Synthesis Techiniques for Ternary Quantum Logic
Sudhindu Bikash Mandal
1
, Amlan Chakrabarti
1
and
Susmita SurKolay
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
GateMultiqutrit Generalized Ternary Gate
Proposed Synthesis MethodologySysthesis of Ternary AdderConclusion
Sudhindu Bikash Mandal
1
, Amlan Chakrabarti
1
and ,
Susmita SurKolay
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 3dimensional Hilbert spaceunder the ﬁeld GF(3)
Sudhindu Bikash Mandal
1
, Amlan Chakrabarti
1
and ,
Susmita SurKolay
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
MS 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 SurKolay
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 SurKolay
2
5