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

Analytical solutions for tight-binding model are presented for 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 1 electron is presented at each separated graphs. Electron(s) quantum dynamical state is described under various electromagnetic circumstances with omission spin degree of freedom. The action of Hadamard and phase rotating gate is given by analytic formulas derived and formulated for any case of physical field evolution preserving the occupancy of two energetic level system. The interface between superconducting Josephson junction and semiconductor position based qubit implemented in coupled semiconductor q-dots is described what 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 can be generalized for the case of interaction semiconductor qubit interacting electrostatically with Field Induced Josephson junction.


INTRODUCTION TO POSITION BASED SEMICONDUCTOR QUBIT
Single electron semiconductor devices has its importance in implementation of quantum computer as well as in implementation of low power consumption CMOS classical computer and was studied by Fujisawa, 4 Petta, 5 Leipold, 1 Giounanlis, 2 Pomorski 7 and many others. On another hand one of the most successful model in condensed matter physics is Hubbard model and its special case known as tight-binding model. 6 We consider two energy level system of position based qubit in tight-binding approach as depicted in Fig.1. Its extensions are given in Fig.4, where one can consider graphs of any shape representing the chain of coupled quantum dots in 2 and 3 dimensions of arbitrary topology. The Hamiltonian of this system is given aŝ The HamiltonianĤ(t) eigenenergies E 1 (t) and E 2 (t) with E 2 (t) > E 1 (t) are given as and energy eigenstates |E 1 (t) and |E 2 (t) have the following form This Hamiltonian is giving description of 2 coupled quantum wells as depicted in Fig.1. In such situation we have real valued functions E p1 (t),E p2 (t) and complex valued functions t s12 (t) = t s (t) = t sr (t) + it si (t) and t s21 (t) = t * s12 (t) what is equivalent to knowledge of 4 real valued time-dependent continuous or discontinues functions E p1 (t), E p1 (2) , t sr (t) and t si (t). The quantum state is superposition of state localized at node 1 and 2 and therefore is given as where |α(t)| 2 (|β(t)| 2 ) is probability of finding particle at node 1 (2) respecitvely what brings |α(t)| 2 + |β(t)| 2 = 1 and obviously 1, 0| x ||1, 0 x = 1 = 0, 1| x ||0, 1 x and 1, 0| x ||0, 1 x = 0 = 0, 1| x ||1, 0 x . In Schroedinger formalisms states |1, 0 x and |0, 1 x are Wannier functions that are parametrized by position x. We work in tight-binding approximation and quantum state evolution with time is given as Last equation has the analytic solution and in quantum density matrix theory we obtain Having Hermitian matrixÂ with real valued coefficients a 11 (t), a 22 (t), a 12r (t), a 12i (t) we observe that Ep2(t1)dt1) (9) Final quantum state at time t evolving from quantum state given at time t 0 can be written as  Ep2(t1)dt1) β(0)) where |c E1 (t)| 2 is probability of occupancy of energetic level E 1 and |c E2 (t)| 2 is probability of occupancy of energetic level E 2 by quantum state |ψ, t and We notice that the quantum state norm N (t) ∈ R is changing with time in accordance to formula what brings the probability of finding the quantum state at node 1 as . (12) and brings the probability of finding the quantum state at node 2 as Phase of wavefunction at node 1 is and phase of wavefunction at node 2 is Phase rotating gate changes . Hadamard gate operation can be defined as operation changing quantum state at time t ψ( , t sr (t ), t si (t ) of certain shape in time t ∈ (t, t 3 ). Having nonorthodox approach we might find class of functions approximately (not exactly) satisfying operation of phase rotating gate or Hadamard gate for the case of 2 energy level system. Obviously going into situation more than 2 energy level system one can find richer class of possible scenarios by considerations of broader class of position based qubit controlling electromagneti signals. It is thus valid subject of study in pursue for better implementation of quantum technologies. Once solution is found in tight-binding model it can be relatively easily transferred into Schroedinger formalism what brings conclusion on quantum circuit controlling procedures. Before moving to the particular physical examples let us examine the properties of evolution of density matrix with time aŝ .

ACTION OF PHASE ROTATING GATE DESCRIBED ANALYTICALLY
Let us consider the situation of single qubit from Fig.1 when we assume the following dependencies: E p1 (t) = E p2 (t) = E p = constant and t s12 (t) = t s21 (t) = t s (t) = constant 1 . In such we have two time-independent eigenenergies E 1 = E p − t s and E p + t s . For simplicity we assume (α(0) ∈ R), (β(0) ∈ R) .The probability of finding electron at node 1 is given by angle Θ at Bloch sphere expressed as and it oscillates periodically with frequency proportional to distance between energetic levels E 2 and E 1 and is given as ω 0 = E2−E1 h . Therefore the same occupancy at node is repeating with periodic time t d = n 2πh E2−E1 for integer n. Obviously probability of finding of particle at node 2 is P 2 = 1 − P 1 . The phase difference between wavefunctions at node 1 and 2 is denoted as φ(t) and can be expressed analytically by formula We recognize that three frequencies are involved h in the dynamics of phase difference of quantum state between nodes 2 and 1. What is more phase difference across position based qubit between nodes 1 and 2 is codepedent on the occupancy of the left and right node as given by equation 18 in the case of time-independent Hamiltonian. Such situation is not taking place in most conventional qubits using energy eigenbases to encode information but takes place in position based semiconductor qubit. The ideal phase rotating gate implemented in position based qubit brings desired phase difference between wavefunctions at nodes 2 and 1 is not changing the occupancy of node 1 and 2. If we want to keep the occupancy from time t=0 we need to consider times t d = n 2πh E2−E1 . At time t=0 phase difference was assumed to be 0. If we want to achieve phase difference φ d the following condition must be meet so positive integer n can be determined in unique way. It simply means that phase imprinted quantum state will have these feature at time t = n s 2πh E2−E1 . In such way action of phase rotating can be implemented in electrostatic position dependent qubit given in Fig.1.

ACTION OF HADAMARD GATE IN POSITION QUBIT
The Hadamard gate is able to conduct the following unitary transformation on quantum state |ψ(t) and is given as It has property U † Hadamard = U Hadamard and U Hadamard U † Hadamard = 1 so double action of Hadamard gate gives U Hadamard U −1 Hadamard = 1. Let us concentrate on the position dependent qubit with time-independent parameters E p1 , E p2 = E p1 = E p , t s ∈ R. In such case we obtain following eigenenrgies E 1 = E p − t s and E 2 = E p + t s . From simple calculations we can notice that two eigenergies E 1 = E p − t s and E 1 = E p + t s have corresponding eigenstates that are orthonormal so 1, 0|1, 0 = 0, 1|0, 1 = 1 and 1, 0|0, 1 = 0, 1|1, 0 = 0. At the same time E 1 |E 1 = E 2 |E 2 = 1 and E 1 |E 2 = E 2 |E 1 = 0. We recognize that formula 21 can be written in the compact form as We recognize that quantum transformation is naturally encoded in transformation from position quantum system eigenbases into energy eigenbases. Quantum logical 0 can be spanned (represented) by state |1, 0 x = |0 L (presence of electron in qubit on the left side in Fig.1) and quantum logical 1 can be spanned (represented) by the state |0, 1 x = |1 R (presence of electron in qubit on the right side). Therefore qubit state shall be defined by Action of Hadamard gate requires We recognize that quantum logical 0 or presence of state (electron) in left well is achieved when there is equal occupancy (given by c E1 ) of energetic level E 1 and E 2 so |c E1 (t)| 2 = |c E2 (t)| 2 . The scheme how to change the complete occupancy of energetic level E 1 into full occupancy of energetic level E 2 is given by formula 67 that is associated with time-dependent Hamiltonian applied to position based qubit. The quantum state is given as Such state will evolve after characteristic time from logical state |0 L into quantum logical |1 L and later into |0 L and so on. We can also set logical quantum state in position space parametrized by x and we can read the results of Hadamard operation action in energy space or reversely. Engineer has the choice of setting qubit state in position space (what is more intuitive if one aims to obtain high integration circuits) or in energy space. By setting the quantum state in position space (as by injecting electron from left side into left well of qubit) one needs to read it by energy space or reversely. Reading quantum state after Hadamard operation (or any other quantum operation) in energy space requires either spectroscopy of occupation of energy levels what basically means that we need to use microwaves in order to populate or depopulate given energy level(s). Alternative method for reading the qubit state after Hadamard operation (or any other quantum operation)is determination the state of neighbouring qubit that interacts with measured qubit in electrostatic way as it is depicted in the right side of Fig.1. Determination of occupancy of energy level E 1 and E 2 will give us the information on the qubit state after Hadamard operation (so presence of at least 2 energy levels in physical system is the requirement) and formally we have In order to trace the situation in physical system we start from consideration of time-independent Hamiltonian and quantum state dynamics given by ) is the probability of occupancy of first energetic level E 1 and |c E2,t | 2 = constans1 = |c E2,t0 | 2 ((since c E2,t = e 1 hi E2(t−t0) c E2,t0 )) is the probability of occupancy of second energetic level E 2 so |c E1,t | 2 + |c E2,t | 2 = 1. We have obtained oscillations of occupancy at node 1 and node 2 given as and those results are the most simple case of solutions for the case of 2 energy level system and can be compared with more general formula 16. We can spot that phase different between coefficients α(t) and β(t) as well as phase difference c E1 (t) and c E2 (t) has sinusoidal periodicity with frequency of oscillation proportional to difference between eigenergies ω 0 = E2−E1 h . Normalization condition for coefficients α t and β t given by |α t | 2 + |β t | 2 = 1 automatically implies normalization condition for c E1,t and c E2,t given by |c E1,t | 2 + |c E2,t | 2 = 1 and reversely. Such reasoning can be easily extended for more than 2 enegetic levels as given by Hamiltonian 4 by 4 describing position dependent qubit.
We refer to the situation from Fig.1 with presence of microwave field that comes from external antenna or from AC voltage component applied to 3 gates controlling the state of position based qubit. In case of lack of time-dependent fields the occupancy of energetic levels E1 and E2 are unchanged with time. The Hamiltonian able to where f 1 (t) and f 2 (t) are time-dependent functions.
Final Hamiltonian under microwave field or time-dependent voltages applied to 3 gates controlling qubit becomeŝ . (31) if one generalizes f 1 (t) and f 2 (t) functions to be complex valued. Therefore finally Hamiltonian can be written in terms of real-valued functionŝ .
The obtained eigenenergies are and energy eigenstate
(34) Heating up of any quantum system can be considered as the increase of occupancy of higher energy levels and depopulation of occupancy of lower energetic levels. In the case of position based qubit described in Fig.1 it simply means that the occupancy of ground energy state E 1 is decreased while occupancy of first excited energy state E 2 is increased. In such situation of constant coefficients E p1 , E p2 , t s12 = t * s21 with time the occupancy of each among two energy levels remains constant with time.
The Hamiltonian increasing the occupancy of eigenenergy state E 2 ("heating up" ) can be described aŝ whereĤ 0 is time independent Hamiltonian and time-dependent part of Hamiltonian is proportional to f 1 (t). We obtain the equations of motion and In case of lack of "heating up" or "cooling down" of quantum state we set f 1 = 0 and consequently what simplify means occupancy of energetic levels Dynamics of quantum state "being heated" follows equation what concludes that the population of |E 2 state increases while population of |E 1 is constant. The only way of avoiding interpretation difficulties is by assumption that state normalization is artifact and what really matters is the ratio between occupancy of excited and occupancy of ground state. This ratio gives the probability of having the excited quantum state. One can determine the one equation whose solutions are present in the second equation by the relations that could be reduced to one equation Setting f 1 = constant and such that f 1 = 0 for t <= 0 and non-zero otherwise we can make first guess as and its derivative is Suddenly turning on function f 1 at t=0 bring discontinuity to the first derivative d dt c E2 (t)(t = 0) of c E2 (t), but preserving continuoity of the occupancy function. It is quite straighforward to evaluate the probability of occupancy of excited state E 2 what is expressed by formula One can refer to different scenarios of P e (t) dependence on time as given by Fig.2. For small values of f 1 we can convert the obtained results to the Hamiltonian matrix in position representation in the following form of We recognize that in case of both real and complex values of function f 1 such matrix is not Hermitian. It simply means that it can be interpreted as the Hamiltonian describing dissipation of quantum state in phenomenological way.

RABI OSCILLATIONS IN GENERAL CASE FOR 2 ENERGY LEVEL SYSTEM
In general case during heating up of q-state or during cooling down of q-state we need to consider the Hamiltonian If we want to have time-dependent only E 1 (t) and onlyE 2 (t) states we need to consider Let us spot the dynamics of quantum state with time so we have f 1 (t), f 2 (t) = 0 for t <= 0 and constant non-zero otherwise (f From first equation we have 1 and we obtain the second equation what gives and it gives After multiplication by f2(t) hi the last equation gives In analogical way we obtain Boundary conditions are given as From later considerations it turns out that where f a (t) and f b (t) are real valued functions. Therefore we can write the equations of motion as In analogical way we obtain Boundary conditions are given as Very special case is when where c, a and b are real valued. In such cases we obtain the equations for the occupancy of energy state E 1 and E 2 expressed as First case is c = 0,h = 1 and solution is where g 1 and g 1 are complex values. Having non-zero c we obtain solutions The simplified case of last formula can be given as and the numerical example of its dependence on time is depicted in Fig.2, where initially energy level E 1 was completly populated and with time the full population of energy level E 2 was achieved while energy level E 1 was completly depopulated. Such dependence can be used for example in the action of Hadamard gate implemented in electrostatic position dependent qubit. If f 1 (t) and f 2 (t) functions have small values one can assume and Hermicity of last Hamiltonian requires that f 1 (t) = f 2 (t) * .

EXTENSION OF 2-ENERGY TIGHT BINDING MODEL INTO N ENERGETIC LEVELS FOR POSITION BASED QUBIT IN ARBITRARY ELECTROMAGNETIC ENVIROMENT
Picture presented before as in equation 1 with N=2 energetic levels can be easily extended for arbitrary number of energy levels E 1 < E 2 < .. < E 2N1=N what is valid in time-independent case. It is worth mentioning that very last chain of inequalities between time depedent eigenenergies does not need to be always valid in the general case of time-dependent Hamiltonian. In most general case we have N = 2N 1 energetic levels among 2 coupled quantum wells controlled electrostatically. Quite obviously we are omitting continuum spectrum of eigenenergies and we only concentrate on the system with electrons confiment by some effective potential. It requires introduction of 2N 1 orthogonal Wannier functionl bases such that |x 1 1 , .., |x 1 N1 , |x 2 1 , .., |x 2 N1 =(|1, 0 E1−E2 ,.., ) and such that x 1 | k (|x 2 m ) = 0 for any m different than k. In such case the quantum state for N 1 = 3 (N = 2N 1 ) is described as The probability of presence of electron at node 1 is P 1 (t) = |γ E1−E2,p1 (t) + γ E3−E4,p1 (t) + γ E5−E6,p1 (t)| 2 and the probability of presence of electrone at node 2 is P 2 (t) = |γ E1−E2,p2 (t) + γ E3−E4,p2 (t) + γ E5−E6,p2 (t)| 2 . The act of measurement on position based qubit is represented by the operator Let us review the Hamiltonian describing system with N = 2N 1 energy levels. Essientially we have 2N 1 coefficients describing energy localized at 2 nodes E p1,1 , E p1,2 , .., E p1,N1 , E p2,1 , E p2,2 , .., E p2,N1 , so we are dealing with E pu,m coefficients, where m=1..N 1 , u is1 or 2 and we have taken into account existence of all N = 2N 1 energetic levels. Let us set N 1 = 3 and in such case the quantum state Hamiltonia in the case of lack of transition between energetic levels corresponding to Fig.4. can be written aŝ It is important to mention that in the case of lack of time-dependent Hamiltonian having any among frequency components E k −E l h for k = l such that (k, l) = 1..6 there is no possibility for the occurence of resonant state and change of probability of occupancy among different energetic levels. In such case (|1, 0 E1,E2 1, 0| E1,E2 )(|1, 0 E3,E4 1, 0| E3,E4 ) = 0. However it is not true if there exists resonant state and if for example Hamiltonian consists following non-zero components with frequencies Now we are moving towards the situation of system with position based qubit with 5 energetic levels, twodifferent potential minima and one occupied localized state on the right side as depicted in Fig.5. We have Hamiltonian of the form with corresponding quantum given as The energetic states parametrized by E 5 , E 4 or E 3 , E 2 can move freely between node 1 and 2 so they are delocalized while the state numerated by E 1 is the particular localized ground state. Specified Hamiltonian structure implies that the ground state cannot be moved to excited states and reversely excited states cannot be moved into ground state .
The coupling between ground state and first excited state at node 2 occurs in the case of modified Hamiltonian of the following form aŝ In particular state it is allowed for the wave-packet in the right-well to undergoe transition from energetic state In such case the projectors (|0, 1 E1,E2 0, 1| E1,E2 )(|0, 1 E1,E3 0, 1| E3,E1 ) are different from zero because of existence of resonant states characterized by frequencies ω 21 and ω 31 . Now we are moving from position based Hamiltonian representation into energy based that is by identity transformation Now we need to specify the energy eigenstates introducingÊ = diag(E 5 , E 4 , E 3 , E 2 , E 1 ) and we obtainÊ acting on It is noticable to recognize that the ground state eigenvector from localized state was converted into delocalized state by the presence of non-zero γ E1,p2 (t)E 1,p1 term in the Hamiltonian .
Also second energergy level eigenvector was changed.
The element t 2→1,p2→p2 is responsible for heating up or cooling down of the localized state. We notice that all other eigenenergy vectors were not changed by the presence of non-zero elements t 2→1,p2→p2 = t * 1→2,p2→p2 in the Hamiltonian 75.
It might happen that potential minima (bottom) in position based qubit can have arbitrary depth so more than one eigenenergy state can be localized. Number of localized states can be arbitrary big both on the left and the right side. In considered example we have only localized on the right state. Localized states can be heated up or cool down so one localized state is transfering into another localized state in the same quantum well. In general k states (as k =2 in reference to the matrix 79) can be localized on the right side among k+m all energetic states (where m=4 is number of delocalized eigenenergy states) so total number of Hamiltonian eigenenergy state k+m is 4+2=6.
We can regonize that term t 1→0,p2→p2 is able to heat up and cool down the localized q-state between 0 and 1 energetic level in q-well p2 and term t 2→0,p2→p2 is describing interaction between 0 and 2 energy level in q-well p2, while term t 2→1,p2→p2 describes the interaction between 1st and 2nd energetic level in second quantum well p2. Now we are describing the situation of 3 localized state in the left well (associated with matrix coefficients in green) and 2 localized states in the right wells (associated with matrix coefficients in red) and 4 states that are delocalized so we are dealing with matrix of 9 states.Ĥ Heating up and cooling down of the localized quantum state in the left q-well is controlled by Hamiltonian coeffcients t 0→−1,p1→p1 , t 1→0,p1→p1 , t 1→−1,p1→p1 and its conjugate counterparts t −1→0,p1→p1 , t 0→1,p1→p1 , t −1→1,p1→p1 . Moving delocalized q-state in the left q-well p1 into delocalized q-state in the left p2 well is by non-zero t 1→2,p1→p1 and its conjugate t 2→1,p1→p1 in orange color. From the point of view of q-mechanics it is also possible to transfer one q-state localized in the left q-well into the q-state localized in the right q-well. It is achieved by the non-zero coefficient t 0→−1,p1→p2 and its conjugate t −1→0,p2→p1 in brown color. All these transfer between states of different energies requires microwave field or AC voltage components. In case of matrix 9 by 9 we can spot (9 2 − 9)/2 processes of transfer from one energetic state into another energetic state in the same q-well or into opposite q-well. In general for N by N matrix one has (N 2 − N )/2 such processes. More detailed knowledge about this processes might be only extracted from Schroediger formalism in 1, 2 or 3 dimensions. In most general case in the case of system with 9 energetic levels are depicted in Fig.6. Now we are describing the most general situation for the system preserving 6 energy levels where position of potential minima and maxima can change in time so localized states can change into delocalized or reversely. It is thus describing the system is placed in outside time-dependent electromagnetic field of any dependence so the matrix of position-based qubitĤ(t) can be written aŝ Such matrix is Hermitian so t * k→s,pk→p l = t * k→s,pk→p l for k and s among 1, 2 and 3 and p k and p l having value p 1 (presence of electron in left quantum well) or p 2 (presence of electron in right quantum well) and having real-valued diagonal elements. The meaning of non-diagonal coefficients is non-trivial.
In general case the eigenvalues of described matrix cannot be determined analytically unless there are some preimposed symmetries as for example E k,p1 =E k,p2 for k=1,2 and 3 and in such case eigenvalues are determined by the roots of polynomial of 3rd order in analytical way. Very last reasoning can be conducted also for the system with 8 energetic levels when one deals with roots of polynomial of 4th order. By proper electromagnetic engineering the system with 6 energetic levels can be controlled by ((36 − 6)/2) + 6 = 15 + 6 = 21 time dependent parameters. In most general case the system of position based qubit having 2 coupled quantum dots with 6 energy levels can be parametrized by 36 real valued functions that are time-dependent. Quite obviously the same system with 2N energetic levels can be parametrized by (2N ) 2 real valued functions under the assumption Figure 5. All possible quantum processes in the system of 2 coupled q-dots in the case of various microwave fields: transitions between delocalized eigen energetic levels (P1), transitions between left localized eigen energies (P2), transitions between right localized eigen energy states (P3), transitions between left and right delocalized eigen energy states (P4), transitions between left localized q-states and delocalized q-states (P5), transitions between right localized q-states and delocalized q-states (P6). One can also distinguish process on injection of electron from outside to 2-qwell sytem (P7) and process of ejection of electron from 2-qwell system to the outside (P8).Six processes P1-P6 are described by the Hamiltonian 81 and its precurson Hamiltonian 80.

CASE OF ELECTROSTATIC QUBIT INTERACTION
We consider most minimalist model of electrostatically interacting two position-based qubits that are double quantum dots A (with nodes 1 and 2 and named as U-upper qubit) and B (with nodes 1' and 2' and named as Llower qubit) with local confinement potentials as given in the right side of Fig.1. By introducing notation |1, 0 x = |1 , |0, 1 x = |2 , |1 , 0 x = |1 , |0 , 1 x = |1 the minimalistic Hamiltonian of the system of electrostatically interacting position based qubits can be written aŝ described by parameters E p1 (t),E p2 (t),E p1 (t),E p2 (t), t s12 (t), t s1 2 (t) and distances between nodes k and l': d 11 ,d 22 ,d 21 ,d 12 . In such case q-state of the system is given as where normalization condiion gives |γ 1 (t)| 2 + ..|γ 4 (t)| 2 . Probability of finding electron in upper system at node 1 is by action of projectorP 1U = 1, 0| U 1, 0| L + 1, 0| U 0, 1| L on q-stateP 1U |ψ so it gives probability amplitude |γ 1 (t) + γ 3 (t)| 2 . On another hand probability of finding electron from qubit A (U) at node 2 and electron from qubit B(L) at node 1 is obtained by projectionP 2U,1L = 0, 1| U 1, 0| L acting on q-state giving ( 0, 1| U 1, 0| L ) |ψ that gives probability amplitude |γ 3 (t)| 2 . Referring to picture from Fig.1 we set distances between nodes as d 11 = d 22 = d 1 ,d 12 = d 21 = (a + b) 2 + d 2 1 and assume Coulomb electrostatic energy to be of the form E c (k, l) = q 2 d kl and hence we obtain the matrix Hamiltonian given aŝ In most general case of 2 qubit electrostatic interaction one has 4 different Coulomb terms on matrix diagonal E c1 = q 2 . In case of 2 qubit interacting electrostatically in any geometrical configuration all eigenenergies and eigenstates can be determined analytically since one can establish matrix 4 by 4 eigenvalues and eigenvectors. Two electrostatically coupled qubits are entangled if hopping elements in each qubit are non-zero. It is worth mentioning that The example of function of eigenenergy spectra of 2 electrostatically interacting qubits on distance is given by Fig.6.
Importing observation is that any element of matrixĤ(t ) for t ∈ (t 0 , t) denoted as H k,l (t ) is transferred to dt (H k,l (t )) of matrixÛ (t, t 0 ). In particular all zeros of matrix H k,l (t ) are are 1 values inÛ (t, t 0 ). We can easily generalize the presented reasoning for the system of N electrostatically coupled electrons confined by some local potentials. However we need to know the position dependent Hamiltonian eigenstate at the initial time t 0 . In case N > 2 finding such eigenstate is the numerical problem since analytical solutions for roots of polynomials of one variable for higher order than 4 does not exist. Using numerical eigenstate at time instance t 0 we can compute the system quantum dynamics in analytical way. This give us the strong and relatively simple mathematical tool giving full determination of quantum dynamical state at the any instance of time. The act of measurement on position based qubit is represented by the operator P Lef t = |1, 0 E1,E2 1, 0| E1,E2 and P Right = |0, 1 E1,E2 0, 1| E1,E2 .

Simplified picture of symmetric Q-Swap gate
Now we need to find system 4 eigenvalues and eigenstates(4 orthogonal 4-dimensional vectors) so we are dealing with matrix eigenvalue problem) what is the subject of classical algebra. Let us assume that 2 double quantum dot systems are symmetric and biased by the same voltages generating potential bottoms V s so we have E p1 = E p2 = E p1 = E p2 = E p = V s and that t s12 = t s1 2 = t s . Denoting E c (1, 1 ) = E c (2, 2 ) = E c1 and E c (1, 2 ) = E c (2, 1 ) = E c2 we are obtaining 4 orthogonal Hamiltonian eigenvectors where c=∓ last two energy eigenstates are not entangled. Situation of c=1 takes place when E c1 = E c2 so when two qubits are inifinitely far away so when they are electrostatically decoupled. Situation of c=0 is interesting because it means that |E 3 and |E 4 are maximally entangled and it occurs when t s = 0 so when two electrons are maximally localized in each of the qubit so there is no hopping between left and right well.
The obtained eigenenergy states correspond to 4 eigenenergies )) 2 + 16t 2 s + 4V s ), We also notice that the eigenenergy states |E 1 , |E 2 ,|E 3 , |E 4 do not have its classical counterpart since upper electron exists at both positions 1 and 2 and lower electron exists at both positions at the same time. We observe that when distance between two systems of double quantum dots goes into infinity the energy difference between quantum state corresponding to |E 3 and |E 4 goes to zero. This makes those two entangled states to be degenerated. Normalized 4 eigenvectors of 2 interacting qubits in SWAP Q-Gate configuration are of the following form It is worth mentioning that if we want to bring two electrostatic qubitsto to the entangled state we need to cool down (or heat-up) the system of interacting qubits to the energy E 1 (or to energy E 2 ). Otherwise we might also wish to disentangle two electrostatically interacting qubits. In such way one of the scenario is to bring the quantum system either to energy E 3 or E 4 so only partial entanglement will be achieved. Other scenario would be by bringing the occupancy of different energetic levels so net entanglement is reduced. One can use the entanglement witness in quantifing the existence of entanglement. One of the simplest q-state entanglement measurement is von Neumann entanglement entropy as it is expressed by formula 105 that requires the knowledge of q-system density matrix with time. Such matrix can be obtained analytically for the case of 2 electrostatically interacting qubits.
It is interesting to spot the dependence of eigenergies on distance between interacting qubits in the general case as it is depicted in Fig.6. Now we are moving towards description the procedure of cooling down or heating up in Q-Swap gate. The proceudure was discussed previously in the case of single qubit. Now it is excercised in the case of 2-qubit electrostatic interaction. For the sake of simplicity we will change the occupancy of the energy level E 1 and energy level level E 2 and keep the occupancy of other energy levels unchanged. We can write the |E 2 E 1 | as Figure 9. Eigenenergies of semiconductor qubit coupled to Josephson junction in dependence on distance in tight-binding minimalitic approach.
calculations. The Hamiltonian of physical system has such structure that allows analytic determination of all eigenenergies since Hamitlonian matrix has many symmetries. In particular we can obtain the spectrum of eigenenergies in dependence on the distance a as depicted in Fig.9 and spectrum of eigeneneries in dependence of superconducting order parameter as given in Fig. 10. One can recognize certain similarities with Fig.2. It simply means that increase of superconducting order parameter strenght brings similar effect as increase of distance between interaction of semiconductor position based qubit and Josephson junction.One of the most interesting feature is tunning the landscape of eigenenergies by applying small voltage (below the size 2e∆) to non-superconducting region of Josephson junction. In such case one obtains the feautures as described in Fig. 11. In the described considerations the spin degree of freedom was omitted in case of Josephson junction as well as in case of semiconductor position based qubit. However they could be easily included but it would increase the size of matrix describing interaction between superconductor Josephson junction and semiconductor electrostatic qubit from 16 by 16 to the size 8*4=32 so one obtains matrix 32 by 32. Adding strong spin-orbit interaction to the Hamiltonian of Josephson junction under the presence of magnetic field allows to describe topological Josephson junction. In such way we can obtain the effective 32 by 32 Hamiltonian for interaction between semiconductor position based qubit and topological Josephson junction in minimalistic way. It shall be also underlined that so far we have used BdGe formalism that is suitable for mean field theory domain. However in our case we have considered very special interactions between individual (electrons, holes) present in area of Josephson junction and specific individual electron present in area of semiconductor qubit. Usage of BdGe formalism is therefore first level of possible approximation and further more detailed study can be attempted in determination of microscopic processes present interacting Josephson junction with semicondutor qubit in more detailed way. It is sufficient to mention that in our case superconductors shall have relatively small size so we are dealing with relatively small number of electrons and holes in non-superconducting area. More detailed considerations are however beyond the scope of this work and requires Density Functional Theory (DFT) methods, etc.

CONCLUSIONS
The obtained results have its meaning in development of single electron electrostatic quantum neural networks, quantum gates as CNOT, SWAP, Toffoli and Fredkin gates as well as any other types of quantum gates with N inputs and M outputs. Single electron semiconductor devices can be attractive from point of view of power consumption and they can approach similar performance as Rapid Single Quantum Flux superconducting circuits 7 having much smaller dimensions than superconducting circuits. In conducted computations spin-degree of freedom was neglected. However it can be added in straightforward way doubling the size of Hilbert space. The obtained results allow us to obtain the entanglement of qubit A (for example) using biparticle von Neumann entrophy S(t) A of qubit A in two electrostatically interacting qubits with time as given by formula where Tr[.] is matrix trace operator andρ A = T r B [ρ] is the reduced density matrix of A qubit after presence of B qubit was traced out with partial trace T r B [.]. The obtained results can be mapped to Schroedinger formalism 10  in order to obtain higher accuracy and resolution in description of quantum state dynamics. One can use the obtained results in determination of quantum transport in the single electron devices or arbitrary topology what can helpful in optimization of devices functionality and sequence of controlling sequences shaping the electron confinement potential. Topological phase transitions as described by Sachdev, 9 Choi,8 Belzig 11 are expected to take place in arrays of coupled electrostatic qubits due to the similarity of tight-binding applied in semiconductor coupled quantum well model to Josephson model in Cooper pair box superconducting qubits. All results are quite straightforward to be generalized for electrons and holes confined in net of coupled quantum dots (what changes only sign of electrostatic energy so q 2 → −q 2 ) under the assumption that recombination processes do not occur.
What is more the interaction between electrostatic position based qubit and Josephson junction was formulated and solved in tight-binding model. In quite straightforwad way one obtains the electrostatically coupled networks of graphs interacting with single Josephson junction in analytical way. It shall have its importance in development of interface between semiconductor CMOS quantum computer and already developed superconducting computer.

ACKNOWLEDGMENT
This work was supported by Science Foundation Ireland under Grant 14/RP/I2921.