Power/Performance/Cost Efficiency of Adiabatic Circuits, as a function of Device On/Off Power Ratios

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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

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Source ITRS 99
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Across Multiple Technologies
Vacuum Tubes
IntegratedCircuits
Mechanical
DiscreteTransistors
ElectromechanicalRelays
Source Kurzweil, The Age of Spiritual Machines,
pp. 22-25
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½CV2 based on ITRS 99 figures for Vdd and
minimum transistor gate capacitance. T300 K
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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
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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
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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
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Adiabatic Charging in CMOS
Exact formula (if R const.)for frequency
factor f ? RC/t
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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!

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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.

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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!

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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
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?
Transformation of local state
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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
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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!
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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)

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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!
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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.

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Minimum Losses w. Leakage
Etot Eadia Eleak
Eleak Pleaktr
Eadia cE / tr
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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

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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.)
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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

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A MEMS Supply Concept

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

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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

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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

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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)
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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.

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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!

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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 ?
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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
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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
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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
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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.

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Pebble Game Representation
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Triangle representation
k 2n 3
k 3n 2
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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)
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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.

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Bennett 89 alg. is not optimal
k 2n 3
k 3n 2
Just look at all the spacetime it wastes!!!
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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
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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

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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
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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)
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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
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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

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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,

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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.

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Spacetime blowup
Energy saved
k
n
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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!

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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.