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

Highlights

• Analytical description of heating up and cooling down in terms of energy level occupancy of interacting position-based qubit describing Q-SWAP gate with 4 energy levels.

• Interface between semiconductor position-based qubit and Josephson junction in tight- binding model with numerical solutions.

• Quantum gates as phase rotating gate, Hadamard gate and Q-Swap for position-based semiconductor qubits in analytical description.

• Procedure of extending the network of coupled quantum dots in tight-binding model forposition-based qubits.

• Quantum state evolution with time expressed by analytical formulas and procedure its derivation for the case of N body electrons confined on N disconnected graphs of coupled q-dots of any topology in 2 and 3-dimensional space.

Abstract

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.