Reversible Gates in
various realization
technologies
Double Rail
Technologies
2.2. DOUBLE- RAIL LOGIC BASED ON NOT, SWAP
AND CONTROLLED SWAP GATES.
• The symbolic notation on the right shows my notation for
single-rail (symbolic) notation of double-rail circuits:
• a’ = (a1, a2’) = (1,0),
• a = (a1’, a2) = (0,1).
a1
a2
Double-rail Inverter
b1
=
a
b
b2
Fig. 2.2.1.: Realization of NOT in double-rail technology
Double-Rail Feynman Gate
• a’ = (a1, a2’) = (1,0),
• a = (a1’, a2) = (0,1).
a1
a
a
b
a b
a2
b1
b
b2
=
• Fig. 2.2.2: Realization of Controlled-Not (Feynman) gate in
double-rail logic.
Fredkin built from Toffoli and CNOT
(a)
(b)
(c)
• Fig. 2.2.3:
• Realization of Fredkin gate using Toffoli and CNOT gates.
• By surrounding the Toffoli gate (Fig. 2.2.3a) with CNOT gates (Fig. 2.2.3b) we
obtain the Fredkin gate (Fig. 2.2.3c).
• Similarly the Toffoli gate can be obtained by surrounding the Fredkin gate using
two CNOT gates.
• This way a Toffoli gate in double-rail logic can be realized, as shown in Fig. 2.2.4.
Toffoli built from Fredkin and
Feynman
0
1
control
0
1
0
1
• Fig. 2.2.4: Realization of Toffoli gate in double-rail
logic using a double-rail Fredkin gate surrounded by
two Controlled NOT gates.
Double-rail Fredkin gate
a1
a2
P1
P2
Q1
b1
b2
c1
c2
b1
Q2
R1
R2
• Fig. 2.2.5.: Detailed circuit of double-rail Fredkin gate
realized with two standard Fredkin gates with the same
control.
Single-rail schematics for double-rail
Fredkin gate
a
P
2
b
c
2
Q
R
• Fig. 2.2.6: The gate from Figure 2.2.5 in another
notation, every signal corresponds now to two wires
from Figure 2.2.5.
New double-rail gates can be created
with single-rail Fredkin gates
(c1,c2) = (a1,a2)
and
(d1,d2) = (not b1, not b2).
d2
a1
a2
b1
b2
P2
c1
c2
d1
d2
• Fig. 2.2.7: Detailed circuit of double-rail “new gate”
realized with three standard Fredkin gates.
Toffoli gate in double-rail
technology
a1
a2
a
P1
P
P2
=
b1
b2
Q1
Q2
b
Q
• Fig. 2.2.8: Realization of CNOT in double-rail technology.
Signal a is represented by a1=0, a2 = 1. Signal a’ (negation
of a) is represented by a1=1, a2 =0.
SWAP in double-rail
a1
c1
a2
c2
b1
d1
b2
d2
=
a
c
b
d
• Fig. 2.2.9: Realization of SWAP in double-rail
technology.
Conclusion on Double-Rail.
• Single-rail SWAP = double-rail NOT
• Basic gates are single-rail SWAP and
controlled-SWAP
• NOT gate can be created.
• Toffoli, Feynman and Fredkin gates can be
created.
• Complex gates can be created.
• Gates are not necessarily conservative.
2.3. DOUBLE-RAIL OPEN/CLOSE
SWITCH REVERSIBLE CMOS LOGIC
OF ALEXIS DE VOS
c
P
c
Q
a
b
a
a
Q
b
R
b
Q
• Fig. 2.3.1: Realization of
Fredkin gate. Notation
for De Vos double-rail
technology from Figure
2.3.2.
c
• Figure 2.3.2. Realization of
function Q in Fredkin gate using
De Vos technology.
Toffoli in De Vos Technology
c
a
a
b
a
b
a
p
p
b
b
c
• Fig. 2.3.3:
• Realization of Toffoli gate in double-rail DeVos technology.
• As we see, this is a kind of double-rail technology.
Toffoli Gate.
• The function realized in Figure 2.3.3 for output p is c’ ab in upper
branch controlled by c’ and c (a’ + b’) controlled by c in the right
branch.
• Thus the function realized for p is the following:
• p = c’ ab + c (a’ + b’) = c’ (ab) + c (ab)’ = ab  c
• Similarly, the function realized for negation of p, denoted by p’ is c’
(a’ + b’) (controlled by c’) and (ab) c controlled by c. Thus the
function for negated p, denoted by p’ is the following:
• p’ = c’ (a’ + b’) + (ab) c = c’ (ab)’ + (ab) c = [ ab  c ]’ = p’ (negation
of signal p).
CNOT gate in De Vos technology
c
Q
a
a
a
a
Q
c
• Figure 2.3.4. Realization of CNOT in De Vos technology.
Regular layout for De Vos technology
Toffoli
C
P
C
P
a
a
b
b
a
a
b
b
• Fig. 2.3.5: DeVos CMOS circuit layout for pass-transistor
diagram from Figure 2.3.3.
Conclusion on De Vos technology.
•
•
•
•
•
•
•
•
•
•
•
•
•
Based on open/close switches.
This is double-rail technology
Parallel and serial connections of switches
Every output of gate has a form:
Variable  Function (other variables)
Single-rail SWAP = double-rail NOT
NOT gate can be created.
Toffoli, Feynman and Fredkin gates can be created.
Complex gates can be created.
Gates are not necessarily conservative.
Creating complex gates is simple.
Realized in CMOS, input-output physically reversible, not only logically
reversible
Requires special power supplies to reduce power
Combined with the idea of adiabatic gate.
2.4. CONSERVATIVE
REVERSIBLE LOGIC BASED
ON 2*3 SWITCHES
a
a
b
ab
ab
a
aba b
CONSERVATIVE REVERSIBLE LOGIC
BASED ON 2*3 SWITCHES
a
a
b
a
ab
ab a b
ab
a
b
a
ab
a
aba b
ab
• Fig. 2.4.1: Realization of CNOT in double-rail switchbased technology
2*3 Switch technologies
ac
c
ac a b
ac
a
b
ab
a
a cab
ab
• Fig. 2.4.2: Realization of Fredkin gate in singlerail switch-based technology.
New Synthesis method using 2*3
switches
•
First Stage:
–
–
–
–
–
–
•
Second Stage:
–
–
–
–
•
When we have the set of products available, next we can make sums of all possible pairs of disjoint products.
Disjoint are products that share at least one variable that is in various polarities in them.
This must be however done always in such a way that finally the number of primary outputs of the circuit
that we create will be the same as the number of primary inputs.
We check that each function being a primary output must be balanced.
Third Stage:
–
•
First we create a sequence of products of literals with more and more literals successively.
We do this in all possible ways.
This is illustrated in the left part of the upper row of gates in Figure 2.4.3 below.
This way, functions a, a’b, ab, a’c and ac are created.
For gates of smaller size the number of such products is quite limited so my method is rather efficient.
We cannot create fan-outs of any signals, including the primary inputs.
Finally we check that the k*k function created by us is reversible.
Comments:
–
–
–
–
Based on a simple heuristic:
This way we always create a k*k gate
(by k*k gate we understand a reversible function with k inputs and k outputs).
The above algorithm can be implemented as a tree search algorithm.
a
a
a
a'b
a'b+ac
b
ab
c
c
a
c
c'a
ca
c'a+cb
New gate in
2*3 switch
technology
b
b
b
c
b'c
bc
b'c+ba
• Fig. 2.4.3: Using Switch gates to build a new
reversible conservative gate.
Fig. 2.4.4: Schematics of the new gate based on the
schematics from Figure 2.4.3.
This gate can realize products, inhibitions, sums and
implication functions on its three outputs.
a
a’b + ac
b
c’a + cb
c
b’c + ba
Methodology to create new gates
using 2*3 switch technology
a
a
b
c
ab
ab
a
ac
ab + a c
ac
a
a b + ac
• Fig. 2.4.5: Fredkin gate with internal swap built from
switch gates
Hierarchical design from 2*3
switches
a
b
P
Q
c
R
d
S
• Figure 2.4.6. Schematic of a new single-rail reversible gate
using Fredkin gates built from 2*3 switches.
P
Q
exor
R
a
b
c
d
New design
ideas
S
• Figure 2.4.7. Another way of drawing the circuit from
Figure 2.4.6 to emphasize its similarity to generalized
gates introduced in Chapter 3.
• Both muxes are controlled by the Exor.
• This circuit, in contrast to circuits in other figures is
drawn from right to left rather than from left to right to
emphasize its reversible nature and to help using the
analysis from Figure 2.4.8.
New design ideas
a u  a z  a (b x  b y )  a (b x  b y )
a
a b xa b yabxab y
b
 ( a  b) x  ( a  b) y
x
y
C
• Fig. 2.4.9: Analysis of
circuit from Fig.2.4.8.
D
U b x b y
Z b x  b y
• Fig. 2.4.8:
• Another notation for the circuit from two Fredkin gates given in
Figure 2.4.6.
• Assume x=c, y=d.
CNOT gate in De Vos technology
c
Q
a
a
a
a
Q
c
a
ac
ac+ a’ c’
c
ac
a
c’
c’a’
a
a c’+ a’ c
c’ a
• Figure 2.3.4. Realization of CNOT in De Vos technology.
Conclusion on 2*3 switch
technology.
• Based on 2*3 switches.
• Realized in optical, fluidic, CMOS and other nano.
• This is not double-rail technology, but double-rail
gates can be created on a higher level.
• NOT gate cannot be directly created.
• Toffoli, Feynman and Fredkin gates can be created.
• Complex gates can be created.
• Gates are not necessarily conservative.
• Creating complex gates is not simple.
• Can create Fredkin, thus all gates from previous
section can be created.
Concluding on 2*3 switches (cont)
• Every conservative reversible gate can be built from 2*3, 3*2
switches and (hierarchically) from k*k conservative gates built
in turn from 2*3 and 3*2 switches.
• Next, by using double-rail technology, every reversible gate can
be built from conservative gates.
• Thus, every circuit discussed in chapters 3 - 6 can on the
bottom of hierarchy be built from only 2*3 and 3*2 switches.
•
The number of such switches is the cost function of the
solution which is used to compare various designs one with
another.
2.5. MAJORITY GATES IN
QUANTUM CELLULAR
AUTOMATA
a
a
b
(ab  a c )  (b  c)  b a  ca  a c  b  ab
c
a(b c) c  ab  ac c  ab  a c
• Fig. 2.5.1: Fredkin gate from Toffoli and CNOT gates with analysis equations.
• The same figure explains also how Toffoli can be created from Fredkin.
• Surrounding with two more Feynmans on left and right and next cancelling
pairs of successive Feynmans would restore the original Toffoli gate.
Miller gate as three majorities is
base of reversible design in
quantum dot cellular automata
a
b
c
maj (a, b , c )
Miller gate
maj ( a, b, c)
maj (a, b, c)
• Fig. 2.5.2: Miller gate is three majorities with various
polarities.
ac
a
(a b  a c  b c)  (a  c)  a b  a c  b c
b
(ab ac bc) (b c) ba c a bc  m(a,b,c)
c
m (a, b, c)
bc
(a  c) (b  c)  c  ab acbc c  c  m(a, b, c)
• Fig 2.5.3: Analysis of Miller’s gate.
• Obviously, by surrounding Miller with two Feynmans with
EXOR up in first bit we will obtain Fredkin gate,
• and next by surrounding again with Feynmans with EXOR up
in second bit we get Toffoli gate.
Miller
Miller
Miller gate
plus inverters
is universal
Miller
Miller
• Fig. 2.5.4: Universality of
Miller gate. Miller with
inverters is universal.
Realization of EXOR needs
ancilla bits
a
b
maj
1
a
A
b
maj
0
B
0
maj
ab
C
Fig 2.5.5: Realization of EXOR using 3
majorities.
Adaptation to reversible design
a
0
1
b
maj
maj
B
maj
ab
C
A
• Fig. 2.5.6: The circuit from Fig.2.5.5 redrawn to a notation
that is more similar to Quantum Array notation.
New gate from Miller gates with no ancilla
bits realizes a new Feynman-like gate
c
ab  ac  b c
a
c
c  a b
ab
b
a
c
( a b  a c  b c)
ba  ca  bc
ac
• Figure 2.5.7. This figure demonstrates that by taking two Miller
gates we can create new reversible functions which are not the
Miller gate with other polarities.
a
b
c
c
a b  c
a
This gate becomes
Feynman gate with c=0
Concluding on Quantum Dot Logic
•
•
•
•
•
•
Basic gates are NOT and majority.
SWAP is cheap.
AND and OR gates can be created with ancilla from majority.
Miller gate is three majorities with various polarities.
Other functions can be created using majorities.
Other functions can be created using Miller and NOT gates
• I believe that I will be able to create many useful gates in various families
based on:
–
–
–
–
Majorities only
Majorities and SWAP
Majorities and NOT
Majorities, NOT and SWAP
• Toffoli, Feynman and Fredkin can be created but are not necessarily the
best way to create big reversible circuits in quantum dot QCA.
Concluding on technologies
•
Concluding. This section showed several ways of realizing reversible gates using
quantum dot based QCAs. All methods from next chapters can be thus used to
synthesize circuits from quantum dots, but I have doubt how cost efficient these
methods will be when compared to the group-theory methods developed by Dr.
Guowu Yang and Prof. Perkowski for the Miller gate. On the other hand, the exact
method of Yang et al is only for 3*3 circuits and my methods work for larger
circuits. The tasks of designing best reversible gates on top of QCA and using such
gates in synthesis are relatively separated and I can work only on one of them, as
shown in next chapters. There is no method to design reversible circuits directly
from majorities. Such method would be very useful for the recent breakthrough
technology of quantum dots. The only paper in the literature is from Yang et al
which has a theoretical analysis and some designs but no heuristic algorithm for
practical design of larger circuits. Another open research topic. I am not
committing myself to solve all these problems. I just want to show here that in
addition to problems that I already solved and I just need to write better software
there are several related tasks for which methods and algorithms should be
created. I will work possible on these topics as well, in addition to my main topic of
implementing software.
Conclusion on all these
technologies.
•
•
•
•
•
•
•
•
•
•
•
•
•
Based on some type of switches.
Realized in optical, fluidic, CMOS and other nano.
Similar to quantum gates.
Double-rail technology often used.
Quadruple-rail technology can be created for adiabatic CMOS.
Universal logic systems.
Some natural some not.
NOT gate can be usually but not always created.
Toffoli, Feynman and Fredkin gates can be always created.
Complex gates can be created.
Gates are not necessarily conservative.
Creating complex gates is simple or not simple, depending on technology.
Very little is published on synthesis of such gates from some type of
specification.
• Many open research problems, in physics and in synthesis methods.
What to remember?
1.
2.
3.
4.
5.
Y gate
Single-rail circuits with Y gates.
Double-rail circuits with Y gates.
What are advantages of double-rail circuits.
Majority gate in Quantum Dot Cellular
Automata.
6. How to use Majority to build Three-input EXOR?
7. Relation between reversible Miller Gate and
Majority – how it can be used in QCA circuits?