random errors. In addition, it was elucidated that the Berry phase in the designed adiabatic evolution may be detected directly via the quantum state tomography developed for superconducting qubits.
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Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies alike. The exotic statistics of anyons realized in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here, we use an all-optical quantum system that simulates the statistical evolution of Majorana fermions. As a result, we experimentally realize non-Abelian Berry phases with the topological characteristic that they are invariant under continuous deformations of their control parameters. We implement a universal set of Majorana-inspired gates by performing topological and nontopological evolutions and investigate their resilience against perturbative errors. Our photonic experiment, though not scalable, suggests the intriguing possibility of experimentally simulating Majorana statistics with scalable technologies.