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Analytical solutions for N interacting electron system confined in graph of coupled electrostatic quantum dots in tight-binding model

Krzysztof Dominik Pomorski

Published Jul 6, 2019• 0 citations
OpenAlex

Abstract

Analytical solutions for a tight-binding model are presented for a position-based qubit and N interacting qubits realized by quasi-one-dimensional network of coupled quantum dots expressed by connected or disconnected graphs of any topology in 2 and 3 dimensions where one electron is presented at each separated graphs. Electron(s) quantum dynamic state is described under various electromagnetic circumstances with an omission spin degree-of-freedom. The action of Hadamard and phase rotating gate is given by analytical formulas derived and formulated for any case of physical field evolution preserving the occupancy of two-energy level system. The procedure for heating up and cooling down of the quantum state placed in position based qubit is described. The interaction of position-based qubit with electromagnetic cavity is described. In particular non-local communication between position based qubits is given. It opens the perspective of implementation of quantum internet among electrostatic CMOS quantum computers (quantum chips). The interface between superconducting Josephson junction and semiconductor position-based qubit implemented in coupled semiconductor q-dots is described such that it can be the base for electrostatic interface between superconducting and semiconductor quantum computer. Modification of Andreev Bound State in Josephson junction by the presence of semiconductor qubit in its proximity and electrostatic interaction with superconducting qubit is spotted by the minimalistic tight-binding model. The obtained results allow in creating interface between semiconductor quantum computer and superconducting quantum computer. They open the perspective of construction of QISKIT like software that will describe both types of quantum computers as well as their interface.

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Analytical solutions for a tight-binding model are presented for a position-based qubit and N interacting qubits realized by quasi-one-dimensional network of coupled quantum dots expressed by connected or disconnected graphs of any topology in 2 and 3 dimensions where one electron is presented at each separated graphs. Electron(s) quantum dynamic state is described under various electromagnetic circumstances with an omission spin degree-of-freedom. The action of Hadamard and phase rotating gate is given by analytical formulas derived and formulated for any case of physical field evolution preserving the occupancy of two-energy level system. The procedure for heating up and cooling down of the quantum state placed in position based qubit is described. The interaction of position-based qubit with electromagnetic cavity is described. In particular non-local communication between position based qubits is given. It opens the perspective of implementation of quantum internet among electrostatic CMOS quantum computers (quantum chips). The interface between superconducting Josephson junction and semiconductor position-based qubit implemented in coupled semiconductor q-dots is described such that it can be the base for electrostatic interface between superconducting and semiconductor quantum computer. Modification of Andreev Bound State in Josephson junction by the presence of semiconductor qubit in its proximity and electrostatic interaction with superconducting qubit is spotted by the minimalistic tight-binding model. The obtained results allow in creating interface between semiconductor quantum computer and superconducting quantum computer. They open the perspective of construction of QISKIT like software that will describe both types of quantum computers as well as their interface.
Analytical solutions for a tight-binding model are presented for position-based charge qubits and N-interacting qubits realized by quasi-one-dimensional network of coupled quantum dots (QD) expressed by connected or disconnected graphs of any topology in two and three dimensions where one electron is present at each separated graph. Electron(s) quantum dynamical state is described under various electromagnetic circumstances with omission spin degree-of-freedom. The action of Hadamard and phase-rotating gates is given by analytical formulas derived and formulated for any case of a physical field evolution preserving the occupancy of a two-energetic-level system. The interface between a superconducting Josephson junction and semiconductor position-based qubit implemented in coupled semiconductor QDs is described, which can be the base for electrostatic interface between superconducting and semiconductor quantum computer. Modification of Andreev Bound State in Josephson junction by the presence of semiconductor qubit in its proximity and electrostatic interaction with superconducting qubit is supported by the minimalistic tight-binding model that also can indicate topological phase transition. The obtained results can be generalized for a case of electrostatic interaction of the semiconductor qubit with the different types of Josephson junctions. The operation of phase rotating gate implemented in semiconductor position-based qubit with use of electrical and magnetic field is presented with use of analytical approach.
The interface between a superconducting Josephson junction and a semiconductor position-based qubit implemented in coupled semiconductor q-dots is described as a possible base for an electrostatic interface between superconducting and semiconductor quantum computers.A modification of an Andreev bound state in a Josephson junction by the presence of a semiconductor qubit in its proximity and electrostatic interaction with a superconducting qubit is spotted by the minimalist tight-binding model.The obtained results allow for the creation of an interface between a semiconductor quantum computer and a superconducting quantum computer.The existence of topological states of matter is indicated.They open a perspective for the construction of QISKIT-like software that will describe both types of quantum computers as well as their interface.
Non-local communication between position-based qubits is described for a system of a quantum electromagnetic resonator entangled to two semiconductor electrostatic qubits via an interaction between matter and radiation by Jaynes-Cummings tight-binding Hamiltonian. Principle of quantum communication between position-dependent qubits is explained. Further prospects of the model development are given. The obtained results bring foundation for the construction of quantum internet and quantum communication networks between position-based qubits that are implementable in semiconductor single-electron devices that can be realized in current CMOS technologies. The case of two semiconductor position-dependent qubits interacting with quantum electromagnetic cavity is discussed and general form of tight-binding Hamiltonian is derived with renormalized tight-binding coefficients. The considerations also describe the situation of mutual qubit-qubit electrostatic interaction.
Analytical solutions describing quantum swap and Hadamard gate are given with the use of tight-binding approximation. Decoherence effects are described analytically for 2 interacting electrons confined by local potentials with use of tight-binding simplistic model and in Schroedinger formalism with omission of spin degree of freedom. The obtained results can be generalized for the case of [Formula: see text] electrostatically interacting quantum bodies confined by local potentials ([Formula: see text]-qubit) system representing any electrostatic quantum gate with [Formula: see text] inputs/outputs. The mathematical structure of system evolution with time is specified.
Generalized electrostatic quantum swap gate implemented in the chain of 2 double coupled quantum dots using single electron in semiconductor is presented in tight-binding simplistic model specifying both analytic and numerical results. The anticorrelation principle coming from Coulomb electrostatic repulsion is exploited in single electron devices. The formation of quantum entanglement is specified and supported by analytical results. The difference between classical and quantum picture is given. The correlations between geometry of quantum structures and entanglement dynamics are shown in analytical way. Effective wavefunction of single electrons is extracted from 2 electron electrostatic interactions. The presented results have its significance in cryogenic CMOS quantum technologies that gives perspective of implementation of semiconductor quantum computer on massive scale. Keyword: electrostatic quantum swap gate, entanglement, anticorrelation, tight-binding model, two-body wavefunction, correlation-anticorrelation crossover, density matrix of 2 electron system, single electron devices, coupled quantum dots
In this study, a formal definition, robustness analysis and discussion on the control of a position-based semiconductor charge qubit are presented. Such a qubit can be realized in a chain of coupled quantum dots, forming a register of charge-coupled transistor-like devices, and is intended for CMOS implementation in scalable quantum computers. We discuss the construction and operation of this qubit, its Bloch sphere, and relation with maximally localized Wannier functions which define its position-based nature. We then demonstrate how to build a tight-binding model of single and multiple interacting qubits from first principles of the Schrödinger formalism. We provide all required formulae to calculate the maximally localized functions and the entries of the Hamiltonian matrix in the presence of interaction between qubits. We use three illustrative examples to demonstrate the electrostatic interaction of electrons and discuss how to build a model for many-electron (qubit) system. To conclude this study, we show that charge qubits can be entangled through electrostatic interaction.