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Synthesis Techniques for Ternary Quantum Logic

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Synthesis of ternary quantum circuits involves basic ternary gates and logic operations in the ternary quantum domain. Works that define ternary algebra and their applications for ternary quantum logic realization, are very few. In this paper, we
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  Synthesis Techiniques for Ternary Quantum Logic Sudhindu Bikash Mandal 1 , Amlan Chakrabarti 1 and Susmita Sur-Kolay 2 1 A K Choudhury School of Information TechnologyUniversity of CalcuttaKolkata - 700009, India acakcs@caluniv.ac.in, sudhindu.mandal@gmail.com 2 Advanced Computing & Microelectronics UnitIndian Statistical InstituteKolkata - 700108, India ssk@isical.ac.in ISMVL - 25 May 2011  Ternary Quantum Computing Motivation for our work Key Contributions Ternary Algebra Ternary Logic Gates Proposed Synthesis Methodology Contents Ternary Quantum ComputingMotivation for our workKey ContributionsTernary Algebra Ternary Projection Operations J and L Ternary Logic Gates Implementation of   J  i   and  L i   operations using Generalized TernaryGate(GTG)A New Ternary  C  2 NOT   GateMulti-qutrit Generalized Ternary Gate Proposed Synthesis MethodologySysthesis of Ternary AdderConclusion Sudhindu Bikash Mandal 1 , Amlan Chakrabarti 1 and ,  Susmita Sur-Kolay 2 2  Ternary Quantum Computing Motivation for our work Key Contributions Ternary Algebra Ternary Logic Gates Proposed Synthesis Methodology Ternary Quantum Computing A ternary quantum system exists in linear superposition of three basisstates: | 0  > ,  | 1  >  and  | 2  > A unit of information is called a qutrit [Muthukrishnan & Stroud (2000)]A pure state of a qutrit can be represented by the Poincar´e sphere,[Klimov et. al. (2004)] | ψ >  = sin( ξ/ 2) cos( θ/ 2) | 0  >  +  e  i  φ 12  sin( ξ/ 2) sin( θ/ 2) | 1  >  +  e  i  φ 13  cos( ξ/ 2) | 2  > where  θ  and  ξ  determine the magnitude of the components of   | ψ > , φ 12 ,  φ 13  are the phases of   | 1  >  relative to  | 2  >  and  | 3  >  respectively.A quantum register of size  m  qutrits can hold 3 m simultaneous values[Zilic & Radecka (2007)]Operations on a qutrit are developed in a 3-dimensional Hilbert spaceunder the field GF(3) Sudhindu Bikash Mandal 1 , Amlan Chakrabarti 1 and ,  Susmita Sur-Kolay 2 3  Ternary Quantum Computing Motivation for our work Key Contributions Ternary Algebra Ternary Logic Gates Proposed Synthesis Methodology Motivation for our work [ Yanget  . al  . (2005)] - a set of universal gates for ternary quantumcomputing, namely ternary NOT, ternary Swap, ternary Toffoli can realize any arbitrary quantum circuit without ancilla bits but   no generalized circuit synthesis rule was proposed [ Gieseckeet  . al  . (2007)] - synthesis using quantum ternary multiplexers method of iterative deepening using depth first search to achieve minimumgate cost  but   at a high computational cost [ Khanet  . al  . (2009)] - synthesis technique for GF(3) based garbage freereversible or quantum logic circuit from its truth values M-S and shift gates as basic building blocks but   no simplification rule to reduce the gate count Sudhindu Bikash Mandal 1 , Amlan Chakrabarti 1 and ,  Susmita Sur-Kolay 2 4  Ternary Quantum Computing Motivation for our work Key Contributions Ternary Algebra Ternary Logic Gates Proposed Synthesis Methodology Key Contributions A new ternary operator - projection operator  L i  Synthesis rules for ternary quantum circuit using SOP based expressionsA new ternary quantum  C  2 NOT   gateSimplification rules for circuit minimization(we assume permutative quantum circuits) Sudhindu Bikash Mandal 1 , Amlan Chakrabarti 1 and ,  Susmita Sur-Kolay 2 5
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