Introduction to Reversible Ckts

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Introduction to Reversible Ckts

• Igor Markov

University of Michigan Electrical Engineering
Computer Science
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Outline

• Historical motivation
• Arbitrary computations via reversible
• Rev. ckts basic definitions examples
• Recent implementations in CMOS
• Reversible synthesis other EDA tasks
• Novel motivations for reversible circuits
• Inherently reversible computations
• Quantum circuits

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Historical Motivation

• Every lost bit causes an energy loss
• C. Bennett, 1973, IBM J. of R D
• the kinetic energy of one molecule in air
• Idea try to avoid those energy costs
• Adiabatic circuits
• Asymptotically energy lossless (Time ? 8 )
• S. Younis and T. Knight, 1994,Workshop on Low
  Power Design

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Implementing Arbitrary Computations via Reversible

• Toffoli 1980, Theorem 4.1Any finite function
  can be writtenas a product of
• trivial encoder ?
• bijection f
• trivial decoder ?
• Constructiveprocedure
• Adds variables

0 0 ?
f
? ? ?
result
argument
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Definitions

• Reversible bit-based computation(e.g., Toffoli
  1980)
• N bits at input
• N bits at output
• Every input output bit-string possible
• Bijection
• These restrictions apply to gates ckts
• Additional restriction no fanout
• Acyclic comb. circuits interesting enough

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Examples
NOT gate

• k-CNOT gate, a.k.a. generalized Toffoli
• (k1)-inputs and (k1)-outputs
• Values on the first k inputs are unchanged
• Last input is negated iff the first k are all 1s
• CNT gate
  library

Toffoli gate
x
x
CNOT gate
y
y
x
x
z
z?xy
y?x
y
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A Reversible Circuit and Truth Table
X
x
x
x
x y z x y z
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 1
0 1 1 0 1 0
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0
y
z?xy? xyz? y
z?xy

• Equiv. to a CNOT gate
• Proof by exhaustivesimulation
• Proof by symbolic arguments

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Implementations in CMOS

• B. Desoete and A. De VosA reversible
  carry-look-aheadadder using control
  gates,Integration, the VLSI Journal,vol. 33
  (2002),pp. 89-104

• Reversible 4-bit adder
• 384 transistors
• no power rails

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Identities for Reversible Ckts
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Temporary Storage / Garbage Bits
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How Much Temporary StorageDo We Need ?

• Toffoli (MIT TR, 1980)
• Odd permutations requireat least 1 line of
  temporary storage
• Shende et al., ICCAD 02
• Even permutations need no temp storage
• Odd permutations need 1 line and no more
• Constructive synthesis procedure (not
  implemented)

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Comb. Synthesis Formulations

• Straightforward
• Given a full truth table, find a circuit
• Shende et al. show an optimal procedure(all
  3-line circuits synthesized in mins)
• With dont cares
• The function of one output bit is restricted
• Iwama et al. (DAC 02) heuristic,transformation-
  based synthesis,may use many lines of temp.
  storage

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Other EDA Tasks

• Fault testing in reversible circuits
• K. Patel et al. (VTS 02) reversible circuits
  require very few test vectors
• Equivalence checking
• Difficulties with empirical validation
• Circuit / gate costs ?
• Circuit benchmarks ?

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New MotivationInherently Reversible Applications

• Information is re-coded,but none is lost or
  added
• Digital signal processing
• Cryptography
• Communications
• Computer graphics
• Micro-processor instructions for
• Bit-permutations
• Butterfly operation from FFT

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New Motivation Quantum Ckts

• Not related to low power
• Quantum circuits operateon linear combinations
  of bit-strings
• E.g., (0gt1gt)/?2, (00gti11gt)/?2
• Linear are expressed by matrices
• Reversibility implied by
  quantum mechanics
• A conventional reversible gate,can be extended
  by linearity,e.g., a quantum inverter is just

0 1 1 0
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Classical Versus Quantum Ckts

• Circuit identities for conventional reversible
  gates (e.g., CNOT and Toffoli)do not change in
  the quantum context
• Conventional techniques applicablewhen there are
  no purely quantum gates
• Conventional subroutines of q. programs
• Purely quantum gates required in apps
• Open problem synthesis withpurely quantum gates

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Thank you!
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Classical Versus Quantum Ckts

• Circuit identities for conventional reversible
  gates (e.g., CNOT and Toffoli)do not change in
  the quantum context
• Conventional techniques applicablewhen there are
  no purely quantum gates
• Conventional subroutines of q. programs
• Purely quantum gates required in apps
• Purely quantum synthesis An Arbitrary Two-qubit
  Computation in 23 Elem. GatesDAC 03, to appear
  in Phys. Review A