Introduction to Reversible Ckts

1. Introduction to Reversible Ckts Igor Markov University of Michigan Electrical Engineering & Computer Science

2. Outline • Historical motivation • Arbitrary computations via reversible • Rev. ckts: basic definitions & examples • Recent implementations in CMOS • Reversible synthesis & other EDA tasks • Novel motivations for reversible circuits • Inherently reversible computations • Quantum circuits

3. Historical Motivation • Every lost bit causes an energy loss • C. Bennett, 1973, IBM J. of R & D • ~ the kinetic energy of one molecule in air • Idea: try to avoid those energy costs • Adiabatic circuits • Asymptotically energy lossless (Time  ∞ ) • S. Younis and T. Knight, 1994,Workshop on Low Power Design

4. Implementing Arbitrary Computations via Reversible • Toffoli 1980, Theorem 4.1:Any finite function can be writtenas a product of • trivial encoder • bijection f • trivial decoder • Constructiveprocedure • Adds variables 0 0  f  ? ? result argument

5. Definitions • Reversible bit-based computation(e.g., Toffoli 1980) • N bits at input • N bits at output • Every input & output bit-string possible • Bijection • These restrictions apply to gates & ckts • Additional restriction: no fanout • Acyclic comb. circuits interesting enough

6. NOT gate Examples • k-CNOT gate, a.k.a. generalized Toffoli • (k+1)-inputs and (k+1)-outputs • Values on the first k inputs are unchanged • Last input is negated iff the first k are all 1s “CNT” gate library Toffoli gate x x CNOT gate y y x x z zxy yx y

7. A Reversible Circuit and Truth Table x X’ x’ x y zxyx’y=zy zxy • Equiv. to a CNOT gate • Proof by exhaustivesimulation • Proof by symbolic arguments

8. Implementations in CMOS • B. Desoete and A. De Vos“A reversible carry-look-aheadadder using control gates”,Integration, the VLSI Journal,vol. 33 (2002),pp. 89-104 • Reversible 4-bit adder • 384 transistors • no power rails

9. Identities for Reversible Ckts

10. Temporary Storage / Garbage Bits

11. How Much Temporary StorageDo We Need ? • Toffoli (MIT TR, 1980) • Odd permutations requireat least 1 line of temporary storage • Shende et al., ICCAD `02 • Even permutations need no temp storage • Odd permutations need 1 line and no more • Constructive synthesis procedure (not implemented)

12. Comb. Synthesis Formulations • Straightforward • Given a full truth table, find a circuit • Shende et al. show an optimal procedure(all 3-line circuits synthesized in mins) • With don’t cares • The function of one output bit is restricted • Iwama et al. (DAC `02): heuristic,transformation-based synthesis,may use many lines of temp. storage

13. Other EDA Tasks • Fault testing in reversible circuits • K. Patel et al. (VTS `02): reversible circuits require very few test vectors • Equivalence checking • Difficulties with empirical validation • Circuit / gate costs ? • Circuit benchmarks ?

14. New Motivation:Inherently Reversible Applications • Information is re-coded,but none is lost or added • Digital signal processing • Cryptography • Communications • Computer graphics • Micro-processor instructions for • Bit-permutations • Butterfly operation from FFT

15. New Motivation: Quantum Ckts • Not related to low power • Quantum circuits operateon linear combinations of bit-strings • E.g., (|0>+|1>)/2, (|00>+i|11>)/2 • Linear: are expressed by matrices • Reversibility implied by quantum mechanics • A conventional reversible gate,can be extended by linearity,e.g., a quantum inverter is just 0 1 1 0

16. Classical Versus Quantum Ckts • Circuit identities for conventional reversible gates (e.g., CNOT and Toffoli)do not change in the quantum context • Conventional techniques applicablewhen there are no purely quantum gates • “Conventional subroutines” of q. programs • Purely quantum gates required in apps • Open problem: synthesis withpurely quantum gates

17. Thank you!