quantum cost calculation of reversible circuit

Quantum Cost Calculation of Reversible Circuit

Sajib MitraDepartment of Computer Science and EngineeringUniversity of [email protected]

OVERVIEW Reversible Logic Quantum Computing Quantum Gates Realization of Quantum NOT Quantum wire and Special Cases Quantum Cost Calculation of RC Conclusion Assignment References

Reversible Logic Equal number of input and output vectors Preserves an unique mapping between input and

output vectors of the particular circuit One or more operation can implement in a single unit

called Reversible Gate (N x N) Reversible Gate has N number of inputs and N

number of outputs where N= {1, 2, 3, …}

NOTA_A

(a) 1x1 Reversible Gate

Reversible Logic (cont…)

Advantage Recovers bit-loss as well as production of heat Adaptable for Quantum Computing Multiple operations in a single cycle Uses low power CMOS technology

A

BA B

3

2

1

0

1

0

INP

UT

VE

CT

OR

(A

, B)

OU

TP

UT

VE

CT

OR

(A

B

)

OU

TP

UT

VE

CT

OR

(P

, Q)

3

2

1

0

2

1

INP

UT

VE

CT

OR

(A

, B)

FGA

B Q=A B

P=A

0

3

(a) Irreversible EX-OR operation (b) Reversible EX-OR operation


Limitation Feedback is strictly restricted Maximum and minimum Fan-out is always one


Most Popular reversible gates are as follows:

Fig. 3x3 Dimensional Reversible gates

F2GA

BC

A

A C

A B

(d) Feynman Double Gate

PGA

BC

AA B

AB C

(b) Peres Gate

FRGA

BC

A

A’C AB

A’B AC

(f) Fredkin Gate

TGA

BC

A

AB C

B

(c) Toffoli Gate

NFTA

BC

AC’ B’CAC’ BC

A B

(e) New Fault Tolerant Gate

FGAA

B A B

(a) Feynman Gate


Most Popular reversible gates are as follows:

Fig. 4x4 Dimensional Reversible gates

D

ABC

P = AQ = R = A B CS = (A B)C AB D

MTSG A B

D

ABC

P = AQ = A’C’ B’R = A’C’ B’ DS = ( A’C’ B’)D AB C

ABC

P = A

D

Q = A BR = AB CS = AB’ D

MIG TSG

(a) Modified IG Gate (b) TSG Gate

(c) Modified TSG Gate

Quantum Computing First proposed in the 1970s, quantum computing relies on

quantum physics by taking advantage of certain quantum physics properties of atoms or nuclei that allow them to work together as quantum bits, or qubits, to be the computer's processor and memory.

Qubits can perform certain calculations exponentially faster than conventional computers.

Quantum computers encode information as a series of quantum-mechanical states such as spin directions of electrons or polarization orientations of a photon that might represent as or might represent a superposition of the two values.

10 or

10 q

Quantum Computing (cont…)

Quantum Computation uses matrix multiplication rather than conventional Boolean operations and the information measurement is realized using qubits rather than bits The matrix operations over qubits are simply specifies by using quantum primitives as follows:

›|B A

|A

|B

|A

›(a) Quantum XOR operation (b) Acting representation

of Quantum XOR

UCN=

1 0 0 00 1 0 00 0 0 10 0 1 0

››


›|B A

››

|A

|B

|A

›Quantum XOR operation

Input Output

A B P Q

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

Input/output

PatternSymbol

00 a

01 b

10 c

11 d


›|B A

››

|A

|B

|A


1 0 0 00 1 0 00 0 0 10 0 1 0

abcd

abdc


›|B A

››

|A

|B

|A


1 0 0 00 1 0 00 0 0 10 0 1 0

abcd

abdc

Input Output

A B P Q

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

Quantum Gates

A A

B VIF (A ) THEN V(B )

ELSE B

A A

B V+ IF (A ) THEN V+(B ) ELSE B

A A

B A B

(b) EXclusive-OR

A A’

(a) NOT

(c) Square Root of NOT (d) Hermitian of SRN

Fig: Quantum Gates are used for realizing Reversible Circuit

Quantum Gates (cont…)

What is SRN?

1 V 1 or 0

But

1 V V 0

Quantum Gates (cont…)

What is SRN?

1 V 1 or 0

But

1 V V 0NOT

But How?

Realization of Quantum NOT

NOT

(a) NOT Operator

CF

(b) Coin Flip Operator

QCF

(c) Quantum Coin Flip Operator

Basic operator for single input line:1. NOT 2. Coin Flip 3. Quantum Coin Flip

Realization of Quantum NOT (cont…)

0 1

1 0

NOT

NOT

0

1

NOT

NOT

NOT

NOT

0

1

1

0

CF00 or 1

CF11 or 0

CF

CF

1 or 0

0 or 1

CF0 0 or 1

CF1 1 or 0

0 0 or 1QCF

1 1 or 0QCF

1 or 00QCF1 QCF

1QCF00 or 1

QCF


1

0 1

10 10

1/21/2

1/2 1/2 1/2 1/2

1/4 1/4 1/4 1/4

Probability of 0 or 1 based on Coin Flip:


1

0 1

10 10

1/21/2

1/2 1/2 1/2 1/2

1/4 1/4 1/4 1/4

Probability of 0 or 1 based on Coin Flip:

So the Probability of P(0)=1/2 P(1)=1/2


Probability of |0> or |1> based on Quantum Coin Flip:

|1>

21

21

21

21

21

21

|0>

|1>

|0>

|0>

|1>

|1>

21

21 2

12

1


Probability of |0> or |1> based on Quantum Coin Flip:

|1>

21

21

21

21

21

21

|0>

|1>

|0>

|0>

|1>

|1>

21

21 2

12

1

So the Probability of P(|0>)=1 P(|1>)=0


NOT operation can be divided into to SRN matrix production

1 V V 0

NOT

1 0

Quantum Cost (QC) of any reversible circuit is the total number of 2x2 quantum primitives which are used to form equivalent quantum circuit.

Quantum Wire and Special Cases (cont…)

A A

B A B

Quantum XOR gate, cost is 1


A A

B B

Two Quantum XOR gates, but cost is 0


A A

B B

Quantum Wire


SRN and its Hermitian Matrix on same line.

VV+= Identity and the total cost = 0

V V+

Quantum Cost of V and V+ are same , equal to one.


SRN and its Hermitian Matrix on same line.

VV+= Identity and the total cost = 0

V V+Y Y

XX


The attachment of SRN (Hermitian Matrix of SRN) and EX-OR gate on the same line generates symmetric gate pattern has a

cost of 1. Here T= V or V+

T(a)

T(b)


A A

B VIF (A ) THEN V(B )

ELSE B

A A

B V+ IF (A ) THEN V+(B ) ELSE B

A A

B A B

(b) EXclusive-OR

A A’

(a) NOT, Cost =0

(c) Square Root of NOT (d) Hermitian of SRN

The cost of all 4x4 Unitary Matrices (b, c, d) and the symmetric gate pattern (e, f, g, h) are unit.

Quantum Cost of F2G

F2Ga

bc

p= a

r= a c

q= a b

F2G, Cost = 2

a

a b

a

bc a c

Quantum Cost of Toffoli Gate

TG, Cost = 5

VV V+

b

aa

b

c ab cTG

ab

c

p= a

r= ab cq= b

But How?

VV V+

a b

aa

b

c ab cPG

ab

c

p= a

r= ab cq= a b


TGab

c

p= aq= b

r= ab c

INPUT OUTPUT

a b r

0 0 c

0 1 c

1 0 c

1 1 c’


V

1

1c V r= c’

INPUT OUTPUT

a b r

0 0 c

0 1 c

1 0 c

1 1 c’


V

1

0c V r= cV+

INPUT OUTPUT

a b r

0 0 c

0 1 c

1 0 c

1 1 c’

V

0

1c V r= cV+

INPUT OUTPUT

a b r

0 0 c

0 1 c

1 0 c

1 1 c’


TGAB

C

P=A

R=AB C

Q=B

Input Output

A B R

0 0 C

0 1 C

1 0 C

1 1 C’

V

a

bc V V+ r=ab c

a

a b

(a)V

a

bc V V+ r= ab c

p= a

q= b

(b)

Have anything wrong?


TGAB

C

P=A

R=AB C

Q=B

Input Output

A B R

0 0 C

0 1 C

1 0 C

1 1 C’

V

A

B

C V V+ R=AB C

P=A

Q=B

Ok

Quantum Cost of Toffoli Gate (cont…)

Alternate representation of Quantum circuit of TG…

b

aa

b

c ab c

TGab

c

p= a

r= ab cq= b

V

a

bc V V+ r= ab c

p= a

q= b

Quantum Cost of Fredkin Gate

V

a

b

c

a

V V+

a’b ac

a’c ab

a

bc

p= a

r=a’c ab

q= a’b acFRG

But How?

V

abc

a

V V+ a’c aba b c

a

bc

p= a

r=a’c ab

q= a b cMUX

Quantum Cost of Fredkin Gate (cont…)

A

BC

P=A

R=A’C AB

Q=A’B ACFRG

P=A

R=AC AB C

Q=AB AC BP=A

R=A(B C) C

Q=A(B C) B


A

BC

FRGP=A

R=A(B C) C

Q=A(B C) B

A

BC

A(B C)C


A

BC

FRGP=A

R=A(B C) C

Q=A(B C) B

A

BC

A

A(B C) C(B C)


a

bc

p= a

r=a’c ab

q= a’b acFRGa

bc

aa(b c) ba(b c) c


V

a

b

c

a

V V+

a’b ac

a’c ab

a

bc

a

a’b aca’c ab


V

a

b

c

a

V V+

a’b ac

a’c ab

V

a

b

c

a

V V+

a’b ac

a’c ab

Quantum Cost of Peres Gate

VV V+

A B

AA

B

C AB C

PG, Cost = 4

PGA

BC

P=AQ=A B

R=AB C

Quantum Cost of NFT Gate

NFT, Cost = 6

VV

A

B

C

V+ AC’ B’C

AC’ BC

A B

NFTA

BC

Q=AC’ B’CR=AC’ BC

P=A B

Quantum Cost of NFT Gate

a

b

c

a bac’ b’cac’ bc

a

bc

p= a b

r=ac’ bc

q= ac’ b’cNFT

V V+V

a

bc

a b

ac’ b’c

ac’ bc

Quantum Cost of MIG Gate

DMIG, Cost = 7

ABC

Peres Gate

A

AB CAB’ D

A B

ABC

P = A

D

Q = A BR = AB CS = AB’ D

MIG

Assignment

D

ABC

P = AQ = A’C’ B’R = A’C’ B’ DS = ( A’C’ B’)D AB C

TSG

Find out cost

D

ABC

P = AQ = R = A B CS = (A B)C AB D

MTSG A B

XAB

C

P=A

R=AB CQ=A’B’ C

About AuthorSajib Kumar Mitra is an MS student of Dept. of Computer Science and Engineering, University of Dhaka, Dhaka, Bangladesh. His research interests include Electronics, Digital Circuit Design, Logic Design, and Reversible Logic Synthesis.

THANKS TO ALL

quantum cost calculation of reversible circuit

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