Penn ESE370 Fall DeHon 1 ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Day 38: December 10, 2010 Energy and Computation

Question Do we have to spend energy in order to compute? –What is the lower-bound on energy required to perform a computation? Penn ESE370 Fall DeHon 2

Minimum Energy Single bit gate output –Set from previous value to 0 or 1 –Reduce state space by factor of 2 –Entropy:  S= k×ln(before/after)=k×ln2 –Energy=T  S=kT×ln(2) Setting a bit costs at least kT×ln(2) Penn ESE370 Fall DeHon 3 Day 32

Today Thermodynamics Information and Energy Reversibility Adiabatic Logic Adiabatic Pipelines Penn ESE370 Fall DeHon 4

Second Law of Thermodynamics Entropy in closed system increases: –  S≥0 –Entropy is a measure of disorder –Move from order to disorder Heat does not move from cold areas to hot areas –Systems tend to equilibrium Penn ESE370 Fall DeHon 5

Entropy Measure of disorder of a system Proportional to – logarithm of the number of microscope states (arrangements of atoms, electrons…) that can give rise to macroscopic observation of state Penn ESE370 Fall DeHon 6

Example 2 electrons Attached to any of 128 atoms –(e.g. conduction band) Equilibrium: –States each could be in any of 128 positions (128*127/2)=8128 –log 2 (8128)=13 Constrain both to be on left 64 –States both on left (64*63/2)=2116 –Smaller Entropy  more order –log 2 (2116)=11 Penn ESE370 Fall DeHon 7

Information Information standpoint –knowing both on left 64 –provides 2 bits of information 6 bits = log(64) to describe each position instead of log(128)=7 In fact we quantify how unknown a bitstream as information entropy Penn ESE370 Fall DeHon 8

Entropy & Information Entropy proportional to Information Content –Pun? Thermodyanic Entropy vs. Information Entropy –Both defined as log(possibilities) If equally likely Reducing the information content –Reduces entropy  requires energy proporational to change in entropy Penn ESE370 Fall DeHon 9

Computation Creates order Take an output at any value and set it to a specific value –Decreases entropy Bit set costs at least kT×ln(2) Penn ESE370 Fall DeHon 10

Discard Information More specifically: Discarding Information is what must cost thermodynamic energy  S proportional to change in information content Penn ESE370 Fall DeHon 11

Idea Don’t discard information Make changes that preserve the size of the state space –Preserve information All state transforms must be reversible Penn ESE370 Fall DeHon 12

Scale Change Going to change scale –From configuration of atoms –To configurations of bits Macroscale information Necessary that macroscale information preservation hold –For microstate information not to shrink Not sufficient by itself Our concern is lower bounds Penn ESE370 Fall DeHon 13

Idea Don’t discard information Make changes that preserve the size of the state space –Preserve information All state transforms must be reversible Penn ESE370 Fall DeHon 14

Reversible Operation Irreversible –C=AND(A,B) 4 states collapse to 2 Reversible –C=XOR(A,B), with D=A –AB: 00  CD: 00 –AB: 01  CD: 10 –AB: 10  CD: 11 –AB: 11  CD: 01 Penn ESE370 Fall DeHon 15

Reversible Operation Irreversible –C=AND(A,B) with D=A Only 3 states –(C=0,D=1) (C=1,D=1) (C=0,D=0) –C=1,D=0 cannot happen Given C=0, D=0, cannot reconstruct B Reversible –C=/A –C=XOR(A,B), with D=A –D=XOR(A&B,C) with E=A, F=B Penn ESE370 Fall DeHon 16

Computational State Transform Need to look at larger state than single result bit –To assess information preservation Typical operations are not information preserving –Group common cases together E.g. AND(A,B), OR(A,B) Penn ESE370 Fall DeHon 17

Penn ESE Spring DeHon 18 Three Reversible Logic Primitives Controlled NOT Controlled Controlled NOT

Penn ESE Spring DeHon 19 Universal Primitives These primitives –Universal –Reversible If keep all the intermediates they produce –Discard no information –Can run computation in reverse

Reversible Half Adder Penn ESE370 Fall DeHon 20 A B 0 A & B XOR(A,B) A

Penn ESE Spring DeHon 21 Cleaning Up Can keep “erase” unwanted intermediates with reverse circuit –Must “uncompute” the value

Reversible Computing In principal –Reversible operations do not need to discard energy Does not violate necessary conditions for energy consumption in thermodynamics Restricting ourselves to reversible operations –Does not limit what we can compute FYI –Reversibility required for Quantum Computing Penn ESE370 Fall DeHon 22

Adiabatic Adiabatic – a thermodynamic process without heat transfer Penn ESE370 Fall DeHon 23

Adiabatic Logic SCRL Split-Level Charge Recovery Logic (Younis and Knight – ISLPED 1994) Penn ESE370 Fall DeHon 24

Penn ESE Spring DeHon 25 SCRL Inverter  ’s, nodes, at V dd /2 P1 at ground Slowly turn on P1 Slow split  ’s Slow turn off P1’s Slow return  ’s to V dd /2

Penn ESE Spring DeHon 26 SCRL Inverter Basic operation –Set inputs –Split rails to compute output adiabatically –Isolate output –Bring rails back together Have transferred input (logic) to output Still need to worry about resetting output adiabatically

Penn ESE Spring DeHon 27 SCRL Controlled NOT Same basic idea works for any gate –Set inputs –Adiabatically switch output –Isolate output –Reset power rails

Penn ESE Spring DeHon 28 SCRL Cascade Cascade like domino logic –Compute phase 1 –Compute phase 2 from phase 1… –Control Clock/power phases How do we restore the output?

Penn ESE Spring DeHon 29 SCRL Pipeline We must uncompute the logic –Forward gates compute output –Reverse gate restore to V dd /2

Penn ESE Spring DeHon 30 SCRL Pipeline P1 high (F1 on; F1 reset (F2 -1 ) off)  1 split: a=F1(a0)  2 split: b=F2(F1(a0)) F2 -1 (F2(F1(a0))=a P1 low – now F2 -1 drives a –But to same value already set no voltage difference F1 restore by  1 converge …restore F2 Use F2 -1 to restore a to V dd /2 adiabatically

Adiabatic Pipeline Drive Forward Hand off control of node to reverse computation from forward path Allows earlier gates to reset for next operation –So can insert next value into pipeline –While previous value still traversing pipe Penn ESE370 Fall DeHon 31

SCRL Pipeline Penn ESE370 Fall DeHon 32 b

Penn ESE Spring DeHon 33 SCRL Rail Timing

Penn ESE Spring DeHon 34 SCRL Requires Reversible Gates to uncompute each intermediate –Macroscopic energy saving does require reversibility we derived for microscale thermodynamics All switching (except IO) is adiabatic Dissipate energy proportional to –Bits discarded at pipeline I/O –Speed of operation

Reversible Processor Pendulum (Vieri) at MIT Preserves enough information so every instruction reversible –E.g. Memory operation is an exchange Penn ESE370 Fall DeHon 35

Critical Questions Same as adiabatic switching –Can contain losses enough to come out ahead? Leakage Resistive losses –High enough Q resonators? Penn ESE370 Fall DeHon 36

Ideas In principal, can compute without energy Costs energy to discard information –So don’t do that –…or do as little as possible Demands reversibility Reversible computation can be universal Can apply idea to CMOS Penn ESE370 Fall DeHon 37

Final Comprehensive – everything Specific things you might expect –Energy and Delay estimation Logic and interconnect Elmore and wire delay –Driving RC wires and C loads –Precharge and Clocking –Memories –Crosstalk and Noise –Variation –Transmission Lines Penn ESE370 Fall DeHon 38

Admin Review Monday –Select Time Monday 5—7pm (likely in Ketterer) Andre office hours Tuesday Penn ESE370 Fall DeHon 39

Feedback Topics –Omitted (hoped to see) –Should have spent more time on –Should have spent less time on Penn ESE370 Fall DeHon 40