effect of bdd optimization on synthesis of reversible and ...€¦ · – low-power design...
TRANSCRIPT
![Page 1: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/1.jpg)
Effect of BDD Optimization Effect of BDD Optimization on Synthesis of Reversible
d Q L iand Quantum Logic
Robert Wille, Rolf Drechsler,
Institute of Computer Science University of Bremen, GermanyUniversity of Bremen, Germany{rwille,drechsle}@informatik.uni-bremen.de
![Page 2: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/2.jpg)
Outline• Motivation and Background• Motivation and Background
• BDD-based Synthesis
• Exploiting BDD-optimization
– Shared Nodes
– Complement EdgesComplement Edges
– Reordering
• Experimental Results
Conclusions
2
• Conclusions
![Page 3: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/3.jpg)
Reversible Logic• Applications in • Applications in
– Quantum ComputingLow Power Design– Low-Power Design
– Optical ComputingDNA C ti– DNA Computing
– …1
1
1
1
Toffoli gate
1 0
3
Toffoli gate
![Page 4: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/4.jpg)
Quantum Logic
• Is inherently reversible• Signals represented by qubits• Signals represented by qubits
(i.e. non-Boolean values)• Value of each qubit is restricted to 0 1 V or V
NOT P f i i
• Value of each qubit is restricted to 0, 1, V0 or, V1
1
0
• NOT: Peforms inversion• CNOT: controled inversion• V: ‘square root’ of NOT
11
10
1V
11
1V0 • V: square root of NOT
• V+: inverse of V VV1 0 V1
V+1 V0
4
![Page 5: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/5.jpg)
Synthesis ProblemGi R f ti t T k Fi d t k • Given: Rev. function to be synthesized
• Task: Find network (i.e. a cascade of gates)
• Previous Work:No fanouts, no feedback
• Often rely on truth table (or similar) descriptionOnly applicable to small functions
5
![Page 6: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/6.jpg)
Outline• Motivation and Background• Motivation and Background
• BDD-based Synthesis
• Exploiting BDD-optimization
– Shared Nodes
– Complement EdgesComplement Edges
– Reordering
• Experimental Results
Conclusions
6
• Conclusions
![Page 7: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/7.jpg)
Binary Decision Diagrams (BDDs)
• Data structure for efficient representation and manipulation of Boolean functions
• Rooted, directed, acyclic graph, which consists of g ap , o s s s odecision nodes and two terminal nodes (leafs)
• Each decision node is labeled by a Boolean variable and has two child nodes (low and high)
7
nodes (low and high)
![Page 8: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/8.jpg)
BDD-based Synthesis #1
1. Build BDD for function f using existing techniques2. Substitute each BDD node by a cascade of gates
8
![Page 9: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/9.jpg)
BDD-based Synthesis #2
9
![Page 10: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/10.jpg)
Example (XOR function)
10
![Page 11: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/11.jpg)
BDD-based Synthesis #3
• Linear worst case behavior regarding run-time g gand space requirements
• Resulting circuits are bounded by BDD size
BDD optimization can be exploited
11
![Page 12: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/12.jpg)
Outline• Motivation and Background• Motivation and Background
• BDD-based Synthesis
• Exploiting BDD-optimization
– Shared Nodes
– Complement EdgesComplement Edges
– Reordering
• Experimental Results
Conclusions
12
• Conclusions
![Page 13: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/13.jpg)
Shared Nodes• Used to represent a sub formula more than once• Used to represent a sub-formula more than once• Need to preserve node values
(requires additional line)(requires additional line)
13
![Page 14: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/14.jpg)
Complement Edges• Allows to represent a function as well as its p
negation by a single node only
14
![Page 15: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/15.jpg)
Reordering
Can be directly yapplied (no further (no further adjustments)
15
![Page 16: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/16.jpg)
Outline• Motivation and Background• Motivation and Background
• BDD-based Synthesis
• Exploiting BDD-optimization
– Shared Nodes
– Complement EdgesComplement Edges
– Reordering
• Experimental Results
Conclusions
16
• Conclusions
![Page 17: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/17.jpg)
Experimental Setup
• Implemented on the top of CUDD • Benchmarks from RevLib (www revlib org) and • Benchmarks from RevLib (www.revlib.org) and
LGSynth package
• Objectives: – Circuit linesCircuit lines– Number of Toffoli gates– Quantum Cost Quantum Cost – Run-time (often negligible)
17
![Page 18: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/18.jpg)
Results (selected)
18
![Page 19: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/19.jpg)
RMRLS G t t l @ TCAD 2006
Comparison to Previous Work
• RMRLS: Gupta et al. @ TCAD, 2006• RMS: Maslov et al. @ TODAES, 2007
• Significant run-time for both RMRLS and RMS• Most of the functions aborted after 500 CPU
19
seconds
![Page 20: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/20.jpg)
Outline• Motivation and Background• Motivation and Background
• BDD-based Synthesis
• Exploiting BDD-optimization
– Shared Nodes
– Complement EdgesComplement Edges
– Reordering
• Experimental Results
Conclusions
20
• Conclusions
![Page 21: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/21.jpg)
Conclusions• BDD-based synthesis has been introducedy
• Effect of BDD optimizationEffect of BDD optimization– Shared Nodes: Always yields better results– Compl. Edges: Better results in most casesCompl. Edges: Better results in most cases– Orderings: Best results with exact ordering,
but Sifting also yields good circuitsg y g
• Comparison to Previous Work: Comparison to Previous Work: – Larger functions can be handled– Significant improvements in quantum cost
21
Significant improvements in quantum cost– More circuit lines needed
![Page 22: Effect of BDD Optimization on Synthesis of Reversible and ...€¦ · – Low-Power Design C–Onlgacomiiputpt – DNA C tiDNA Computing –… 1 1 1 1 Toffoli gate 1 0 3. Quantum](https://reader033.vdocuments.net/reader033/viewer/2022060518/604c17daab946b70ba2e4be2/html5/thumbnails/22.jpg)
Effect of BDD Optimization Effect of BDD Optimization on Synthesis of Reversible
d Q L iand Quantum Logic
Robert Wille, Rolf Drechsler,
Institute of Computer Science University of Bremen, GermanyUniversity of Bremen, Germany{rwille,drechsle}@informatik.uni-bremen.de