Synthesis - Quantum Computing - Lecture Slides, Slides of Computer Science

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SYNTHESIS OF REVERSIBLE CIRCUITS

WITH NO ANCILLA BITS

FOR LARGE REVERSIBLE FUNCTIONS SPECIFIED

WITH BIT EQUATIONS

Outline

• WHAT
  • Reversible Circuits
  • Logic Synthesis:

1. Synthesis with no ancilla bits (MMD - n*n)

2. Synthesis with small number of ancilla bits

(Perkowski/Mishchenko, Khlopotine –

(n+m)*(n+m))

3. Synthesis with very many ancilla bits (Dreschler,

Wille)

• HOW
  • MMD {0,1}^ & company
  • New ideas – Multiple Pass (MP) algorithm
    • Generalized Ordering for MMD Algorithm
    • No truth tables are needed.
      • Truth tables reduce the size of functions that can be handled

Reversible Gates

a a

b^ f^ =a © b

f =a² b © c

b b

c

a a

Toffoli Gate

Fredkin Gate = controlled SWAP

a b

b a

Plain Ole NOT

Feynman Gate

a a

Graph on how over time energy wasted by physics of device will be less than wasted by loss of information Energy lost for physical design reasons when one bit of information is lost (switching) Energy lost for information theory reasons when one bit of information is lost

Year 2012 - 2020

Here reversible logic will become critical

Every future technology must be reversible – IBM and Sandia study

Real-world Applications of

Reversible Logic

• Technologies:
  • Low Power design – adiabatic CMOS
  • Y gates and ballistic circuits
  • Quantum Dots and Quantum Cellular Automata
  • Quantum Computing (truly quantum phenomena)
  • Optical computing
  • DNA
• System Ideas:
  • Digital signal processing
  • Cryptography
  • Computer graphics
  • Network congestion

a b a f 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0

Logic Synthesis & Minimization

• Transform the in/out relation into logic

gates,

Where:

• Every Input generates

corresponding Output

While Optimizing for:

• Minimum number of gates
• Minimum number of control

lines.

• Minimum quantum cost (various

definitions)

a a

b^ f^ =a © b

Presented results use quantum cost but they can be extended for other reversible technology Docsity.com

Previous work on reversible synthesis

with no ancilla bits - MMD

• Reversible logic for quantum computing is recently flourishing research

area, not for others (optical, ballistic, quantum dots).

• The MMD algorithm (Miller, Maslov and Dueck) is currently the leading

reversible logic synthesizer

• MMD assumes a reversible function specification as data and it uses no

ancilla bits.

• MMD software is reasonably fast and it distinguishes itself among other

programs of this type because it achieves (theoretical) 100%

convergence regardless the problem size.

• This program is therefore the current benchmark for the evaluation of

programs for reversible circuit synthesis.

Previous work on reversible synthesis

(methods that assume no ancilla bits)

• 2002-present Perkowski et al use of

complexities of ESOPs, FPRMs and Maitra cascades in the cost functions that evaluate the search results.

• 2004 Agrawal and Jha’s algorithm uses the

number of terms in the Positive Polarity Reed- Muller (PPRM) expansion of synthesized functions as its cost function.

• 2004 Kerntopf’s algorithm uses complexity of

SBDD’s as its cost function.

MMD Background

• The MMD algorithm transforms step-by-step a reversible function to its identity

function.

• The function is arranged in a natural binary code order by inputs assignments.
• Each iteration adds a gate in order to correctly transform the outputs to equal

the inputs without changing any of the previously assigned output patterns

(minterms).

• Gates are chosen to reduce the cost function such as a Hamming distance of

the gate choice function to the original function or to identity function.

• In some variants the gates can be added bi-directionally, at the beginning and

the end of the cascade.

• Once a complete circuit is generated, the original template matching approach

is applied to reduce the gate cost, which is a variant of local optimization

method.

The Basic Algorithm

in 000 001 010 011 100 101 110 111

out 001 000 011 010 101 111 100 110

S 000 001 010 011 100 110 101 111

S 000 001 010 011 100 110 101 111

S 000 001 010 011 100 110 101 111

S 000 001 010 011 100 111 101 110

S 000 001 010 011 100 101 111 110

S 000 001 010 011 100 101 111 110

S 000 001 010 011 100 101 110 111

Final circuit

c b a

From Miller, Maslov and Dueck

MMD “Transformation Based Algorithm for Reversible Logic

Synthesis”

• Single Input Sequence
• Bidirectional Application
• Post Template Matching

Round 1

The weaknesses of MMD include:

1. For n-variable functions it uses a permutation vector of length 2 n^ as its input data which precludes from using it for large circuits.
2. It works only with completely specified functions, thus excluding initial specifications being relations or incompletely specified functions.
3. It does not allow to create arbitrary orders of output functions, which would be one more degree of freedom and is useful in some problems of quantum layout- level optimization.
4. It needs template matching method to optimize its results because only one order of realizing minterms is used in it and the initial result may be far from minimum.
5. It does not allow to investigate the trade-off between the number of ancilla bits and the cost (length of quantum cost or a gate cost) of the cascade.
6. Below we present our research on improving the MMD’s weaknesses.

MMDS = Stedman: “Synthesis of reversible circuits with small ancilla bits for large irreversible incompletely specified multi-output Boolean functions”

• Why should “I” Limit my

Input order?

The idea to extend from Natural Ordering of

Minterms to more general orderings.

MMD has natural order

MMDS has many other orders

MMDS Ordering

• Without any backtracking, bi-directional search or

template matching the MMD algorithm with the new ordering uses multiple MMDS input orders to produce better results than the original MMD ordering.

• It can be used with any number of inputs and has

larger gains compared to MMD when the number of inputs increases.

• Our interest is in what orders converge always?