Introduction to Reversible Ckts

Igor Markov

University of Michigan

Electrical Engineering & Computer Science

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

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

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

00

f

??

argument

result

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

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

NOT gate

CNOT gate

Toffoli gate

zxy

y

xx

y

zx

y

xyx

A Reversible Circuit and Truth Table

x y z x y z

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 1

0 1 1 0 1 0

1 0 0 1 0 0

1 0 1 1 0 1

1 1 0 1 1 1

1 1 1 1 1 0

Equiv. to a CNOT gate Proof by exhaustive

simulation Proof by symbolic

arguments

zxy

yX’ x’ x

zxy x’y=z y

x

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

Identities for Reversible Ckts

Temporary Storage / Garbage Bits

How Much Temporary StorageDo We Need ?

Toffoli (MIT TR, 1980) Odd permutations require

at 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)

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

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 ?

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

New Motivation: Quantum Ckts Not related to low power Quantum circuits operate

on 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 11 0

Classical Versus Quantum Ckts Circuit identities for conventional

reversible gates (e.g., CNOT and Toffoli)do not change in the quantum context Conventional techniques applicable

when there are no purely quantum gates “Conventional subroutines” of q. programs

Purely quantum gates required in apps Open problem: synthesis with

purely quantum gates

Thank you!