Analytic view on N body interaction in electrostatic quantum gates and decoherence effects in tight-binding model

Analytical solutions describing quantum swap and Hadamard gate are given with the use of tight-binding approximation. Decoherence effects are described analytically for two 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 N electrostatically interacting quantum bodies confined by local potentials (N-qubit) system representing any electrostatic quantum gate with N1/N-N1 inputs/outputs. The mathematical structure of system evolution with time is specified. Index Terms: quantum computation, entanglement, single-electron devices, position-dependent qubit, Q-Swap Gate, quantum CMOS


I. TECHNOLOGICAL MOTIVATION
Systematic progress in implementation of superconducting quantum computer by IBM, Google and D-wave companies is obtained but it faces the key challenges due to the fact that superconducting qubits are controlled by magnetic and RF fields. The usage of magnetic field is main obstacle in scalability of such structures so very high integration quantum circuits are not expected to take place. What is even more important the usage of low temperature superconductors that has superconducting coherence lenght of range of 300nm (it thus proportional to the size of Cooper pair) brings the limitation in further miniaturization of those structures [10], [15]. Suprisingly semiconductor technologies have no such limitations and most advanced CMOS transistors has the channgel lenght between source and drain of 3nm. Therefore we have no longer temperature activation of electric carriers as it is the case of standard CMOS technologies. What is more important in case of such small strucures the big gradient of electric field takes place and it serves as electric field activation of electric carriers even at mK temperatures. It motivates us to study the physics of quantum information processing in such devices. First study of such structures was conducted by Fujisawa [2], Petta [3] and continued by many others as Pomorski [6], [18], Imran [1], Panagiotis, Leipold [5], [4].  Fig. 1. Basic concept of position based qubit [6], [13] and its correspondence to Bloch sphere [18].
The last expressions can be written in a compact form Setting t si (t) = 1,t sr (t) = 0 and E p1 (t) = E p2 (t) = E p we ob- which brings Hadamard matrix as relating q-state in the position base and in the energy base, where E 1(2) n = 1 √ 2 E 1 (2) . If we associate logic state 0 with occupancy of node 1 (spanned by |x 1 ) and logic state 1 with occupancy of node 2 spanned by |x 2 , then Hadamard operation on logic state 0 brings occupancy of E 2 (so it is spanned by |E 2 ) and Hadamard operation on logic state 1 brings the entire occupancy of energy level E 1 (that is spanned by |E 1 ).
It shall be underlined that in the most simple case of position-based qubit E p1 = E p2 = E p = const 1 and t s12 = |t| = const 2 and we obtain |ψ(t) = 1 It implies an oscillation of probabilities for the electron presence at node 1 (quantum logical 0) and 2 (quantum logical 1) with frequency 2|t| is the probability for the quantum state to be in the ground (excited) state. It is possible to determine the qubit state under any evolution of two eigenergies E 1 (t) and E 2 (t) that are dependent on E p1 (t), E p2 (t),t s12 (t) = t sr (t) + t si (t)i. Simply, we have the state at any time instant given by We notice that in case of qubit the evolution operator is given aŝ Here, c e1 (t 0 ) and c e2 (t 0 ) describe the qubit in the energy representation at the initial time t 0 , so |c e1 (t 0 )| 2 +|c e2 (t 0 )| 2 = 1. Such presented evolution of position-based qubit is under the circumstances of small adiabatic changes in t s (t) and in E p1 (t), E p2 (t). It is not the case of a qubit subjected to the rapid AC field that will support the existence of resonant states [6]. The presented tight-binding approach can be seen as simplified version of Hubbard model when omission of spin was done [7].
1) Double Q-Dot in external non-uniform time-dependent potential: We can introduce external time-dependent potential acting on points 1 and 2 such as V (1,t) and V (2,t). The Hamiltonian of the system is then given as If external time-dependent potential is weak than t 1,2 (t) ≈ t 1,2 and t 2,1 (t) ≈ t 2,1 .
We have 2 different time dependent eigenenergy values (E 1 (t), E 2 (t)) given as with eigenvalues and At any time instant the quantum state is given as where |c1(t)| 2 + |c2(t)| 2 = 1 and normalized eigenstates are denoted as |ψ 1n (t) > and |ψ 2n (t) >. Equivalently we obtain The equation of motion can be written as |ψ(t + dt) >= |ψ(t) > + dt ih (H(t)|ψ(t) >). Equivalently we obtain 2 coupled recurrent relations for coefficients c1(t + dt) and c2(t + dt) depending on coefficients c1(t) and c2(t). We have (14) and In case of symmetric wells E p1 = E p2 = E p ,t 1,2 = |t| = t 2,1 and time-independent Hamiltonian we have Adding and substracting those two equations give us relations |t|-hopping (kinetic) term In analogy to the evolution operator in Schroedinger equation we recognize two analytic solutions By adding or substracting 2 equations and multiply by 1/2 we obtain c 1 (t) or c 2 (t) given as Presence of weak-time dependent potential present in 1 and 2 will change the analytic solutions into The last two equations are equivalent to In quite analogical way we deal with coefficient c 2 (t) whose analytic solution can be written as

B. Position qubit in resonant state and Rabi oscillations
We can define the general form of energy evolution of 2 level qubit preserving its two level states. It will have the formĤ and one obtains the following energy eigenstates E 1m (t) = E 1 (t) that are in first order approximation can be approxi- 12 (t)| 2 and for small values of resonant field |E 12 | << E 1 (t), E 2 (t) we can assume E 1 (t) ≈ E 1m (t) and E 2 (t) ≈ E 2m (t). In general case one shall assume the following evolution matrix of qubit system Now we analyze the heating effects in time-dependent 2 level qubit Hamiltonian. We have We observe that from knowledge of t s21 (t) and t s21 (t) we can extract the knowledge on E 12 (t) function. We have and we finally can obtain the unique formula for Re(E 12 (t) and Im(E 12 (t) functions that control the Rabi oscillations of switching the occupancy between E g and E g level.
In particular we can observe that changing phase imprint by means of t s12 (t) = |t s |e α(t) = |t s | cos(α(t)) + i|t s | sin(α(t)) with t s = constant and non-constant . Therefore it is possible to obtain the heating or cooling down the quantum state only by change of phase imprint expressed by α(t). In particular case we when E 1 (t) and E 2 (t) are time independent we have Rabi oscillations with constant frequency The term E 12 (t) |E 1 E 2 | describes the flow of energy from excited energy level into ground level (or lower energy level) what practically means that qubits system is cooling down. In quite analogical way term E 12 (t) |E 1 E 2 | describes the flow of energy from ground energy level (or lower energy level) into excited energy level (or higher energy level) what practically means that qubits system is heating up. Let us investigate the case when |E 1 (t)|, |E 2 (t)| >> |E 12 (t)| 2 , |E 21 (t)| 2 so Hamitonian term responsible for energy flow between considered energy levels is small.

III. QUANTUM ELECTROSTATIC SWAP GATE
We consider the situation as depicted in Fig. 3. We have 2 separated in space systems of double quantum dots U (Upper) and L (Lower). One electron is in L system and one electron is in U system. If U system is far away from L system than the quantum states of U is independent from L quantum system and in such case we can write |ψ > U = c 1 |1, 0 > U +c 2 |0, 1 > U and |ψ > L = c 3 |1, 0 > L +c 4 |0, 1 > L and normalization conditions |c 1 | 2 +|c 2 | 2 = 1 and |c 3 | 2 +|c 4 | 2 = 1. In case of separated systems we can write the total Hilbert space by factorization so |ψ >= |ψ > U |ψ > L . However it is not true when we bring L and U systems sufficiently close so Coulomb interaction has no longer perturbative character. In such case we have the most general form of quantum state given as Since we have 2 electrons we can write |c a | 2 + |c b | 2 + |c c | 2 + |c d | 2 = 1. In rare and special case we can write If we are dealing with non-entangled state since total quantum state can be factorized as product of 2 quantum subsystems. Such case is rare and in most cases our system depicted in Fig.1 is entangled what especially takes place if U and L are not far away.
Now we need to write down the system Hamiltonian from Fig.1. One electron from U can be in points 1 and 2 and second electron can be in points 1' and 2'. We make assumptions that U and L have the same physical structure and are symmetric.
We have total system Hamiltonian as the sum of all potential and kinetic energies given as The last 4 terms are Coulomb interaction terms between one electron confined in L system and one electron confined have values Coulomb classical energy between points (1,1'),(2,2'),(2,1'),(1,2'). Two double quantum dots are geometrically parametrized by constants d1, d, a, b and thus we have distances d 1, . The electron kinetic energy in U system is parametrized by t U and in L system by t L . In simplified case we have t U = t L = t. We also denoted I L = (|0, 1 >< 0, 1| L + |1, 0 >< 1, 0| L ) and I U = (|0, 1 >< 0, 1| U + |1, 0 >< 1, 0| U ). I L is identity operator and is projection of L state on itself. The same is with I U operator that is identity operator and is projection of state U on itself. It is convenient to express total system Hamiltonian in matrix representation. We have 4 by 4 matrix given as 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 corresponding to 4 eigenenergies We observe that setting quantum state to vector V 1 we obtain the state and such state is indeed entangled. Setting the quantum state to vector V 2 we also obtain the entangled state We also notice that the state V 3 and V 4 does 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 V 1 and V 2 goes to zero. This makes those two entangled states to be degenerate. We notice that vectors We observe that |v 1 | 2 = 2, |v 2 | 2 = 2 and that Thus normalized 4 eigenvectors are of the following form 1) Case of 2 double quantum dots on the line: We consider the situation as depicted in Fig.III.
We have the following Coulomb interaction terms 1 ). We assume t L = t R = |t|. All energies at the nodes 1,2, 1' and 2' are controlled with biasing voltage V s . We obtain the following Hamiltonian of the system The quantum state of the system can be written as The coefficients |c a | 2 + |c b | 2 + |c c | 2 + |c d | 2 = 1 since < ψ|ψ >= 1. We assume that In such case the system Hamiltonian is given as We notice that E c (2, 2 ) = E c (1, 1 ) since two double symmetric qubits are on the same line. Now we place the dependence of Coulomb energy on geometry. We obtain where we set all hoping coefficients to |t| and we set E p (1) = E p (2) = E p (1 ) = E p (2 ) = V s .
2) Case of 2 perpendicular double quantum dots: It is important to consider the situation as depicted in Fig.1A.
In highest simplified case we have Now we need to be able to extract one body wavefunction from 2-body wavefunction. Let us extract the wavefunction for U system. Thus we need to apply the following projection operators P1 U and P2 U on general quantum state (U, L) to obtain only U wavefunction subcomponents: and (52) Consequently the whole U 1-body wavefunction is given as follows In quite analogical way we can introduce the following projection operators P1 L and P2 L on general quantum state (U, L) to obtain only L wavefunction subcomponents: and Consequently the whole L: 1-body wavefunction is given as follows C. The action of strong measurement on one of the subsystems L and U Making the measurement determining the position of particle from U system on the left side is represented by the projection Thus after the determination of the state of electron in U to be on the left side we have the total quantum state after measurement |ψ > expressed by the state before measurement |ψ > to be of the form Making the measurement determining the position of particle from U system on the right side is represented by the projection Thus after the determination of the state of electron in U to be on the left side we have the total quantum state after measurement |ψ > expressed by the state before measurement |ψ > to be of the form Quite obviously instead of measurement of position of electron from U system we can make the determination of electron state from the L system to be on the left what we obtain with the projection operator PL U that is represented as following (61) In similar way we can introduce the projection measurement determining the position of electron to be on the right side of system L in the way as In similar way as before the state of the quantum system after determination of particle position in L subsystem to be on the left side is PL L |ψ > while the state of the total quantum system after measurement determination of particle position in L subsystem to be on the right side is PR L |ψ >.

D. Combined approach of tight-binding + integro-differential equations for Q-CNOT gate
The effect of 1-st quantum system from Fig.4 on the second quantum system (1-qubit:double Q-Dot system) can be accounted by the following Hamiltonian. We assume that d 3 > d 2 . The Hamiltonian of 2-nd quantum system that is CNOT output in its functional dependence from Hamiltonian of 1-st quantum system [Q-SWAP gate]. 2-nd quantum system Hamiltonian has the matrix and |c 1 | 2 , |c 2 | 2 , |c 1 | 2 , |c 2 | 2 are the probabilities of occupancies of nodes 1,1',2,2' by electrons. Thus we need to know dynamics of system 1 to determine dynamics of system 2. We need to use formulas ?? and ??. The terms H 1→2 [1], H 1→2 [2] can be treated as the perturbation to H 2,non−interaction . Analyzing more precisely we can introduce pertubation to the system 1 coming from system 2. This perturbation of system 1 from system 2 is later affecting the dynamics of system 2 as well. However in the first level of approximation we can recognize that only system 1 is affecting system 2 and that system 2 is no-having impact on the system 1.

PROPAGATOR IN TIGHT BINDING MODEL
Without presence of microwave field we have the Hamiltonian H given as Here f 1 (t) and f 2 (t) are time-dependent signals and (|1, 0 >= w L (x), |0, 1 >= w R (x)) are position based functions (Wannier functions), while H|e >= E 2 |e > and H|g >= E 1 |g >. There is continous unitary transformation from (|1, 0 >, |0, 1 >) to (|g >, |e >) bases. Spectral representation of system Hamiltonian with no external microwave field is given as and for the symmetric case we have The system eigenergies are given as (E p − ε) 2 − |t| 2 = 0 what brings ε 1 = E p − |t| and ε 2 = E p + |t|. From the condition We have b a = ε−E p t that gives either -1 or 1 for ε 1 and ε 2 . We have the following eigenvectors (67) and and such that H |g = ε 1 |g and H |e = ε 2 |e , where |g > denotes the quantum system energy ground eigenstate and |e > denotes quantum system excited state. Zero tunneling case between 1 and 2 or 2 and 1 is by presence of infinite barrier between 1 and 2. In such case all kinetic and potential energy components are encoded in E p (1) and in E p (2) Hamiltonian components as components describing two insulated quantum systems.
(69) with eigenvector of energies We have Last expression can be written as or equivalently Controlling the occupancy of |g > and |e > by external microwave signal that is encoded in c g and c e coefficients we can obtain the occupancy of the left and right well by the coefficients 1 √ 2 (c e + c g ) and 1 √ 2 (c e − c g ). We have normalization condition that is Now we need to evaluate the time dependent Hamiltonian that is about evaluation of the terms and Final Hamiltonian of 2 symmetric qdots that is under microwave field becomes The most general Hamiltonian becomes The hopping coefficient becomes renormalized as well as E p coefficients. Let us find the eigenvalues in the simplified case. We have It is equivalent to The last equation has two solutions Therefore we have the renormalized |t| function by means of f 1 and f 2 functions. The system Hamiltonian with no presence of microwave field can be written as Adding the microwave field we obtain Let us assume that at given time instant the quantum system has the equations |ψ(t) >= c e (t)|e > +c g (t)|g > If f 1 (t) = f 2 (t) = 0 we have c e (t) = c e (0)e − ī h t , c g (t) = c g (0)e − ī h t with normalization condition |c e | 2 + |c g | 2 = 1. Adding time dependent functions f 1 and f 2 to Hamiltonian brings ih dt (c e (t + dt) − c e (t))|e > +(c g (t + dt) − c g (t))|g >= We have ih dt (c e (t + dt) − c e (t)) = (E p + |t|)c e (t) + f 1(t)c g (t) = 0 (88) and ih dt (c g (t + dt) − c g (t)) = (E p − |t|)c g (t) + f 2(t)c e (t) = 0. (89) Having f 1 (t) = f 2 (t) = 0 we have two analytic solutions c e (t) = c e (0)e − ī h t(E p +|t|) , We notice that in such case we have |c e | 2 = const1 and |c g | 2 = const2. Now we are dealing with time-dependent case. We Adding two equations we get and substracting we have The last two equations can be rewritten to the operator format. We obtain (c e (t) + c g (t)) = O(c e (t) + c g (t)) = +|t|(c e (t) − c g (t)) = 0 and . It is interesting to observe that equations have propagator format. Let us define the object We apply the operator [ih d dt − E p − f 1 (t)] to the G (1, 2). We obtainÔ = |t|(c e (t) − c g (t))(c e (t) − c g (t)) * + |t|(c e (t) + c g (t))(c e (t) − c g (t)) * = |t|. (98) Final propagator can be written as We can extend our definition of propagator to the form It is interesting to observe that set of equations can be solved by applying additional operatorÔ to each of sides. We obtain using new introduced functions u 1 (t) = c e (t)+c g (t) and u 2 (t) = c e (t) − c g (t): In similar fashion we obtain (c e (t) + c g (t)) = |t| 2 (c e (t) − c g (t)) = |t| 2 u 2 (t).
We need to evaluate the operator We therefore end up in and If f 1 = sin(γt + φ ) both functions u 1 (t) and u 2 (t) have the analytic solutions. We also notice that |u 1 (t)| 2 + |u 2 (t)| 2 = 2 at any time instant t.

A. Describing energy flow between 2 interacting qubits
The Hamiltonian for 2 electrostatically interacting qubits A and B (or in general any interacting physical systems A and B) is of the form Such structure of Hamiltonian is postulated and is fundamentally justified. At first we assume that there is no electron-electron interaction between two qubits. In such case in tight-binding model we have localized E p1A(B) , E p2A(B) and delocalized energy t 12A(B) ,t 21A(B) associated with qubit A and with qubit B. Now we need to interlink this Hamiltonian structure with turning on Coulomb interaction. Coulomb interaction is responsible for energy exchange between qubit A and B. It is propagated by the photons exchange that have discrete values. However the electrostatic energy has the limited value so it shall bring certain renormalization to the initial qubit eigenstates. It is easier and methodologically justified to start from Coulomb interaction that has the following form for electron A and electron B at nodes (x k , x l ) so we have (1, 1 ) → |x 1 |x 1

total Coulomb Hamiltonian assoctated with qubit A and B electrostatic interaction is expressed by the Hamiltonian
(109) This Hamiltonian is responsible for generating entanglement between qubit A and B since Hilbert space of 2 non-interacting qubits is its tensor product of Hilbert space of qubit A and qubit B. It is instructive to notice that we can control the entanglement between qubits A and B in electrostatic way by having time-dependent control on coefficients a A (t), b A (t), c A (t), d A (t) and a B (t), b B (t), c B (t), d B (t) (8 complex value parameters) that are function of 6 voltages applied to qubit A and B (3 voltages for each qubit). We start from spectral decomposition of operator |x 1 |x 1 q 2 d(1,1 ) x 1 | x 1 | into eigenergy represenation of qubits A and B and we obtain , We identify 4 types of renormalization coming to the Hamiltonian of non-interacting qubits A and B from interacting term |x 1 |x 1 q 2 d(1,1 ) x 1 | x 1 |. Renormalization 1 denoted by r1 is describing the change of total energy in the non-interacting qubit due to the apperance of Coulomb interaction. Renormalization 2 describes the process in which qubit A energy is unchanged and qubit B populates or depopulates energy levels E 1b and E 2b so qubit B is either heated up or cooled down. Renormalization r3 is describing the same as renormalization 2 but in the case when qubit A is heated up or cooled down while energy of qubit B is unchanged. Finally renormalization 4 describes the process when both qubit A and B are heated up or cooled down by mutual exchange of energy due to Coulomb interaction. What is more processes r2-r4 are describing decoherence of qubit A and B due to existence of Coulomb interaction in analytical way due to internal system dynamics. Here we have omitted the interaction of external world with our qubit systems. Furhtermore we can think about quantum state hybridization that is taking place in the presence of very strong Coulomb interaction. It be omitted in this work and it is the subject of future works. Similarly as it was done before we have spectral decomposition of operator |x 2 |x 2 q 2 d(2,2 ) x 2 | x 2 | and we obtain Again we identify 4 types of renormalization coming to the Hamiltonian of non-interacting qubits A and B from interacting term |x 2 |x 2 q 2 d(2,2 ) x 2 | x 2 |. They have the same interpretation as it was in case of operator |x 1 |x 1 It is important to notice resonant states coming from changes of voltages at each separated qubits as in infite distance forms the matrix Now we are making spectral decomposition of operator |x 1 |x 2 q 2 d(1,2 ) x 1 | x 2 | and we obtain and We can easily identify 4 renormalized eigenenergies to be of the form under assumption that decoherence effects described by terms spanned by E k,l E s,w | are small as in comparison with and under assumption that δ k,s δ l,w = 0. Now we are going to identify decoherence matrix arrising from natural non-dissipative equations of motion.
We are have the following decoherence matrix describing heating of qubit B and keeping unchanged qubit A as . The decoherence terms can be expressed by the matrix where Similar reasoning can be conducted for 3 and more interacting qubits and thus we have hint how to describe the decoherence processes in the system with N-N1 inputs (qubits) and N1 outputs (qubits) having 2 energy levels (or more). Decoherence effects are due to qubit-qubit interaction and they arise from Coulomb interaction and they are inevitable. Finally we arrive to the Hamiltonian of the system of the following form Particular simple analytical form can be obtained if |a A (t)| 2 = |b A (t)| 2 = |c A (t)| 2 = |d A (t)| 2 and |a B (t)| 2 = |b B (t)| 2 = |c B (t)| 2 = |d B (t)| 2 = |a A (t)| 2 that implies E p1,A (t) = E p2,A (t) = E p1,B (t) = E p2,B (t) = E p (t) ∈ R and |t s12,A (t)| = |t s12,B (t)| = |t s12 (t)| so a (A)B = 1 Having such simplifications we immediately recognize that all 4 renormalized eigenenergies brings the same renormalized value to each among 4 eigenenergies in the way as We also have and this brings Finally the quantum state is the subject to the evolution with time of the following form where normalization condition takes place |γ 1 (t 0 )| 2 + |γ 2 (t 0 )| 2 + |γ 3 (t 0 )| 2 + |γ 4 (t 0 )| 2 = 1 and γ coeffients determine initial state of quantum system. It is therefore quite straighforward to obtain density matrix of the quantum system with time that has the structure ρ(t) = |ψ (t) ψ| (t) = exp From the above considerations we recognize that evolution of the quantum state is equivalent to rotation by 4 real values angles α 1 (t), α 2 (t), α 3 (t) and α 4 (t) that correspond to 4 eigenergies of the system and by 6 complex valued angles Θ 12 (t), Θ 13 (t) , Θ 14 (t), Θ 23 (t), Θ 24 (t), Θ 34 (t) that correspons to transition between eigenenergies of non-interacting system.

VI. INTRODUCTION TO SPECTRAL REPRESENTATION OF COULOMB ENERGY IN SCHROEDINGER FORMALISM
We have two weakly electrostatically interacting particles that are confined in separate wells by potentials V p1 (x 1 ) and V p2 (x 2 ). We neglect spin presence. In order to avoid certain infinities we set small and non-zero d (|d| << 1). Then sightly mishaped Coulomb operator is given In formalized way we haveV