Realization of Single Qubit Operations Using. Coherence Vector Formalism in. Quantum Cellular Automata

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1 Adv. Studies Theor. Phys., Vol. 6, 01, no. 14, Realization of Single Qubit Operations Using Coherence Vector Formalism in Quantum Cellular Automata G. Pavan 1, N. Chandrasekar and Narra Sunil Kumar 3 School of Electrical and Electronics Engineering SASTRA University, Thanjavur , India paone.nano@gmail.com 1, nchandra@ece.sastra.edu, nsunil497@gmail.com 3 Abstract Use of Quantum Dots is wide spread in Nanoelectronics, Photonics and in Quantum information sciences. The Quantum-dot Cellular Automata (QCA), constructed from quantum dots, can also perform quantum computational operations by using the concept of coherence. In this letter we show that the basic single qubit operations of NOT gate, Phase-flip gate and Hadamard transformation can be realized using four-quantum dot QCA with control electrical pulses given to the cell electrodes. Keywords: Quantum-dot Cellular Automata, Coherence, Quantum Computing, Hadamard gate, NOT Gate, Phase-flip Gate 1. Introduction Quantum computing involves the manipulation of zero and one bits with the quantum mechanical property of superposition principle operating in the all the manipulations of qubits. Therefore in any state of the system both zero bits and one bits exist simultaneously with certain probabilities.these two stables states are known as basis and they are used to represent the state of vector in quantum computer. The superposition principle and quantum entanglement results in what is known as quantum parallelism. This involves, for example, evaluation of functional values of a given function for all values of independent variable simultaneously in single operation. Consequently information processing is done in quantum computers in a new way which is not possible, even in principle, in

2 698 G. Pavan, N. Chandrasekar and N. S. Kumar classical computers. Here we consider Quantum-dot Cellular Automata (QCA) to implement quantum NOT, Phase shift and Hadamard operations. QCA concept was proposed as a transistorless alternative for digital circuit technology at nanoscale [6]. A QCA cell consists of four quantum dots in a square array, coupled with tunnel junctions and two electrons as shown in Fig. 1. γ P Figure 1.Model of a QCA Cell used for quantum computing. Each cell is coupled to its neighboring cells and electrodes. P and γ are and tunneling energy respectively. Due to columbic repulsion [], the electrons move to the diagonal dots resulting in two stable states(polarized states).qca cells used for classical computation applications are fully polarized( P = ± 1) during the operation. In the case of quantum computing the cells are not completely polarized. The polarization can continually vary from P = 1 to P = + 1 [4]. Unlike classical digital applications, quantum computing needs coherence for correct operation [5]. Here we use the physical parameters (potential barrier heights, driver polarization) for realizing the single qubit operations in quantum computing using the Bloch sphere representation.. Model of a QCA Cell The parameters that are controlled externally for a QCA cell are the tunneling energy γ and potential P.The barrier heights decide the tunneling energy. Each cell haselectrostatic coupling to its left and right neighbors and to the electrodes. Thus the tunneling energy γ and P are the inputs to a cell[3]. The state of a cell with superposition of the two basis states 0 and 1 is given as ψ = α 0 + β 1 and its polarization is P = α β. A four quantum dot cell can be modeled as a coupled two state system. By considering the coupling energy of the cell with the neighboring cells ( E ) surrounding and the electrodes( E = Esurrounding Psurrounding + EP ), tunneling energy of an electron( γ ) and polarization of neighboring cells, the Hamiltonian for a N-QCA system as a two state system can be written as [1] N N 1 E x i z z (1) i= 1 i= 1 Hˆ = γ ˆ σ () ˆ σ ( i) ˆ σ ( i+ 1). This Hamiltonian equation is same as the Ising spin chain. Here the interaction energy and the external energy are the kink energy and tunneling energy respectively. The tunneling barriers are connected to the electrodes and their heights are controlled externally by voltage sources. If the barriers are raised, electron tunneling is suppressed and the cell is latched. Decoupling Eq. (1), the

3 Realization of single qubit operations 699 Hamiltonian of N-QCAsystem to Hamiltonian of N single systems using the mean field theory gives the Hamiltonian of single QCA cell ˆ E H γσ = x + P ˆ σ z. The dynamics of a cell is given by coupled single cell time-dependent Schrödinger th equations. The Schrödinger equation of the i cell in the system is ih ψi = Hi ψi t. The density operator of a system in a two-dimensional Hilbert 1 ( ) space can be decomposed as ρ = Î + σˆ where σ and represent three elements of Pauli spin matrices and coherence vectors respectively.the density matrix notation can also represent the mixed states where as the state vectors represent only the pure states. Instead of state vector or density vector the coherent vector can be used to represent the state of a system. This coherence vector contains the same information as the x density matrix. The three elements of the coherence vector are calculated by the expectation values of the Pauli spin matrices ( x = σx, y = σy, z = σz ). The polarization of a cell is P = ˆ σ and the z coherence vector( x, y, z) of a completely polarized cell is written as ( 0, 0, + 1) for P = 1 and ( 0, 0, 1) for P =+ 1. doˆ Using Heisenberg s picture i [ Oˆ, Hˆ h = ], the time dependence of the Pauli dt spin operator and coherence vector are respectively ˆ σx 0 E 0 ˆ σx x 0 E 0 x d ˆ σ y = E 0 γ ˆ σ d h y and h dt y = E 0 γ y () dt ˆ σz 0 γ 0 ˆ σz z 0 γ 0 z The time-dependence of coherence vector can also be written in an expressive form as d r dt = r r h (3) r T where = Tr[ ˆ σi, H ] = [ γ 0 E ], i = x, y, z. Eq.(3) is the dynamical equation of coherence vector giving the time dependence of single r cell density matrix. It around the r vector in describes the precession of the coherence vector ( ) coherence vector space. The coherence vector ( r ) describes the state of the cell while r describes the influence of environment on the same cell. The inter dot barrier height corresponding to x and the coupling of the cell with neighboring cells corresponding to z are the two main influences on the cell from the environment.

4 700 G. Pavan, N. Chandrasekar and N. S. Kumar 3. Quantum computing with QCA The r vector will be used to manipulate the coherence vector by setting the inter dot barrier heights and the polarization of the driver cell. Rotation around -axis: Consider the case when potential barriers are lowered ( γ = ) and the input is zero( P = 0). For implementation perspective, we take γ >> E. From these conditions the coherence vector becomes r 1 = ( γ,0,0) T h (4) This causes r to precess around negative x-axis as shown in Fig. (a). The coherence vector rotates around r vector with an angle φ. The physical significance of the rotation angle φ is that it represents the time period φ h Δ t = r = φ for which the input conditions are maintained. The dynamics of γ the stable state is given by the time-dependent Schrödinger equation. Integrating time-dependent Schrödinger equation we get ψ( T) = Uˆ ψ(0) where U ˆ iuφ is a unitary operator [7] and satisfies e = (cos φ) I i(sin φ) U. The rotation operator representing the rotation around the and axes of the Bloch sphere can be achieved from the exponentiation of Pauli matrices.the unitary time-evolution operator for the single qubit rotation around the - axis is given by iσ φ x cos( φ ) i sin ( φ ) φ φ ˆ x, φ = = cos sin σ x = i sin ( φ ) cos( φ ) Rˆ e I i (5) r r (a) (b) (c) Figure.(a).Precession of r around r. (b).rotation around axis.(c).rotation around axis. Rotation around -axis: By raising the potential barriers ( γ = 0) and applying P >> + 1 we make coherence vector( r ) to precess around axis. r for these 1 T 0, 0, E k h φ h Δ t = = φ E conditions is given as = ( ) the angle φ is given by r. The duration of rotation corresponding to.

5 Realization of single qubit operations 701 The unitary time evolution operation for rotation around - axis is given by φ i iσ φ z e 0 ˆ φ φ Rz, φ e cos I isin ˆ σ = = z = φ i (6) 0 e The Hamiltonian of the QCA cell (Eq. (1)) does not contain any σ y terms explicitly. However rotation around the axis can be realized by a series of rotations around and axes. cos( φ ) sin( φ ) Rˆ ˆ ˆ ˆ y, φ = R π R x, φr 3π =, z, sin ( φ ) cos( φ ) (7) 4. Single qubit gates from rotation operations Gate, an abstraction that represents information processing is a unitary operator in quantum computing. In this section we construct single qubit operations in quantum computing using a single QCA cell. Three main single qubit gates implemented here are NOT gate, Phase-flip gate and Hadamard gate. NOT Operation: Applying NOT operator to a state vector, exchanges the probabilities of between the two basis states. NOT operator is physically implemented in QCA by lowering the potential barriers ( γ = 0) and applying a φ h P >> 1 for a time period Δ t = r = π. In coherence vector space, this γ event is represented by rotation of the coherence vector ( r ) by an angle π around the -axis (Fig. 3(b)). The rotation operator for a NOT-Operation is obtained by substituting φ = π in Eq. (5). iσ ( π x ) 0 ˆ ˆ π π i Unot = R x, φ= π = e = cos I isin ˆ σ x = = i σ x i 0 (8) Hadamard Operation: Hadamard gate creates superposition states in quantum algorithms. When it is applied to a qubit, interference between the basis states occurs. ˆ α + β α β U H ψ = (9) There are several ways of implementing a Hadamard gate in a Bloch sphere. One way of implementation is by rotating about the axis followed by reflection in the - plane (Fig 3(c)). Operator for Hadamard gate is i 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ U H = R R R R R R x, π y, π = x, π z, π x, π z, 3 π = 1 1 (10) Phase flip Operation: The Phase flip gate takes the qubit ψ = α 0 + β 1 to the state ψ = α 0 β 1. In Bloch sphere it is represented by rotation of around

6 70 G. Pavan, N. Chandrasekar and N. S. Kumar -axis by an angle π (Fig 3(d)). Physically this is implemented by raising the potential barrier γ << E and applying higher ( P >> + 1) as input condition. The operator π i ˆ e Rz, φ= π = = i π i e (11) represents the phase flip operation. The phase flip state obtained at the output cannot be physically distinguished from the input because the probability of the output state is same as that of the input state. However when this state interacts with other cells it might affect their states and cause measurable changes. Input Qubit (a) (b) (c) (d) Figure 3. (a).bloch sphere representing input state. Resulting qubits for (b).not Operation. (c).hadamard Operation. (d).phase flip Operation. 4. Conclusion We have proposed a method by which the single qubit operations can be realized using QCA cell. The parameters that control the environment of a QCA cell are given in coherence space representation and quantum operations are performed. The change in parameters is represented as change of angle and phase in the Bloch sphere representation. The three basic single qubit operations namely NOT, Hadamard, and Phase flip are realized using the rotation operators. References [1] Craig S. Lent, P. Douglas Tougaw, Nanotechnology, 4 (1993) [] David McMahon, Quantum Computing Explained John Wiley & Sons, Inc., Publication, New Jersey (008). [3] Geza Toth and Craig S. Lent Phy. Rev. A, Vol. 63, [4] Geze Toth, Craig S. Lent, Superlattices Microstruct., Vol-0, No. 4, [5] G.L. Snider, A.O.Orlov, I. Amlnni, G. H. Bernstein, C. S. Lent, J.L Merz and Q.Porod; Solid-State Electron., Vol-4,7-8.pp [6] K. Goser, P. Glosekotter, V. Dienstuhl, Nano Electronics and nano systems, springer, 004. [7] R. Bose, H.T. Johnson, Microelectron. Eng., 75 (004) Received: February, 01 Not Operation Hadamard Operation Phase-flip Operation