QUANTUM LOGIC

SYNTHESIS

Srikanth Bitra (10881A0451)

Under guiedence of C.Padmini

ECE dept.

Contents

Introduction

Differences between traditional and quantum circuits

Reversible computation

Reversible gates NOT

C-NOT

Toffoli

RSWAP

Fredkin

Synthesis frame works RMRLS

DDS

RMDDS

Features of quantum computer:

Super position

Interference

Entanglement

Measurement

Quantum vs Classical

Quantum Computation

Bijective or Reversible

Input and output vectors have same bit-width

No information loss

Classical Computation

Irreversible

Different input and output bit-width

Loss of information

Need for reversibility

Fundamental physics dictates that energy

must be dissipated when information is

erased, in the amount KT ln2 per bit erased,

where K is Boltzmann constant(k=1.38x10-

23JK-1) and T is absolute temperature in K

NOT

C-NOT (Feymann gate)

CCNOT (Toffoli) gate

Fredkin(CSWAP)

Logic:p=a, q=if(a=1) then c else b, r=if(a=1) then b else c

Synthesis methods of quantum

circuits

QBC synthesis

Reed muller reversible logic synthesis

Decision diagram synthesis

RMRLS + DDS RMDDS

Reed Muller Reversible Logic

Synthesis(RMRLS)

Any Boolean function can be described by an exclusive-OR sum-of-products (ESOP) expression

ex: for a boolean expression y= a+b’c

SOP=a+b’c

ESOP=a a’b’c

Two types of expressions:

PPRM :All variables are un-complemented

PPRM=abc ac bc a c

FPRM:Either x or x’ appear throughout

FPRM=a’b’c a’ 1

Reed Muller Reversible Logic

Synthesis (cont..)

Synthesis flow

Find factors for any outptut a0 without literal a (target) EX: a0 = a bc 1 ac

Valid factors :bc,1

Target : a

Build a node in a search tree, where: #PPRM terms is reduced by Toffoli gate

Create a child node

Insert the node into Priority Queue (PQ) Sort Priority queue

Priority α # PPRM terms eliminated

Pop the queue ,repeat above steps

Decision Diagram Synthesis

Boolean functions are represented by a DD

(Decision Diagrams)

DD:acyclic graph G=(V, E) where

decompositions are applied to each node v ϵ V

Shannon decomposition(S) : f=xi’.fxi=0 xi.fxi=1

Positive Davio decomposition(pD): f=xi’.fxi=0

xi.fxi=2

Negative Davio decomposition(nD): f=xi’.fxi=1

xi.fxi=2

fxi=0 and fxi=1 are co-factors w.r.t. xi’ and xi

fxi=2 = fxi=0 fxi=1

Decision Diagram Synthesis

(DDS)

Synthesis flow

Build DD(Decision Diagrams)

Depth first search

Map each gate to Toffoli gate

Note:

DDS method automatically transforms an

irreversible specification into reversible

0

a a

a

a a

b

a a

b

0

0

0

0

0

0

0

0

1

1

1

1 1

1

1 1

1

Low(f1)

Low(f1)

Low(f1)

Low(f3)

Low(f3)

Low(f3) high(f3)

high(f2)

high(f2)

high(f3)

high(f2)

high(f3)

b

high(f3)

Low(f3)

Low(f1)

a

0

high(f2)

Low(f3)

high(f3)

b

Low(f1)

a

0

high(f2)

Low(f3)

high(f3)

b

Low(f1)

a

0

high(f2) g

g

g

g

g

f2

f1

f1

f1

f1

f2

f2

f2

f3

f3

f3

RMRLS vs DDS

RMRLS DDS

Pros Fewer qubits(lines)Low quantum cost

Low synthesis timeAble to synthesize large circuitsCan synthesize irreversible specifications

Cons High synthesis timeOnly able to synthesize small circuits

Large no. of garbage outputs

Reed-Muller Decision Diagram

Synthesis (RMDSS)

Hybrid of RMRLS and DDS

A flexible and efficient reversible circuit

synthesizer

Qubits can be traded off for QC

User defined time limit

User

constriants

Applications of quantum gates

Efficient white light LED’s

Processor’s designed for complex scientific

calculations

Artificial intelligence

cryptography

Acknowledge

Susmita Sur-Kolay (Indian Statistical Institute)

References

D. P. Di Vincenzo, “Quantum Computation,”

Science, 270, 1995, pp. 255-256.

K. Iwama, Y. Kambayashi, S. Yamashita,

“Transformation Rules for Designing CNOT-

Based Quantum Circuits,” Design Automation

Conference,2002, pp.419-424.

P. A. M. Dirac, The Principles of Quantum

Mechanics, Oxford University Press, 1st

Edition, 1930.