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arXiv:1410.5891

Factorization of unitary matrices induced by 3D anisotropic Ising interaction

Francisco Delgado

Published Oct 22, 2014• 2 citations
arXivHTMLPDF

Abstract

Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements being involved. This work shows how anisotropic Ising interactions, written in a non local basis, lets reproduce elementary operations in terms of which unitary processes can be factorized. In this sense, gates could be written as a sequence of pulses ruled by that interaction driven by magnetic fields, stating alternative results in quantum gates design for magnetic systems.

DOI
10.48550/arxiv.1410.5891

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Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements being involved. This work shows how anisotropic Ising interactions, written in a non local basis, lets reproduce elementary operations in terms of which unitary processes can be factorized. In this sense, gates could be written as a sequence of pulses ruled by that interaction driven by magnetic fields, stating alternative results in quantum gates design for magnetic systems.
Quantum information and quantum computation are emerging research areas based on the properties of quantum resources, such as superposition and entanglement. In the quantum gate array version, the use of convenient and proper gates is essential. While these gates adopt theoretically convenient forms to reproduce computational algorithms, their design and feasibility depend on specific quantum systems and physical resources used in their setup. These gates should be based on systems driven by physical interactions ruled by a quantum Hamiltonian. Then, the gate design is restricted to the properties and the limitations imposed by the interactions and the physical elements involved. This work shows how anisotropic Heisenberg-Ising interactions, written in a non-local basis, allow the reproduction of quantum computer operations based on unitary processes. We show that gates can be generated by a pulse sequence of driven magnetic fields. This fact states alternative techniques in quantum gate design for magnetic systems. A brief final discussion around associated fault tolerant extensions to the current work is included.
Control of quantum entanglement has been considered as elemental physical resource for quantum applications in Quantum Information and Quantum Computation. Control of entangled states on a couple of atoms, ions or quantum dots are milestones in almost all quantum applications towards a scalable spin-based quantum computers or quantum devices. For magnetic systems, Ising model is an interaction which generates and modifies entanglement properties of quantum systems based on matter. In addition, when this interaction includes driven magnetic fields, it can be controlled to sustain, characterize or modify entanglement and other quantum properties. In this work, recent results about evolution in a general anisotropic three dimensional Ising model including an inhomogeneous magnetic field is considered to obtain some general quantum control effects for their sustainability, programmed evolution or transformation: Evolution loops and Exchange operations. This control is achievable through a set of physical parameters, whose prescriptions are reported. The use of a non local basis in the model to express time evolution lets take advantage to describe and control the system, in particular with those issues associated with entanglement and operations mentioned before. Finally, some analysis about equivalent gates based on our development is made including an example with teleportation, using one of the gates constructed.
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Entanglement is considered a basic physical resource for modern quantum applications as Quantum Information and Quantum Computation. Interactions based on specific physical systems able to generate and sustain entanglement are subject to deep research to get understanding and control on it. Atoms, ions or quantum dots are considered key pieces in quantum applications because they are elements in the development toward a scalable spin-based quantum computer through universal and basic quantum operations. Ising model is a type of interaction generating entanglement in quantum systems based on matter. In this work, a general bipartite anisotropic Ising model including an inhomogeneous magnetic field is analyzed in a non-local basis. This model summarizes several particular models presented in literature. When evolution is expressed in the Bell basis, it shows a regular block structure suggesting a [Formula: see text] decomposition. Then, their algebraic properties are analyzed in terms of a set of physical parameters which define their group structure. In particular, finite products of pulses in this interaction are analyzed in terms of [Formula: see text] covering. Thus, evolution denotes remarkable properties, in particular those related potentially with entanglement and control, which give a fruitful arena for further quantum developments and generalization.
Quantum information and quantum computation is a rapidly emergent field where quantum systems and their applications play a central role. In the gate version of quantum computation, the construction of universal quantum gates to manipulate quantum information is currently an intensive arena for quantum engineering. Specific properties of systems should be able to reproduce such idealized gates imitating the classically inspired computational gates. Recently, for magnetic systems driven by the bipartite Heisenberg-Ising model a universal set of gates has been realized, an alternative easy design for the Boykin set but using the Bell states as grammar. Exact control can be then used to construct specific prescriptions to achieve those gates. Physical parameters impose a challenge in the gate control. This work analyzes, based on the worst case quantum fidelity, the associated instability for the proposed set of gates. An strong performance is found in those gates for the most of quantum states involved.