Conceptual aspects of geometric quantum computation

Abstract

Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic evolution, controlled by slowly changing parameters, this form of quantum computation can as well be realized at high speed by using nonadiabatic schemes. Recent advances in quantum gate technology have allowed for experimental demonstrations of different types of geometric gates in adiabatic and nonadiabatic evolution. Here, we address some conceptual issues that arise in the realizations of geometric gates. We examine the appearance of dynamical phases in quantum evolution and point out that not all dynamical phases need to be compensated for in geometric quantum computation. We delineate the relation between Abelian and non-Abelian geometric gates and find an explicit physical example where the two types of gates coincide. We identify differences and similarities between adiabatic and nonadiabatic realizations of quantum computation based on non-Abelian geometric phases.

Revised version; journal reference added

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Cited by in corpus (17)

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• Accelerating geometric quantum gates through non-cyclic evolution and shortcut to adiabaticity
• Topological Uhlmann phase transitions for a spin-j particle in a magnetic field
• Exploring Non-Abelian Geometric Phases in Spin-1 Ultracold Atoms
• Conditions for Equivalent Noise Sensitivity of Geometric and Dynamical Quantum Gates
• Designing dynamically corrected gates robust to multiple noise sources using geometric space curves
• Universal non-adiabatic geometric manipulation of pseudo-spin charge qubits
• Non-Abelian Geometric Phases Carried by the Spin Fluctuation Tensor
• Effective Hamiltonian theory of the geometric evolution of quantum systems
• Fast Evolution of Single Qubit Gate in Non-Adiabatic Geometric Quantum Computing
• Estimate of the time required to perform a nonadiabatic holonomic quantum computation
• Geometric phase through spatial potential engineering
• Observable-geometric phases and quantum computation
• Isoholonomic inequality and tight implementations of holonomic quantum gates
• Parallel transport in rotating frames and projective holonomic quantum computation
• Geometric and holonomic quantum computation