Minimization Techniques
for
Reversible Logic Synthesis
Outline
•
•
•
•
•
•
•
Why reversible logic?
The building blocks
The synthesis problem
Some solutions
Optimization
Finding identities
Remaining problems
Reversible Logic Synthesis
Slide 2
Reversible Logic
inputs
circuit
network
outputs
• from the output you can determine the input (bijection)
• Why would we want this?
• Landauer's principle: every bit of information lost
consumes energy
kT ln2
• k is the Boltzman constant
• T is the temperature
21 Synthesis
Logic
2.9  10
Joule
• energy loss is small Reversible
Slide 3
Applications
• Quantum Computing
– necessarily reversible
• Low power CMOS
– In adiabatic circuits, current is restricted to flow
across devices with low voltage drop and the
energy stored on their capacitors is recycled
• Optical Computing
• Nano-technologies
– Billiard Ball Model (BBM)
Reversible Logic Synthesis
Slide 4
Standard Gates
• which ones are reversible?
and
not
Embedding a
non-reversible
function into a
reversible one
How many ways?
xor
or
Reversible Logic Synthesis
Slide 5
Reversible Gates
• Feynman gate (controlled not)
x
x
y
x y
• Toffoli gate
x
x
y
y
z
xy  z
Synthesis
• Generalized ToffoliReversible
gateLogic
more control lines
Slide 6
Reversible Gates
• Fredkin gate (controlled swap)
x
x
y
yx’ + zx
z
zx’ + yx
• Generalized Fredkin gate  more control lines
Reversible Logic Synthesis
Slide 7
Restrictions
• No fan-out
• No back-feeds
Reversible Logic Synthesis
Slide 8
Restrictions
• No fan-out
• No back-feeds
Cascade of Gates
inputs
g1
g2
… gi gi+1 … gn-1 gn
Reversible Logic Synthesis
output
Slide 9
Function Representation
• How do we represent a reversible function?
– truth table
– BDDs
• Is there an easy check to see that the function is
reversible?
Reversible Logic Synthesis
Slide 10
Synthesis
• Given a reversible function find a network of gates
that realize the function
• Cost should be near minimal
• Possible cost assumptions:
– Each gate has the same cost
– The cost of each gate reflects its actual
implementation cost
Reversible Logic Synthesis
Slide 11
Approach 1
inputs
Reversible
transformation
Reversible Logic Synthesis
outputs
Slide 12
Approach 1
Reversible
Transformation
T1
inputs
inputs
G
A
T
E
Reversible
Transformation
T2
Reversible Logic Synthesis
outputs
outputs
Slide 13
Conditions
• T2 should be “simpler” than T1
• How do you measure simplicity?
– Hamming distance
– Spectral techniques
• Questions:
– Will it converge?
– How good is the result?
• Improvements
– Look ahead
– Backtracking
Reversible Logic Synthesis
Slide 14
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
Reversible Logic Synthesis
Slide 15
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
These are correct
Reversible Logic Synthesis
Slide 16
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
These are correct
Step 1: move up
100 to 110
Reversible Logic Synthesis
Slide 17
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
These are correct
Step 1: move up
100 to 110
a
a
b
b
c
c
Reversible Logic Synthesis
Slide 18
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
These are correct
Step 1: move up
100 to 110
a
a
b
b
c
c
Reversible Logic Synthesis
Slide 19
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
100
011
111
101
010
110
These are correct
Step 1: move up
100 to 110
a
a
b
b
c
c
Reversible Logic Synthesis
Input
000
001
010
011
100
101
110
111
Output
000
001
110
011
101
111
010
100
Slide 20
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
110
011
101
111
010
100
These are correct
Step 2: move down
110 to 010
Reversible Logic Synthesis
Slide 21
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
110
011
101
111
010
100
These are correct
Step 2: move down
110 to 010
a
a
b
b
c
c
Reversible Logic Synthesis
Slide 22
Approach 2
Input
000
001
010
110
100
101
110
111
Output
000
001
110
011
101
111
010
100
These are correct
Step 2: move down
110 to 010
a
a
b
b
c
c
Reversible Logic Synthesis
Input
000
001
010
011
100
101
110
111
Output
000
001
010
111
101
011
110
100
Slide 23
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
110
011
101
111
010
100
Step 2: move down
110 to 010
a
a
b
b
c
c
Reversible Logic Synthesis
Input
000
001
010
011
100
101
110
111
Output
000
001
010
111
101
011
110
100
Slide 24
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
010
111
101
011
110
100
Reversible Logic Synthesis
Slide 25
Approach 2
Input
000
001
010
011
100
101
110
111
Output
000
001
010
111
101
011
110
100
c
c
b
b
a
a
Reversible Logic Synthesis
Slide 26
Advantages/Disadvantages
•
•
•
•
•
•
Always converges
Fast
No look ahead
Several optimizations are possible
May not be optimal
Worst case?
Reversible Logic Synthesis
Slide 27
Approach 3
• Local transformation
• Replace a sequence of gates with another
• For example replace 3 gates with 2
Reversible Logic Synthesis
Slide 28
Reversible Logic Synthesis
Slide 29
Approach 3
• Use Identity circuits for local transformations
inputs
g1
g2
… gi gi+1 … gn-1 gn
Reversible Logic Synthesis
identity
Slide 30
Approach 3
inputs
g1
g2
… gi gi+1 … gn-1 gn
identity
F-1(X)
F(X)
Reversible Logic Synthesis
Slide 31
Approach 3
inputs
g1
g2
… gi gi+1 … gn-1 gn
identity
F-1(X)
F(X)
Replace this sequence
of gates
Reversible Logic Synthesis
Slide 32
Approach 3
inputs
g1
g2
… gi gi+1 … gn-1 gn
identity
F-1(X)
F(X)
with this sequence
of gates
Replace this sequence
of gates
Reversible Logic Synthesis
Slide 33
Template Description
• As a class - an identity that is not reducible by another
template
• A template may be rotated
• A template may be applied in reverse order
• Size 2 ==> Duplicate gates (may be deleted)
• Size 3 ==> There are none
• Size 4 ==> Passing rule
Reversible Logic Synthesis
Slide 34
Template Class — Size 5
A line may
be
duplicated
A line may
be
removed
Reversible Logic Synthesis
Slide 35
How many classes of
identities are there?
Size
Identities
Classes
1
0
2
1
3
0
4
2
5
1
6
3
7
2
8
16
9
56
Reversible Logic Synthesis
Slide 36
The Templates: Application
Reversible Logic Synthesis
Slide 37
The quest for identities
• Exhaustive searches are not feasible
• A feasible approach (to find size n identities):
– Find all functions of size n/2
• With limited number of lines
• With some canonical order
– Pair each function with its inverse
– If no reduction is possible ==> we found a new
template
Reversible Logic Synthesis
Slide 38
A new approach
• Start with an identity that has target lines only
• For any subset of gates that preserve the identity a
new control line may be added
• Example:
Reversible Logic Synthesis
Slide 39
Recent Advances
• Automatic template encoding
• Application of iterative minimization heuristics
• Progress in calculating “real quantum cost”
– Using gates NOT, CNOT, V, and V+
– Quantum templates
• Using SAT to find exact results
Reversible Logic Synthesis
Slide 40
Future Work
• Handling don’t cares (can be done with SAT)
• Billiard ball implementation of reversible circuits
(via cellular arrays)
• Better representation for reversible functions
– Truth table is not adequate
– Some form of BDD (possibly Davio)
• Minimization with aid of group theory
Reversible Logic Synthesis
Slide 41