An Alternative Geometry for Quantum Cellular Automata

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VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 549-553Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

An Alternative Geometry for Quantum Cellular AutomataPAUL G. KRAUSE, RACHEL M. MUELLER, P. DOUGLAS TOUGAW*

and JANELLE M. WEIDNERDepartment of Electrical and Computer Engineering, Valparaiso University, Valparaiso, IN 46383 USAWe examine an alternative layout geometry for the quantum cellular automata (QCA)architecture. In the traditional QCA geometry, all of the cells are placed in a singleplane, so that each cell interacts with a particular neighbor only along one of its edges.By rotating the cells out of the plane, we make it possible for neighbors to interact alongall four edges at once. This increased interaction leads to a more bistable cell-cellresponse function and a 50% higher excitation energy. We also present a majority logicgate designed using three-dimensional cells.

Keywords." Quantum cellular automata, computer architecture, quantum-dot molecules

1. INTRODUCTIONThe quantum cellular automata (QCA) architec-ture, which is composed of arrays of Coulombi-cally coupled quantum dot molecules, is arevolutionary new paradigm for quantum comput-ing [1]. QCAs offer new possibilities for comput-ing at the nanometer scale, since they can be usedto hierarchically implement any logical function[2-4].

In this paper, we present an alternative layoutgeometry for the QCA architecture by rotating thecells out of their plane. In the next section, webriefly review the QCA paradigm in two dimen-sions, while in Section 3 we examine the QCAarchitecture in three dimensions. By generalizingQCA cells to three dimensions and computing the

resulting cell-cell response curve, we show thatthe three-dimensional cells are more bistable thanthe traditional two-dimensional cells.

In Section 4, we examine a three-dimensionalmajority logic gate. We show that two-dimen-sional and three-dimensional majority gates giveidentical functional results.

2. TWO-DIMENSIONAL QCA DEVICESFigure shows the basic building block in theQCA paradigm, the quantum dot molecule cell.Four quantum dots are spaced close enough forelectrons to tunnel between neighboring pairs ofdots. Two stable states are possible with thisconfiguration. The first, corresponding to an

*Corresponding author: [email protected]


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FIGURE A schematic of the basic QCA cell. Electronsinteract Coulombically and tunnel among the four sites. FIGURE 2 The cell-cell response function of two neighbor-

ing QCA cells. The bistable nature of the response functionallows us to encode a binary bi t in each cell.

electron occupying the upper right hand and lowerleft hand dots, represents a polarization of + 1.The other, corresponding to an electron in theupper left hand and lower right hand dots,represents a polarization of-1. These twopolarizations can represent binary values of "1"and "0", respectively.The Hamiltonian used to describe each cell

includes Coulombic interaction between each pairof sites, near-neighbor tunneling, next-near-neigh-bor tunneling, on-site energies and on-site char-ging costs. The on-site energy of each site will beaffected by the polarization of its neighboring cells.Thus, Coulombic forces cause the polarizationof one cell to affect the polarization of neighboringcells. The cell-cell response curve shown in Figure 2is computed by solving the two-particle, time-independent Schr6dinger equation for the first cellin the presence of the second cell. The curve showsthat the response of such cells is very bistable,which means that for even very small polarizationsin one cell, the second cell is completely polarized.This feature is very attractive, since it allows thesecells to encode a single binary bi t in their quantumstate. Furthermore, when propagating signals, thesignal level is restored to the maximum polariza-tion at every stage.The most useful logic structure in the QCA

paradigm is the three-input majority gate. A

majority gate is one in which the output agreeswith the majority of the inputs. This logic structureis essential because we can produce both AND andOR gates from it . A two-input OR gate can beproduced by setting one of the three inputs equalto 1, while a two-input AND can be produced bysetting one of the three inputs equal to 0.

Figure 3 shows a schematic for the coplanarQCA majority gate. In this figure, inputs A, B andC are used as inputs. The output is propagateddown the line of cells acting as a binary wire.The ground state of this multiple-cell structure was

Input BInput A

Output

Device cell 1 Input CFIGURE 3 The coplanar majority voting gate. The device cellalways matches a majority of the three fixed neighbors.


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4. A THREE-DIMENSIONALMAJORITY GATE

As mentioned in the previous section, the mostimportant logical device of the QCA paradigm isthe majority logic gate. Figure 6 demonstrates thelayout of a QCA majority gate in three dimen-sions. Cells 1 and 3 are coplanar to cell 4, so theytend to drive it to match their polarization. On theother hand, cell 2 drives cell 4 to an oppositepolarity, since it interacts using the three-dimen-sional cell-cell response function. This can becancelled by using an extra cell in that input, or itcan be used to implicitly invert that input.

In order to ensure that the three inputs "vote"with equal weighting, the distances between cellswere chosen such that when all four neighbors ofthe device cell are voting in opposite directions, thedevice cell exhibits no polarization.To find this distance, the three coplanar cells

(the device cell and two inputs) were placed adistance 60 nm apart from each other and the twocoplanar drivers were set with P= + 1. The othertwo neighbors were then se t with P=-I andmoved away from the device cell until it exhibitedzero polarization. At that point, we were certainthat the horizontal and vertical neighbors werehaving the same impact on the polarization of thedevice cell. The distance between the non-coplanarcells at which this occurred was 1.7435 times larger

Cell 4Cell 5

Cell 1 Cell 3Cell 2

FIGURE 6 A schematic representation of the three-dimen-sional majority logic gate. Cells 1, 2 and 3 are drivers, while cell4 is the device cell and cell 5 is the output. Cells 2 and 4 tend toanti-align, so the input at cell 2 is implicitly inverted.

than the distance between the coplanar cells, sincethe non-coplanar cells interact more strongly witheach other.We used the Intercellular Hartree Approxima-

tion to find the ground state of this multiple-cellsystem. The three input cells were fixed at a specificpolarity (either + or -1), while the device celland the output cell were allowed to change.We tested each of the eight combinations ofdriver polarizations and found that in each case,the ground state corresponds to the correct resultfor a majority logic gate. The state of the outputalways matches the state of a majority of the threeinputs, assuming the non-coplanar input has beenimplicitly inverted.

5. CONCLUSIONSWe have proposed an alternative geometry for theQCA computing architecture and we have shownthat this geometry achieves a better cell-cellresponse than the two-dimensional implementa-tion. We have also shown that an inverter and amajority logic gate can be constructed three-dimensionally, so an y combinational functioncan be constructed using this alternative geometry.

AcknowledgementsThe authors would like to acknowledge thesupport of the Valparaiso University College ofEngineering, as well as many productive conversa-tions with the University of Notre Dame Nano-electronic Devices Group.

References[1] Lent, C. S., Porod, W. and Tougaw, P. D. (1993). Appl.

Phys. Lett., 62, 714.[2 ] Lent, C. S., Tougaw, P. D. , Porod, W. and Bernstein, G.(1993). Nanotechnology, 4, 49.[3 ] Tougaw, P. D. , Lent, C. S. and Porod, W. (1993). J. Appl.

Phys., 74 , 3558.[4 ] Lent, C. S. and Tougaw, P. D. (1993). J. Appl. Phys., 74,

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Authors BiographiesPaul G. Krause is a graduate student at the Un-iversity of Minnesota, where he is pursuing an ad-vanced degree in Biomedical Engineering. Heearned his B.S.E.E. degree from Valparaiso Uni-versity.

Rachel M. Mueller is currently employed byMotorola, where she is working as a softwareengineer in the iDEN group. She earned herB.S.Cp.E. degree from Valparaiso University.

P. Douglas Tougaw is an assistant professor atValparaiso University. He received his B.S.E.E.degree from Rose-Hulman Institute of Technologyand his Ph.D. degree from the University of NotreDame.

Janelle Weidner is currently an undergraduatestudent of electrical engineering at ValparaisoUniversity. She is participating in a cooperativeeducation experience with the National SecurityAgency.

An Alternative Geometry for Quantum Cellular Automata

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