a general decomposition for reversible logic m. perkowski, l. jozwiak#, p. kerntopf+, a. mishchenko,...
DESCRIPTION
Atom-scale computation: What are the difficulties in trying to build a classical computing machine on such a small scale? One of the biggest problems with the program of miniaturizing conventional computers is the difficulty of dissipated heat. by heat dissipation.As early as 1961 Landauer studied the physical limitations placed on computation by heat dissipation.TRANSCRIPT
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A General Decomposition A General Decomposition for Reversible Logicfor Reversible Logic
M. Perkowski, L. Jozwiak#, P. Kerntopf+, A. Mishchenko, A. Al-Rabadi, A. Coppola@, A. Buller*, X. Song,
M. Md. Mozammel Huq Azad Khan&, S. Yanushkevich^, V.Shmerko^, M. Chrzanowska-Jeske
Portland State University, Portland, Oregon 97207-0751#Technical University o f Eindhoven, Eindhoven, The Netherlands, + Technical
University of Warsaw, Warsaw, Poland, @ Cypress Semiconductor Northwest and Oregon Graduate Institute, Oregon, USA , * Information Sciences Division, Advanced
Telecommunications Research Institute International (ATR), Kyoto, Japan, & Department of Computer Science and Engineering, East West University, Bangladesh, ,
^ Technical University of Szczecin, Szczecin, Poland
Year 2001
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Atom-scale computation:Atom-scale computation:• What are the difficulties in trying to build a classical
computing machine on such a small scale?• One of the biggest problems with the program of
miniaturizing conventional computers is the difficulty of dissipated heat.
• As early as 1961 Landauer studied the physical limitations placed on computation by heat by heat dissipation. dissipation.
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• Plot showing the number of dopant impurities involved in logic with bipolar transistors with year. – (Copyright 1988 by International Business Machines
Corporation, reprinted with permission.)
R. W. Keyes, IBM J. Res. Develop. 32, 24 (1988).
Computing at the atomic scale:Computing at the atomic scale:
a survey made by Keyes in 1988
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Information loss = energy loss• The loss of information is associated with laws of
physics requiring that one bit of information lost dissipates k T ln 2 of energy, where k is Boltzmann’ constant and T is the temperature of the system.
• Interest in reversible computation arises from the desire to reduce heat dissipation, thereby allowing:– higher densities– speed
R. Landauer,R. Landauer, “Fundamental Physical Limitations of the Computational Process”, Ann. N.Y. Acad.Sci, 426, 162(1985).
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When will When will ITIT happen? happen?
201020202001
k T ln 2
Power for switching one bitLogarithmic scale
Related to information loss
Assuming Moore Law
works In our lifetime
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Reversible LogicReversible Logic• Bennett showed that for power
not be dissipated in the circuit it is necessary that arbitrary circuit can be build from reversible gates.
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Information is Physical• Is a minimum amount of energy required
per computation step?
• Rolf Landauer, 1970. Whenever we use a logically irreversible gate we dissipate energy into the environment.
A
BA B
A
B
A
A Breversible
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Information is PhysicalInformation is Physical• Charles Bennett, 1973: There are no
unavoidable energy consumption requirements per step in a computer.
• Power dissipation of reversible circuit, under ideal physical circumstances, is zero.
• Tomasso Toffoli, 1980: There exists a reversible gate which could play a role of a universal gate for reversible circuits.
A
B
C
AReversible and universal
BC AB
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Reversible computation:Reversible computation:• Landauer/Bennett: almost all operations required in
computation could be performed in a reversible manner, thus dissipating no heat!
• The first condition for any deterministic device to be reversible is that its input and output be uniquely retrievable from each other. – This is called logical reversibility.
• The second condition: a device can actually run backwards then it is called physically reversible– and the second law of thermodynamics guarantees that it
dissipates no heat. Billiard Ball Model
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Reversible logicReversible logicReversible are circuits
(gates) that have one-to-one mapping between vectors of inputs and outputs; thus the vector of input states can be always reconstructed from the vector of output states.
000 000
001 001
010 010
011 011
100 100
101 101
110 110
111 111
INPUTS OUTPUTS
2 4
3 6
4 2
5 3
6 5
(2,4)(365)
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Reversible logicReversible logicReversible are circuits
(gates) that have the same number of inputs andand outputs and have one-to-one mapping between vectors of inputs and outputs; thus the vector of input states can be always reconstructed from the vector of output states.
000 000
001 001
010 010
011 011
100 100
101 101
110 110
111 111
INPUTS OUTPUTS
Feedback not allowed
Fan-out not allowed
2 4
3 6
4 2
5 3
6 5
(2,4)(365)
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Reversible logic constraintsReversible logic constraintsFeedback not allowed in combinational part
Fan-out not allowed
In some papers allowed under certain conditions
In some papers allowed in a limited way in a “near reversible” circuit
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• To understand reversible logic, it is useful to have
intuitive feeling of various models of its realization.
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Conservative Conservative Reversible Reversible
GatesGates
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DefinitionsDefinitions• A gate with k inputs and k outputs is called
a k*k gate.• A conservative circuit preserves the
number of logic values in all combinations. • In balanced binary logic the circuit has half
of minterms with value 1.
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Billiard Ball Model Billiard Ball Model DEFLECTION
SHIFT
DELAY
• This is described in E. Fredkin and T. Toffoli, “Conservative Logic”, Int. J.Theor. Phys. 21,219 (1982).
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Billiard Ball Model Billiard Ball Model (BBM)(BBM)
Input output
A B 1 2 3 4
0 0 0 0 0 00 1 0 1 0 01 0 0 0 1 01 1 1 0 0 1
A and BA
B A and B
B and NOT A
A and NOT B
This is called interaction gate
This illustrates principle of conservation (of the number of balls, or energy) in conservative conservative logic.
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Interaction gateInput output
A B z1 z2 z3 z4
0 0 0 0 0 00 1 0 1 0 01 0 0 0 1 01 1 1 0 0 1
Z1= A and BA
BZ4 = A and B
Z2 = B and NOT A
Z3 = A and NOT B
A
B
Z1= A and B
Z2 = B and NOT A
Z3 = A and NOT B
Z4 = A and B
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Inverse Interaction gateInverse Interaction gateoutputinput
A Bz1 z2 z3 z4
0 0 0 0 0 00 1 0 0 0 10 0 1 0 1 01 0 0 1 1 1
Z1= A and BA
BZ4 = A and B
Z2 = B and NOT A
Z3 = A and NOT B
Other input combinations not allowed
z1z3
z2z4
A
B
Designing with this types of gates is difficult
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Billiard Ball Model (BBM)
Input output
A B z1 z2 z3
0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1
1A
B
2
3
Z1 = NOT A * B
Z2 = A * B
Z3 = A
switchswitch
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Priese Switch Gate
Input output
A B z1 z2 z3
0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1
1A
B
2
3
Z1 = NOT A * B
Z2 = A * B
Z3 = A A
B
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Inverse Priese Switch GateInverse Priese Switch Gate
outputinput
A B z1 z2 z3
0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1
z1A
B
z2
z3
Z3 Z1
A
BZ2
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Inverter and Copier Gates from Priese Gate
Z1 = NOT A * 1
garbage
garbage A
1garbage
V2 = B * 1
V3 = B B
1 Garbage outputs shown in green
Inverter realized with two garbages
Copier realized with one garbage
Input constants
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• The 2*2 Feynman gate, called also controled-not or quantum XOR realizes functions P = A, Q = A B, where operator denotes EXOR..
• When A = 0 then Q = B, when A = 1 then Q = B.
• Every linear reversible function can be built by composing only 2*2 Feynman gates and inverters
• With B=0 Feynman gate is used as a fan-out gate.fan-out gate. (Copying gateCopying gate)
Feynman GateFeynman Gate
+
A B
P Q
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Feynman Gate from PrieseZ1 = NOT A * B
Z2 = A * B
Z3 = A A
BV1 = NOT B * A
V2 = B * A
V3 = B B
A
B
AB
Fan-out > 1
Garbage Garbage outputs Z2 and V2 shown in green
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Fredkin GateFredkin Gate– Fredkin Gate is a fundamental concept in
reversible and quantum computingreversible and quantum computing. – Every Boolean function can be build
from 3 * 3 Fredkin gates: P = A, Q = if A then C else B, R = if A then B else C.
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Notation for Fredkin GatesNotation for Fredkin Gates
A
0 1 0 1
C B
PQ R
A
0 1
P
B C
Q R
A circuit from two multiplexers
Its schemataThis is a reversible gate, one of many
C AC’+B
BC’+AC
CA
B
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Operation of the Fredkin gate
A 0 B
A B 1
A 0 1
C A B
C AC’+B
BC’+AC
A AB
A A+B
A A A’
0 A B
1 A B
C A B
1 B A
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A 4-input Fredkin gateX
A
B
C
0 A
B
CA
B 0 1
1 A
B
C
X
AX’+CX BX’+AX CX’+BX
0 A
B
C1 C
A
B
A A+B ABA’
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Optical Conservative Optical Conservative Reversible and Reversible and
Nearly Reversible Nearly Reversible GatesGates
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Integrated optics
• Integrated optics offers a particularly interesting candidate for implementing parallel, reversible computing structures
• These structures operate in closer correspondence with the underlying microphysical laws which presume non-dissipative interactions and global interconnections.
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Reversible, Conservative, Optical Computers
• Zero energy can be dissipated internally• Dissipation in such circuits would arise only in
reading the output, which amounts for clearing the computer for further use.
• Total decoupling of computational and thermal modes.
• Decoupling is achieved by:– reversing the computation after the results have been
computed,– restoring the circuit to its initial configuration
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Requirements for gates
• No distinction can be made between the inputs. • Each must be of the same type (in this case optical)
and at the same level• The unrestricted type of gate permits a significant
reduction in circuit’s complexity.• The circuit must be both optically reversible and
information-theory (logic) reversible.
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The device
• An optical four-state nonlinear interface switching configuration is derived from the symmetry of an information-losless three-port structure
• The device is:– bit-conservative– optically reversible– logically reversible– with dissipation related to the Kramers-Kronig inverse of
the index of refraction
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The deviceThe device• The device inherently possesses three-terminal
characteristics:– insensitivity to line-noise fluctuations (maintains high
contrast between transmitted and reflected beams)– cascadability through bit conservation,– fan-out by pumped transparentization,– free-space optical fan-in,– pumped (total internal) reflected inversion.
• Planar lattice-regularized layouts for binary adders
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The Dual Beam Nonlinear interface with polarized signal beams of intensity I0 incident at glancing angle
0 or I0
0 or I0
2I0 or 0
n 2 = n 10 - n L + n 2NL(I)
I0 or 0
n 1 = n 10
• n 1 = n 10
• n 2 = n 10 - n L + n 2NL(I0)
• 90o inc sin-1 (n10 - n L + n 2NL(I0) )/ n10
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Fig. 3. Signal replication circuits consisting of an RNI and a half-wave plate.
• Note that:– P or Q is the degraded signal– P and Q is the restored signal
RNI = Reversible Non-linear Interface
Pv.Qh
PQv.PQh
PQ’v.P’Qh
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Fig. 3. Signal replication circuits consisting of an RNI and a half-wave plate.
• Note that:– P or Q is the degraded signal– P and Q is the restored signal
PQv.PQh
Pv.Qh PQ’v.P’Qh/2
1 h
P v
Ph
Pv
Pv
Ph
1 v
Q h
Qh
Qv
Qv
Qh
RNI
Half-wave plate
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Interaction gate implemented with a Fabry-Perot cavity
n = n0 + n 2NL(I)
Q
P
PQ
PQPQ’
P’Q
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Interaction GateInteraction Gate
A
B
AB
A’BAB’AB
A
B
AB
A’BAB’AB
In this gate the input signals are routed to one of two output ports depending on the values of A and B
Interaction GateInteraction Gate Inverse interaction Inverse interaction GateGate
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Minimal Full Adder using Interaction Gates
C
AB
A’B
AB’
AB
A
B
AB
A’B
AB’
AB
Carry
00
Sum
A B
Garbage signals shown in greenGarbage signals shown in green 3 garbage bits
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Reversible logic:Reversible logic:GarbageGarbage
• A k*k circuit without constants on inputs which includes only reversible gates realizes on all outputs only balanced functions, therefore it can realize non-balanced functions only with garbage outputs.
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Priese Switch GatePriese Switch Gate
In this gate the input signal P is routed to one of two output ports depending on the value of control signal C
Priese SwitchPriese Switch Inverse Priese SwitchInverse Priese Switch
P
C
CP
C’P
C P
C
CP
C’P
C
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Minimal Full Adder Using Priese Switch GatesMinimal Full Adder Using Priese Switch Gates
A
B
AB
A’B
B
carry
Csum
Garbage signals shown in greenGarbage signals shown in green 7 garbage bits
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Minimal Full Adder Using Fredkin GatesMinimal Full Adder Using Fredkin Gates
In this gate the input signals P and Q are routed to the same or exchanged output ports depending on the value of control signal C
C
AB
carry
10 sum
3 garbage bits
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RNI Half-Adder
v
hhorizontal polarization mirror
Vertical polarization mirror
vh
vv,h
hv
hv,h
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Ah
Bv
ABv + ABh
A’Bv + AB’h
Ah
Bv
ABv
A’BvAB’hABh
One Beam with Two polarizationsOne Beam with Two polarizations
Logical versus physical Logical versus physical realization of signal in realization of signal in
opticsoptics
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v
RNI Half-Adder
h
h
v
vAh
Bv
Sum = (A’B+A’B)h
Bv(A’B+A’B)h
Removes v
1h
AB’h
Carry = ABh
ABv
ABv
AB’h
A’Bv
A’Bh
A’Bv
ABv + ABh
A’Bv + AB’h
A’Bv 1h
We created similar lattice structures for optical realizations of symmetric, threshold, unate and other circuits
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Fredkin GateFredkin Gate
In this gate the input signals P and Q are routed to the same or exchanged output ports depending on the value of control signal C
Fredkin GateFredkin Gate Inverse Fredkin GateInverse Fredkin Gate
PC
CP+C’Q
C’P+CQ
C
Q
PC
CP+C’Q
C’P+CQC
Q
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Fredkin Gate from Priese Switch GatesFredkin Gate from Priese Switch Gates
Q
C
P
C
CP+CQ
CP+ CQ
CP
CP
CQ
CQ
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Operation of a circuit from Priese SwitchesOperation of a circuit from Priese Switches
Q=0
C
P=1
C=1
CP+CQ
CP+ CQCP
CP
CQ
CQ
Red signals are value 1 Two red on inputs and outputs
One red on inputs and outputsConservative property
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Fredkin Gate from Interaction GatesFredkin Gate from Interaction Gates
P
Q
C
C
CP+ CQ
CP+ CQ
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Concluding on the Billiard Ball Model• The interaction and Priebe gates are reversible and
conservative, but have various numbers of inputs and outputs
• Their inverse gates required to be given only some input combinations
• From now on, we will assume the same number of inputs and outputs in gates.
• INVERTER, FREDKIN and FEYNMAN gates can be created from Billiard Ball Model.
• There is a close link of Billiard Ball ModelBilliard Ball Model and Optical gates and other physical models on micro level
• Many ways to realize universal optical gates, completely or partially reversible but conservative
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Quantum versus reversible computing• Quantum ComputingQuantum Computing is a coming revolution – after recent
demonstrations of quantum computers, there is no doubt about this fact. They are reversible.
• Top world universities, companies and government institutions are in a race.
• Reversible computingReversible computing is the step-by-step way of scaling current computer technologies and is the path to future computing technologies, which all happen to use reversible logic. – DNA– biomolecular – quantum dot– NMR– nano-switches
• In addition, Reversible Computing will become mandatory because of the necessity to decrease power consumptionpower consumption
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What to remember?What to remember?1. Importance of reversible logic2. Importance of conservative logic3. Billiard Ball model of computing.4. Gates of Billiard Ball model.5. Are they all reversible in traditional sense? –
different type of reversibility?6. Priese or Switch gate.7. Interaction gate8. Realization of reversible gates in Billiard Ball
model.9. Optical realization of reversible gates.