- Nano Express
- Open Access

# Two-layer synchronized ternary quantum-dot cellular automata wire crossings

- Iztok Lebar Bajec
^{1}Email author and - Primož Pečar
^{1}

**7**:221

https://doi.org/10.1186/1556-276X-7-221

© Lebar Bajec and Pečar; licensee Springer. 2012

**Received:**17 January 2012**Accepted:**16 April 2012**Published:**16 April 2012

## Abstract

### Abstract

Quantum-dot cellular automata are an interesting nanoscale computing paradigm. The introduction of the ternary quantum-dot cell enabled ternary computing, and with the recent development of a ternary functionally complete set of elementary logic primitives and the ternary memorizing cell design of complex processing structures is becoming feasible. The specific nature of the ternary quantum-dot cell makes wire crossings one of the most problematic areas of ternary quantum-dot cellular automata circuit design. We hereby present a two-layer wire crossing that uses a specific clocking scheme, which ensures the crossed wires have the same effective delay.

## Keywords

- Quantum-dot cellular automata
- Ternary processing
- Wire crossing
- Multi-layer design
- 85.35.Be
- MCS: 68Q80
- 03B50

## Background

Since the first introduction of quantum-dot cellular automata (QCA), an interesting nanoscale computing paradigm by Lent et al. in 1993 [1], many researchers have embraced its simple concept and potential as a future processing platform [2–6]. In recent years, a group of researchers have presented a generalization of the basic QCA cell, namely the ternary QCA (tQCA) cell [7–10], which enables ternary computation. Their principal motivator was the premise that future processing platforms should not disregard the advantages of multi-valued processing [11–15]. The group presented the basic ternary building blocks, the inverter, majority gate, wire, corner wire and fan-out, and more recently also, a functionally complete set of ternary logic functions, based on Post Logic, and a memorizing tQCA circuit [16, 17]. Due to the specifics of the tQCA cell, wire crossings seem to be the principal drawback before a more widespread acceptance of tQCA circuitry. Wire crossings are one of the most used steps in systematic logic design. In the classic, binary QCAs, wires can be crossed either in a coplanar fashion by using rotated QCA cells for one of the wires or in a multilayer fashion where two intermediate layers are used to prevent any possible crosstalk between the two crossing lines [18]. Although the multilayer approach proves to be more robust [19], the majority of designs employ the coplanar one; that is in fact one of the most praised features of classic QCA. Since coplanar crossings tend to be prone to robustness issues, much research has been devoted to its increase, even to the extents of altering the design of QCA logic gates [20–22], or as in the case of molecular implementations, through the elimination of crossings by logic gate duplication [23, 24]. Another approach exploits the pipelined nature of QCAs and uses parallel-to-serial converters and a specialized clocking scheme to design a coplanar crossbar network [25]. Coplanar crossings with rotated tQCA cells are not possible, but multilayer crossings are, as it has been reported recently [26]. Here, we go a step further by presenting a wire crossing that is synchronized, i.e., the two wires employ such clocking schemes that the outputs of the two wires have the same effective delay. In addition, the clocking schemes allow for a two-layer design, in other words, removing the requirement for additional layers, whose sole purpose is to prevent possible crosstalk. The article is organized as follows. We first present the overview of a ternary QCA cell, its architecture and inter-cell interaction. We follow by presenting a two-layer design, where we first show the inter-layer interaction followed by presenting the design and clocking scheme of a two-layer synchronized wire crossing.

### The ternary QCA cell

It turns out that placing a well-polarized cell (i.e., a cell with electrons fixed in one of the four arrangements) nearby causes one of the four states to become the favored one even in the observed cell. When the well-polarized cell is on the same plane, but to the left, right, above, or below the observed cell, then the state A or B in the well-polarized cell induces the same state even in the observed cell. However, state C induces state D, and the same goes for state D, which induces state C in the observed cell. If one interprets state A, B, C, and D as balanced ternary logic values, so that state A represents logic value −1, state B logic value 1, and states C and D both logic value 0, then inter-cell interaction causes the logic value of the well-polarized cell to be transferred to the observed cell.

If the well-polarized cell is placed diagonally to the observed cell, then state A in the well-polarized cell induces state B in the observed cell, state B induces state A, state C induces state C, and state D induces state D. When the states are interpreted as logic values, this translates to logic negation. With specific planar arrangements of cells, it is thus possible to mimic the behavior of interconnecting wires as well as logic gates [27]. By interconnecting such building blocks, more complex devices capable of processing can be constructed.

The reliability of the logic value transfer throughout a QCA device (i.e., a spatial arrangement of QCA cells) depends foremost on the reliability of the switching process, i.e., the transition of a cell’s state that corresponds to one logic value to a state that corresponds to another. The reliability is ensured via the adiabatic switching concept [9, 28], where a cyclic signal, namely adiabatic clock, is used to control the switching dynamics. The cyclic signal is comprised of four phases. The switch phase serves the cell’s gradual update of the state with respect to the neighbors. The hold phase is intended for the stabilization of the cell’s state when it is to be passed on to the neighbors that are in the switch phase. The release phase and the relax phase support the cell’s gradual preparation for a new switch.

## Methods

Simulations were conducted following the same methods as outlined in the work of Pečar et al. [10]. Inter-layer interaction was analyzed as a two-cell system of one well-polarized cell and one observed cell placed directly one over the other but on separated layers. Through a series of sequential steps, a transition from the initial state to neutral and then from the neutral state to the final state is applied to the well-polarized cell, and the response of the observed cell is computed. The response is computed by numerical diagonalization of a tight-binding Hubbard-type Hamiltonian, where quantum dots are represented as sites, and the degrees of freedom internal to the quantum dots are ignored. The same set of parameters was used as in the work of Pečar et al. [10], with the inter-layer distance equal to that between the neighboring cells.

The behavior of the wire crossing was assessed using the intercellular Hartree approximation, as in the work of Pečar et al. [10]. A single simulation consists of the circuit’s total delay time discrete time steps. We use 200 time steps per clock cycle. At each time step, the ground state of the QCA is found by iteratively solving for the ground state of each cell. The ground state of a cell (observed cell) is calculated under the influence of the states of all other cells in the QCA which are momentarily treated as well-polarized. In turn, each of the QCA cells is chosen as the observed cell, so their states change. This process is iterated until the QCA relaxes, and no further change in any of the cells is observed (i.e., until the QCA reaches its ground state at the corresponding time step). At every simulation, an initial state is applied to cells marked as input cells (X1,X2) and the simulation run for the corresponding total delay time of the QCA circuit. This is when the cells marked as output cells (Y1,Y2) are in the hold phase, and their states are treated as valid. We simulated all possible combinations of initial states.

## Results and discussion

### Multi-layer interaction

QCA processing is based on inter-cell interaction, where the state of a cell influences the states of its neighbors and vice versa. The same applies for inter-layer interaction. The cell that is closest to the observed cell has the largest influence on the observed cell’s state. In a multi-layer case, two cells are closest when placed directly one over the other, i.e., on the same location but on separate layers.

A negation of the transferred logic value occurs by moving from one layer to the other. When designing a wire crossing, this has no real effect as eventually the transferred logic value will be negated once more upon moving back to the original layer. In the case when processing is to be performed on different layers, however, this fact has to be kept in mind. For states C and D, it presents no real problem, as they both represent the same logic value, and alternating between the two states is achieved through simple addition of another adjacent cell. For states A and B, which represent two opposite logic values (−1 and 1, respectively), this, however, means adding an inverter (which in its simplest form could be just one cell displaced diagonally) or designing the processing element based on an inverted input value.

### Synchronized two-layer wire crossing

The total delay of the crossing is one clock cycle. The four phases are used so as to keep the distance between active cells (neighboring cells that are currently in the hold or switch phase) as large as possible, as well as to achieve robust inter-cell transfers. Active cells on the two layers are never directly one over the other, although this would not present a real issue as long as enough cells are active in the same instant. Reducing the number of active cells makes them more susceptible to inter-layer crosstalk, all due to the highly nonlinear inter-cell interaction.

The line, marked X1-Y1, travels in a diagonal fashion upwards on layer L1. This is achieved in one clock cycle (four phases), with blocks of two cells, so that the same state that is input to the first cell, marked X1, appears on the last cell, marked Y1, after a delay of one clock cycle.

The line, marked X2-Y2, travels first vertically from layer L1 to layer L2, then in a diagonal fashion downwards, and back vertically from layer L2 to layer L1, again all in one clock cycle. This ensures that the state that is input to the first cell, marked X2, appears on the last cell, marked Y2, after a delay of one clock cycle.

## Conclusions

Due to the specifics of the ternary quantum-dot cell, the basic building block of ternary quantum dot cellular automata, coplanar wire crossings are not possible. In this article, we present a two-layer synchronized wire crossing; a wire crossing that uses such clocking schemes that the effective latency is equal for both wires (one clock cycle). In addition, the clocking schemes allow for a two-layer design. They override the requirement for additional layers, whose sole purpose is to prevent possible crosstalk.

Our current research is devoted to the study of synchronized two-layer wire crossings that consume fewer clock cycles as well as tile-based solutions, what we find to be one of the more promising approaches for QCA design in general.

## Author’s contributions

ILB designed the two-layer synchronized wire crossing and the corresponding clocking scheme. He also prepared this manuscript. PP helped in the design of the clocking scheme, performed the simulations, and provided assistance through literature research for the preparation of the manuscript. Both authors read and approved the final manuscript.

## Declarations

### Acknowledgements

This work was funded in part by the Slovenian Research Agency through the Pervasive Computing research programme (P2-0359).

## Authors’ Affiliations

## References

- Lent CS, Tougaw PD, Porod W, Bernstein GH: Quantum cellular automata.
*Nanotechnology*1993, 4: 49. 10.1088/0957-4484/4/1/004View ArticleGoogle Scholar - Lent CS, Tougaw PD: Lines of interacting quantum-dot cells: a binary wire.
*J Appl Phys*1993, 74(10):6227. 10.1063/1.355196View ArticleGoogle Scholar - Niemier MT, Kontz MJ, Kogge PM: A design of and design tools for novel quantum dot based microprocessor. In
*DAC ’00: Proceedings of the 37th conference on Design automation*. ACM Press, New York; 2000:227–232.View ArticleGoogle Scholar - Niemier MT, Rodrigues AF, Kogge PM: A potentially implementable FPGA for quantum dot cellular automata.
*1st Workshop on Non-Silicon Computation (NSC-1)*2002, 38–45.Google Scholar - Huang J, Momenzadeh M, Lombardi F: Design of sequential circuits by quantum-dot cellular automata.
*Microelectron J*2007, 38(4–5):525. 10.1016/j.mejo.2007.03.013View ArticleGoogle Scholar - Vankamamidi V, Ottavi M, Lombardi F: A serial memory by quantum-dot cellular automata (QCA).
*IEEE Trans Comput*2008, 57(5):606.View ArticleGoogle Scholar - Lebar Bajec I, Mraz M: Towards multi-state based computing using quantum-dot cellular automata. In
*Unconventional Computing 2005: From Cellular Automata to Wetware*. Edited by: Teucher C, Adamatzky A. Luniver Press, Beckington; 2005:105–116.Google Scholar - Lebar Bajec I, Zimic N, Mraz M: The ternary quantum-dot cell and ternary logic.
*Nanotechnology*2006, 17(8):1937. 10.1088/0957-4484/17/8/023View ArticleGoogle Scholar - Pečar P, Mraz M, Zimic N, Janež M, Lebar Bajec I: Solving the ternary QCA logic gate problem by means of adiabatic switching.
*Jpn J Appl Phys*2008, 47(6):5000. 10.1143/JJAP.47.5000View ArticleGoogle Scholar - Pečar P, Ramšak A, Zimic N, Mraz M, Lebar Bajec I: Adiabatic pipelining: a key to ternary computing with quantum dots.
*Nanotechnology*2008, 19(49):495401. 10.1088/0957-4484/19/49/495401View ArticleGoogle Scholar - Porat DI: Three-valued digital systems.
*Proc IEE*1969, 116(6):947.Google Scholar - Frieder G: Ternary computers: Part 1: motivation for ternary computers. In
*MICRO 5: Conference record of the 5th annual workshop on Microprogramming, September 1972; Urbana, Illinois*. ACM, New York; 1972:83–86.View ArticleGoogle Scholar - Rine DC (Ed):
*Computer science and multiple-valued logic: theory and applications*. Elsevier/North-Holland, Amsterdam; 1984.Google Scholar - Dubrova E, Jamal Y, Mathew J: Non-silicon non-Binary computing: why not? In
*1st Workshop on Non-Silicon Computation (NSC-1), February 2002*. , Boston, Massachusetts; 2002:23–29.Google Scholar - Fitting M, Orlowska E (Eds):
*Beyond two: theory and applications of multiple-valued logic*. Physica-Verlag, Heidelberg; 2003.Google Scholar - Pečar P, Janež M, Zimic N, Mraz M, Lebar Bajec I: The ternary quantum-dot cellular automata memorizing cell. In
*Proc. of ISVLSI 2009*. IEEE Computer Society, Los Alamitos, CA; 2009:223–228.Google Scholar - Pečar P, Lebar Bajec I: The key elements of logic design in ternary quantum-dot cellular automata.
*Lecture Notes Comput Sci*2011, 6741: 177.Google Scholar - Walus K, Schulhof G, Jullien GA: High level exploration of quantum-dot cellular automata (QCA). In
*Proc. of Signals, Systems & Computers 2004*. IEEE Computer Society, Los Alamitos, CA; 2004:30–33.Google Scholar - Schulhof G, Walus K, Jullien GA: Simulation of random cell displacements in QCA.
*ACM J Emerging Technol Comput Syst*2007, 3: 2:1.Google Scholar - Bhanja S, Ottavi M, Lombardi F, Pontarelli S: QCA Circuits for robust coplanar crossing.
*J Electron Test*2007, 23: 193. 10.1007/s10836-006-0551-yView ArticleGoogle Scholar - Sen B, Dalui M, Sikdar BK: Introducing universal QCA logic gate for synthesizing symmetric functions with minimum wire-crossings. In
*Proc. of ICWET 2010*. ACM, New York, NY; 2010:828–833.Google Scholar - Sen B, Sengupta A, Dalui M, Sikdar BK: Design of testable universal logic gate targeting minimum wire-crossings in QCA logic circuit. In
*Proc. of DSD 2010*. IEEE Computer Society, Los Alamitos, CA; 2010:613–620.Google Scholar - Chaudhary A, Chen DZ, Hu XS, Whitton K, Niemier M, Ravichandran R: Eliminating wire crossings for molecular quantum-dot cellular automata implementation. In
*Proc. of ICCAD 2005*. IEEE Computer Society, Washington, DC, USA; 2005:565–572.Google Scholar - Chaudhary A, Chen DZ, Hu XS, Niemier MT, Ravichandran R, Whitton K: Fabricatable interconnect and molecular QCA circuits.
*IEEE Trans Comput Aided Des Integr Circuits Syst*2007, 26(11):1978.View ArticleGoogle Scholar - Graunke CR, Wheeler DI, Tougaw D, Will JD: Implementation of a crossbar network using quantum-dot cellular automata.
*IEEE Trans Nanotechnol*2005, 4: 435. 10.1109/TNANO.2005.851278View ArticleGoogle Scholar - Pečar P: Introducing interconnection crossing in ternary quantum-dot cellular automata. In
*Proc. of ICQNM 2011*. IARIA; 2011:1–5.Google Scholar - Tougaw PD, Lent CS: Logical devices implemented using quantum cellular automata.
*J Appl Phys*1994, 75(3):1818. 10.1063/1.356375View ArticleGoogle Scholar - Tougaw PD, Lent CS: Dynamic behaviour of quantum cellular automata.
*J Appl Phys*1996, 80(8):4722. 10.1063/1.363455View ArticleGoogle Scholar - Niemier MT, Kogge PM: Problems in designing with QCAs: layout = timing.
*Int J Circuit Theory Appl*2001, 29: 49. 10.1002/1097-007X(200101/02)29:1<49::AID-CTA132>3.0.CO;2-1View ArticleGoogle Scholar

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