Physical Limits of Computing Dr. Mike Frank CIS 6930, Sec.

1
Physical Limits of ComputingDr. Mike Frank CIS
6930, Sec. 3753XSpring 2002

• Lecture 25Limits on Adiabatics Friction,
  Leakage, Clock/Power SuppliesFri., Mar. 15

2
Administrivia Overview

• Dont forget to keep up with homework!
• We are ?8 out of 14 weeks into the course.
• You should have earned ?57 points by now.
• Course outline
• Part III, Background, Fundamental Limits - done
• Part III, Future of Semiconductor Technology -
  done
• Part IV, Potential Future Computing Technologies
  - done
• Part V, Classical Reversible Computing
• Adiabatic electronics CMOS logic families, -
  Mon. Wed
• Limits of adiabatics Friction,Leakage,Power
  supplies. TODAY
• RevComp theory I Emulating Irreversible Machines
  - Fri. 3/15
• RevComp theory II Bounds on Space-Time Overheads
  - Mon. 3/18
• (plus 7 more lectures)
• Part VI, Quantum Computing
• Part VII, Cosmological Limits, Wrap-Up

3
Structured Systems

• A structured system is defined as a system about
  whose state we have some knowledge.
• Some of its physical information is known.
• ? Its entropy is not at a maximum (by defn.).
• ? It is not at equilibrium (by defn.).
• For states with a given energy E,
• we say the systems energy is distributed among
  those states, in proportion to their probability.

All statesof the abstractsystem havingenergy E
The systemsenergy isin here
States w.prob. gt 0
4
Desired Trajectories

• Any structured systemwe build to servesome
  purposehas somedesiredtrajectory, or set
  oftrajectories, through its configuration space
  that we would ideally like it to follow at all
  times.
• Think of any given state as having a specific
  desirability at any given time.

Time
Config-uration
Desired trajectories
5
Energy Losses

• Energy dissipation can be viewed as a departure
  of part of the systems energy away from the
  systems desired trajectory.
• E.g., 1 of 106 electronsleaks out of aDRAM cell
  systems energy hasdeparted from
  desiredtrajectory (all 106 stay)by a small
  amount

Time
Config-uration
Energy that hasdeparted from desiredtrajectories
6
Limits of Adiabatics IFriction
7
Generalized Friction

• Any force leading to departure from desired
  trajectory that obeys the adiabatic principle
• I.e., force strength ( total energy loss) is
  proportional to velocity along trajectory at low
  velocities
• Examples
• Ordinary sliding friction
• Fluid viscosity
• Electrical resistance
• Forces causing electromagnetic radiative losses
• Forces causing losses in inelastic collisions

8
Ways to Quantify Friction

• Normal friction measures referring to length,
  mass, etc. may not apply to all processes.
• For a given mechanism executing a specified
  process (i.e., following a specified desired
  trajectory or -ies) 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!
9
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/dox ? ?, C ?
  ?, and V ? ?, so cE ? ?3/? ?4 ?Elost cE/t
  ? ?4/? ?3 (like CV2
  energy)

10
Degree of Reversibility of CMOS

• What is the Q of a min-size CMOS transistor?
• Q Efree/?Ediss
• Efree/(cE/t) ½CV2/(C2V2R/t)
  ½(t/RC) ½ s (s slowdown factor)
• Note Using transistors wider than minimum-size
  (larger C, smaller R) wouldnt change RC or Q,
  and would increase overall dissipation by
  increasing cE.

11
Lower Bounds on Friction?

• No general (technology-independent) lower bounds
  on friction coefficients for interesting types of
  processes (e.g. computation) are currently known.
• Clever engineering may eventually reduce the
  friction in desired processes to values as small
  as is desired.
• Some ways
• Reduce number of moving parts (or particles)
• Isolate moving parts of system from unwanted
  interactions w. environment

12
Entropy coefficients of some reversible logic
gate operations

• From Frank, Ultimate theoretical models of
  nanocomputers (Nanotechnology, 1998)
• 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!
13
Is Adiabatic Limit Achievable?

• Even if there is some lower bound on cS, it seems
  we can have ?Smade? 0 as t ? ?.
• What factors may prevent this?
• Any lower bound gt0 on the number of irreversible
  bit-operations performed. (Each has ?Smade ? 1.)
• Fortunately, the lower bound can always be made
  0.
• Any lower bound on the rate of energy leakage,
  even when system is completely stopped.
• Any upper bound on the Q of the clocking
  synchronization system.
• The system dissipates Efree/Q on every cycle.
• No technology-independent upper bounds on Q known

14
Limits of Adiabatics IILeakage
15
Some Synonyms

• Leakage of energy or (equivalently) probability
  mass out of a desired configuration or
  trajectory.
• Occurrence of errors in the desired analog or
  digital state of a system. (Motion away from
  desired states.)
• Decay of structure of a structured system. (The
  state departs from desired state.)

Leakage Error Decay
16
Perfect Mechanisms?

• If a structured system is perfectly closed,
• I.e. non-interacting with other systems, at all!
• And if its internal interactions are perfectly
  known,
• Then, and only then, is its (von Neumann) entropy
  going to be a constant.
• Otherwise, its entropy will continuously increase
  as we lose track of its state.
• In this case, no mechanism is perfect, in that
  some of its energy (i.e. some probability mass)
  is always leaking away from the desired
  trajector(y/ies) at some nonzero base rate, even
  when the rate of systems progress along its
  trajectory is zero.

17
Leakage Limits

• Claim No real, structured system can have
  absolutely zero rate of energy leakage out of its
  desired trajectories, even if not moving.
• However No general,technology-independentlower
  bound onleakage ratesis known (otherthan
  zero.)
• Engineering advances mightmake leakage as small
  as desired.

Time
Config-uration
Energy that hasleaked from desiredconfiguration
18
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.

19
Ways to Decrease Leakage

• Have high potential-energy barriers
• slows down thermally excited leakage
  exponentially
• Have thick potential-energy barriers
• slows down quantum tunneling exponentially
• Example Older generations of CMOS.
• Mechanical (clockwork) systems have high
  potential energy barriers, for their size
• Decay may require atoms to diffuse out of
  tightly-bonded spots.
• Mechanisms that avoid making/breaking contacts
  (e.g. buckled logic) avoid losses due to stiction.

20
Minimum Losses w. Leakage
Etot Eadia Eleak
Eleak Pleaktr
Eadia cE / tr
21
Minimum Loss Derivation
22
Leakage in CMOS

• See transparencies.

23
Limits of Adiabatics IIIClock/Power Supplies

• See transparencies.