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Power/Performance/Cost Efficiency ofAdiabatic

Circuits, as a function ofDevice On/Off Power

Ratios

- Michael P. FrankCISE Department / ECE

Dept.Brown Bag Seminar - Tue., Mar. 26

(No Transcript)

Source ITRS 99

Across Multiple Technologies

Vacuum Tubes

IntegratedCircuits

Mechanical

DiscreteTransistors

ElectromechanicalRelays

Source Kurzweil, The Age of Spiritual Machines,

pp. 22-25

½CV2 based on ITRS 99 figures for Vdd and

minimum transistor gate capacitance. T300 K

Information Entropy

1 2 3

Example System with 3 two-state subsystems,such

as quantum spins.

Ruled outby someknowledge

Informational Spin label Status

1 Entropy 2 Known

information 3 Entropy

238 states

Illustrating Landauers principle

Before bit erasure

After bit erasure

State ofbit to beerased.

s0

0

0

s??0

State ofrest ofsystem(thermalmodes, c.)

Nstates

sN-1

s??N-1

0

0

Unitary(1-1)evolution

2Nstates

s?0

s??N

1

0

Nstates

s??2N-1

0

s?N-1

1

Conventional Gates are Irreversible

- Logic gate behavior (on receiving new input)
- Many-to-one transformation of local state!
- Required to dissipate bT by Landauer principle
- Incurs ½CV2 dissipation in 2 out of 4 cases.

Transformation of local state

Example Static CMOS Inverter

in

out

(No Transcript)

Adiabatic Charging in CMOS

Exact formula (if R const.)for frequency

factor f ? RC/t

Adiabaticity is Fundamental

- Adiabatic (dissipation ? quickness) processes

can occur in any type of system. - Cf. Adiabatic theorem of quantum mechanics.
- Specific adiabatic logics have been described for

many proposed future device technologies - Superconducting (Likharev 82, Averin et al. 01)
- Nanomechanical (Drexler 92, Merkle mid-90s)
- Quantum-dot (Lent Tougaw, mid-90s-present)
- Quantum computing implementations (inherently)
- Claim Work on architectures analysis for

adiabatic CMOS will still apply post-CMOS!

Adiabatic Rules for Transistors

- Rule 1 Never turn on a transistor if it has a

nonzero voltage across it! - I.e., between its source drain terminals.
- Why This erases info. causes ½CV2 disspation.
- Rule 2 Never apply a nonzero voltage across a

transistor even during any on?off transition! - Why When partially turned on, the transistor has

relatively low R, gets rel. high PV2/R

dissipation. - Corollary Never turn off a transistor when it

has a nonzero current going through it! - Why As R gradually increases, the VIR voltage

drop will build, and then rule 2 will be violated.

Adiabatic Rules continued

- Transistor Rule 3 Never apply a large voltage

across any on transistor. - Why So transition will be more reversible

dissipation will approach CV2(RC/t), not ½CV2. - Adiabatic rules for other components
- Diodes Dont use them at all!
- There is always a built-in voltage drop across

them! - Resistors Avoid moderate network resistances.
- e.g. stay away from range gt10 k? and lt1 M?
- Capacitors Minimize, reliability permitting.
- Note Adiabatic dissipation scales with C2!

Transistor Rules Summarized

Legal transitions in green. (For n- or

p-FETs.)Dissipative states and transitions in

red.

off

high

low

off

off

high

high

low

low

off

high

low

on

on

high

low

high

low

on

on

low

low

high

high

?

Transformation of local state

Simple Reversible CMOS Latch

- Uses a standard CMOS transmission gate
- Sequence of operation
- (1) input initially matches latch contents

(output), - (2) input changes?output changes, (3) latch

closes, (4) input removed.

b

a

Before Input Inputinput arrived removedin out

in out in outa a a a a a b b a b

P

in

out

b

a

Generic Frictional Coefficients

- Normal defs. of friction (coeff. of sliding

friction, viscosity, etc.) may not apply to all

processes. - For a given mechanism executing a specified

process (i.e., following a specified desired

trajectory or -ies) adiabatically over a time t - Energy coefficient cE ?Elostt ?Elost/q
- Energy dissipated from traj. per unit of

quickness - Note quickness q 1/t has units like Hz
- Entropy coefficient cS ?Smadet ?Smade/q
- New entropy generated per unit of quickness
- Note that cE cST at temperature T.

What matters!

Energy Coefficient in Electronics

- For charging capacitive load C by voltage V

through effective resistance R cE ?Elostt

(CV2RC/t)t C2V2R - If the resistances are voltage-controlled

switches with gain factor k controlled by the

same voltage V, then effective R ? 1/kV cE

C2V/k - In constant-field-scaled CMOS, k ? 1/hox ? ?, C ?

?, and V ? ?, so cE ? ?3/? ?4 ?Elost cE/t

? ?4/? ?3 (like CV2

energy)

Entropy coefficients of some reversible logic

gate operations

- From Frank 98, Ultimate theoretical models of

nanocomputers (Nanotechnology journal) - SCRL, circa 1997 1 b/Hz
- Optimistic reversible CMOS 10 b/kHz
- Merkles quantum FET 1.2 b/GHz
- Nanomechanical rod logic .07 b/GHz
- Superconducting PQ gate 25 b/THz
- Helical logic .01 b/THz

How low can you go? We dont really know!

Quantifying Leakage

- For a given structured system
- Leakage power Pleak dEleak / dt
- Spontaneous entropy generation rate Sleak

dSleak / dt - Again, note Pleak Sleak T at temperature T.

Minimum Losses w. Leakage

Etot Eadia Eleak

Eleak Pleaktr

Eadia cE / tr

Min. energy Roff/Ron ratio

- Note that cE C2V2Ron and if dominant leakage

is source/drain Pleak V2/Roff - So cEPleak C2V4/(Roff/Ron) Emin

2(cEPleak)1/2 2CV2(Roff/Ron)?1/2 - So Qmax ½CV2 / (2CV2(Roff/Ron)?1/2)

¼(Roff/Ron)1/2 ¼(Ion/Ioff)1/2

Clock/Power Supply Desiderata

- Requirements for an adiabatic timing signal /

power supply - Generate trapezoidal waveform with very flat

high/low regions - Flatness limits Q of logic.
- Waveform during transitions is ideally linear,
- But this does not affect maximum Q, only energy

coefficient. - Operate resonantly with logic, with high Q.
- Power supply Q will limit overall system Q
- Reasonable cost, compared to logic it powers.
- If possible, scale Q ? t (cycle time)
- Required to be considered an adiabatic mechanism.
- May conflict w. inductor scaling laws!
- At the least, Q should be high at leakage-limited

speed

(Ideally,independentof t.)

Supply concepts in my research

- Superpose several sinusoidal signals from

phase-synchronized oscillators at harmonics of

fundamental frequency - Weight these frequency components as per Fourier

transform of desired waveform - Create relatively high-L integrated inductors via

vertical, helical metal coils - Only thin oxide layers between turns
- Use mechanically oscillating, capacitive MEMS

structures in vacuo as high-Q (10k) oscillator - Use geometry to get desired wave shape directly

A MEMS Supply Concept

- Energy storedmechanically.
- Variable couplingstrength -gt customwave shape.
- Can reduce lossesthrough balancing,filtering.
- Issue How toadjust frequency?

Summary of Limiting Factors

- When considering adiabaticizing a system
- What fraction of system power is in logic? fL
- Vs. Displays, transmitters, propulsion.
- What fraction of logic is done adiabatically? fa
- Can be all, but w. cost-efficiency overheads.
- How large is the Ion/Ioff ratio of switches?
- Affects leakage minimum adiabatic energy.
- What is the Qsup of the resonant power supply?
- What is the relative cost of power logic? r
- E.g. decreasing power cost by r by increasingHW

cost by ? r will not help. Power premium

Minimizing cost/performance

- P Cost of power in original system
- H Cost of logic HW in original system
- P rH H P/r
- For cost-efficiency inverse to energy savings
- tot,min Pr-1/2 Hr1/2 2 Pr-1/2
- tot,orig P H (1r)H ((1r)/r) P
- tot,orig/tot,min ½(1r)r-1/2 ?

½r1/2 for large r

Summary of adiabatic limits

- Cost-effective adiabatic energy savings factor
- Sa Econv / Eadia in cost-effective adiabatic

system - Some rough upper bounds on Sa Sa ?

1/(1?fL) Sa ? 1/(1?fa) Sa ? ¼(Ion/Ioff)1/2

Sa ? Qsup Sa ? r1/2 - Discussion ignores benefits from adiabatics of

denser packing smaller communications delays in

parallel algorithms.

(worse than thesefor non-idealcomputations)

Motivation for this study

- We want to know how to carry out any arbitrary

computation in a way that is reversible to an

arbitrarily high degree. - Up to limits set by leakage, power supply, etc.
- We want to do this as efficiently as possible
- Using as few device ticks as possible

(spacetime) - Minimizes HW cost, leakage losses
- Using as few adiabatic transitions as possible

(ops) - Minimizes frictional losses
- But, a desired computation may be originally

specd in terms of irreversible primitives.

General-Case vs. Special-Case

- Wed like to know two kinds of things
- For arbitrary general-purpose computations,
- How to automatically emulate them in a fairly

efficient reversible way, - w/o needing new intelligent/creative design work

in each case? - For various specific computations of interest,
- What are the most efficient reversible

algorithms? - Or at least, the most efficient that we can find?
- Note These may not look anything like the most

efficient irreversible algorithms!

The Landauer embedding

- The obvious embedding of irreversible ops into

expanding reversible ones leads to a linear

increase in space through time. (Landauer 61) - Or, increase in width of an input-consuming

circuit

Expandingoperations(e.g., AND)

Desiredoutput

Garbagebits

input

Circuit depth, or time ?

Lecerf Reversal

- Lecerf (63) was interested in the group-theory

question of whether an iterated permutation of

items would eventually return to initial item. - Proved undecidable by reducing Turings halting

problem to this question, w. a reversible TM. - Reversible TM reverses direction instead of

halting. - Returns to initial state iff irreversible TM

would halt. - Only problemNo useful output data!

Desiredoutput

f

f ? 1

Garbage

Copy ofInput

Input

The Bennett Trick

- Bennett (73) pointed out that you could simply

fan-out (reversibly copy) the desired output

before reversing. - Note O(T) storage is still temporarily needed!

Desired output

f

f ? 1

Copy ofInput

Input

Garbage

Triangle Representation

- Represents any use of Bennett 73 embedding

State ofirrev. comp._at_ time ti?ti

Time in irreversiblesystem

AdiabaticProcess

?ti

Reversephase

Forwardphase

State ofirrev. comp._at_ time ti

Mass on anyvertical line space usage_at_ that

time

Time in reversiblesystem

Improving Spacetime Efficiency

- Bennett 73 transforms a computation taking

spacetime ST to one taking ?(ST2) in the worst

case. - Can we do better?
- Bennett 89 Described a technique that takes

spacetime - Actually, can generalize slightly and arrange for

exponent on T to be 1?, where ??0 (very slowly) - Lange, McKenzie, Tapp 97 Space ?(S) is

possible, if you use time ?(exp(?(S))) - Not any more spacetime-efficient than Bennett.

Pebble Game Representation

Triangle representation

k 2n 3

k 3n 2

Analysis of Bennett Algorithm

- n of recursive levels of algorithm
- k of lower-level iterations to go forward 1

higher-level step - Tr of reversible lowest-level steps

executed c(2k?1)n (c a small

constant, e.g. 2) - Ti of irreversible steps emulated kn
- So, n logk Ti, and so Tr c(2k?1)log Ti/log k

celog(2k?1)log(Ti)/log k cTilog(2k ?1)/log k

(n1 spikes)

E.g. k2 Tr 2Tilog(3)/log(2)

Cost-Efficiency Analysis

- Total cost of doing a computation includes
- Spacetime costs (storage used, integrated over

time) - Includes time-amortized manufacturing cost
- Includes cost of total energy leakage
- leakage from any in-use storage element
- Irreversibility costs (energy loss from irrev.

ops) - Total number of irreversible bit-erasures, CV2 gt

kT each. - Adiabatic costs (energy loss from reversible

ops.) - Proportional to number na of adiabatic ops

performed,times ce, divided by time top of a

single op.

Bennett 89 alg. is not optimal

k 2n 3

k 3n 2

Just look at all the spacetime it wastes!!!

Parallel Frank02 algorithm

- We can simply squish the triangles closer

together to eliminate the wasted spacetime! - Resulting algorithm is linear time for all n and

k and dominates Ben89 for time, spacetime,

energy!

k3n2

k2n3

Emulated time

k4n1

Real time

Setup for Analysis

- For energy-dominated limit,
- let cost equal energy.
- c energy coefficient, r r(min) leakage

power - i energy dissipation per irreversible

state-change - Let the on/off ratio Ron/off r(max)/r(min)

Pmax/Pmin. - Note that c ? itmin i (i / r(max)),

so r(max) ? i2/c - So Ron/off ? i2 / cr(min) i2 / cr

Time Taken

- There are n levels of recursion.
- Each multiplies the width of the base of the

triangle by k. - Lowest-level triangles take time ctop.
- Total time is thus ctopkn.

k4n1

Width 4 sub-units

Number of Adiabatic Ops

- Each triangle contains k (k ? 1) 2k ? 1

immediate sub-triangles. - There are n levels of recursion.
- Thus number of adiabatic ops is c(2k ? 1)n

k3n2

52 25little triangles(adiabaticoperations)

Spacetime Usage

- Each triangle includes the spacetime usage of all

k ? 1 of its subtriangles, - Plus,additional spacetime units, each

consisting of 1 storage unit, for time

topkn?1

k5n1

1 state of irrev. mach. Being stored

1

2

Time top kn-1

3

Resulting recurrence relationST(k,0) 1 (or

c)ST(k,n) (2k?1)ST(k,n?1) (k2?3k2)kn?1/2

123 units

Reversible Cost

- Adiabatic cost plus spacetime cost r a r

(2k-1)nc/t ST(k,n)rt - Minimizing over t gives r 2(2k-1)n

ST(k,n) c r1/2 - But, in energy-dominated limit, c r ? i2 /

Ron/off, - So r 2i (2k-1)n ST(k,n) / Ron/off1/2

Tot. Cost, Orig. Cost, Advantage

- Total cost i for irreversible operation

performed at end of algorithm, plus reversible

cost, gives tot i 1 2(2k-1)n

ST(k,n) / Ron/off1/2 - Original irreversible machine performing kn ops

would use cost orig ikn, so, - Advantage ratio between reversible irreversible

cost,

Optimization Algorithm

- For any given value on Ron/off,
- Scan the possible values of n (up to some limit),
- For each of those, scan the possible values of k,
- Until the maximum R(i/r) for that n is found
- (the function only has a single local maximum)
- And return the max R(i/r) over all n tried.

Spacetime blowup

Energy saved

k

n

Asymptotic Scaling

- The potential energy savings factor scales as

R(i/r) ? Ron/off0.4, - while the spacetime overhead goes only as

R(i/r) ? R(i/r)0.45, or Ron/off0.18. - E.g., with an Ron/off of 109, you can do

worst-case computation in an adiabatic circuit

with - An energy savings of up to a factor of 1,200 !
- But, this point is 700,000 less

hardware-efficient!

Conclusions

- A new, more spacetime-efficient

energy-efficient algorithm for doing arbitrary

computations adiabatically has been described. - The energy savings in worst-case computations

goes as the 0.4th power of device on/off ratio. - Best case computations 0.5th power.
- However, the reduction in spacetime efficiency

scales with energy savings to the 1.6th power. - Still much faster than we would like!
- Adiabatics can be generally cost-effective, but

still only for heavily energy-dominated apps.

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