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Quantum Cost Calculation of
Reversible Circuit

      Sajib Mitra
      Department of Computer Science and
      Engineering
      University of Dhaka
      sajibmitra.csedu@yahoo.com
OVERVIEW
 Reversible Logic
 Quantum Computing
 Quantum Gates
 Realization of Quantum NOT
 Quantum wire and Special Cases
 Quantum Cost Calculation of RC
 Conclusion
 Assignment
 References
Reversible Logic
    Equal number of input and output vectors
    Preserves an unique mapping between input and output
     vectors of the particular circuit
    One or more operation can implement in a single unit
     called Reversible Gate
    (N x N) Reversible Gate has N number of inputs and N
     number of outputs where N= {1, 2, 3, …}
Reversible Logic       (cont…)

  Advantage
    Recovers bit-loss as well as production of heat
    Adaptable for Quantum Computing
    Multiple operations in a single cycle
    Uses low power CMOS technology
Reversible Logic    (cont…)

  Limitation
    Feedback is strictly restricted
    Maximum and minimum Fan-out is always one
Reversible Logic       (cont…)

Most Popular reversible gates are as follows:




            Fig. 3x3 Dimensional Reversible gates
Reversible Logic       (cont…)

Most Popular reversible gates are as follows:




            Fig. 4x4 Dimensional Reversible gates
Quantum Computing
   First proposed in the 1970s, quantum computing relies on
    quantum physics by taking advantage of certain quantum
    physics properties of atoms or nuclei that allow them to work
    together as quantum bits, or qubits, to be the computer's
    processor and memory.
   Qubits can perform certain calculations exponentially faster
    than conventional computers.
   Quantum computers encode information as a series of
    quantum-mechanical states such as spin directions of
    electrons or polarization orientations of a photon that might
    represent as 0 or 1 or might represent a superposition of the
    two values.
                        q =α 0 + β 1
Quantum Computing                  (cont…)

   Quantum Computation uses matrix multiplication rather than
    conventional Boolean operations and the information
    measurement is realized using qubits rather than bits The matrix
    operations over qubits are simply specifies by using quantum
    primitives as follows:
Quantum Computing   (cont…)




  Input   Output        Input/output   Symbol
 A    B   P   Q          Pattern
 0    0   0   0               00         a
 0    1   0   1               01         b
 1    0   1   1               10         c
 1    1   1   0               11         d
Quantum Computing   (cont…)
Quantum Computing   (cont…)


                              Input   Output
                              A   B   P   Q
                              0   0   0   0
                              0   1   0   1
                              1   0   1   1
                              1   1   1   0
Quantum Gates




    Fig: Quantum Gates are used for realizing Reversible Circuit
Quantum Gates      (cont…)

   What is SRN?




                     But
Quantum Gates      (cont…)

   What is SRN?




                      But


                    NOT

                             But How?
Realization of Quantum NOT
 Basic operator for single input line:
        1. NOT
        2. Coin Flip
        3. Quantum Coin Flip
Realization of Quantum NOT   (cont…)
Realization of Quantum NOT                    (cont…)
 Probability of 0 or 1 based on Coin Flip:



                             1
                       1/2          1/2


                  0                       1
            1/2       1/2          1/2        1/2


            0          1           0           1
           1/4         1/4         1/4         1/4
Realization of Quantum NOT                   (cont…)
 Probability of 0 or 1 based on Coin Flip:



                   1
             1/2          1/2         So the Probability of
                                           P(0)=1/2
        0                       1          P(1)=1/2
  1/2       1/2           1/2       1/2


  0           1          0           1
 1/4         1/4         1/4         1/4
Realization of Quantum NOT                                 (cont…)
 Probability of |0> or |1> based on Quantum Coin Flip:


                                     |
                                     1>
                        1                 1
                            2                     2

                   |                                  |
                   0>                                 1>
          1             − 1               1                 1
               2                 2            2                     2


          |             |                 |                 |
          0>            1>                0>                1>
           1            −1                1                 1
               2             2                2                 2
Realization of Quantum NOT                                    (cont…)
 Probability of |0> or |1> based on Quantum Coin Flip:


                            |
                            1>
               1                 1
                   2                     2        So the Probability of
           |                                  |
                                                       P(|0>)=1
          0>                                 1>        P(|1>)=0
  1            − 1               1                1
      2                 2            2                    2


   |           |                  |               |
  0>           1>                0>               1>
  1            −1                1                1
      2             2                2                2
Realization of Quantum NOT           (cont…)

   NOT operation can be divided into to SRN matrix
    production


            1
                        NO               0
                         T
   Quantum Cost (QC) of any reversible circuit is the
    total number of 2x2 quantum primitives which are
    used to form equivalent quantum circuit.
Quantum Wire and Special Cases       (cont…)




       Quantum XOR gate, cost is 1
Quantum Wire and Special Cases     (cont…)




    Two Quantum XOR gates, but cost is
                  0
Quantum Wire and Special Cases   (cont…)




           Quantum Wire
Quantum Wire and Special Cases           (cont…)

Quantum Cost of V and V+ are same , equal to one.




     SRN and its Hermitian Matrix on same
                     line.
      VV+= Identity and the total cost = 0
Quantum Wire and Special Cases       (cont…)




    SRN and its Hermitian Matrix on same
                    line.
     VV+= Identity and the total cost = 0
Quantum Wire and Special Cases          (cont…)




   The attachment of SRN (Hermitian Matrix of
     SRN) and EX-OR gate on the same line
   generates symmetric gate pattern has a cost
                      of 1.
                Here T= V or V+
Quantum Wire and Special Cases                     (cont…)




    The cost of all 4x4 Unitary Matrices (b, c, d) and the
        symmetric gate pattern (e, f, g, h) are unit.
Quantum Cost of F2G
Quantum Cost of Toffoli Gate




                           But How?
Quantum Cost of Toffoli Gate


                         INPUT       OUTPUT
                         a       b     r
                         0       0     c
                         0       1     c
                         1       0     c
                         1       1     c’
Quantum Cost of Toffoli Gate

    INPUT       OUTPUT
    a       b     r
    0       0     c
    0       1     c
    1       0     c
    1       1     c’
Quantum Cost of Toffoli Gate
        INPUT       OUTPUT
    a           b      r
    0           0      c
    0           1      c
    1           0      c
    1           1      c’

    INPUT           OUTPUT
   a        b         r
    0           0     c
    0           1     c
    1           0     c
    1           1     c’
Now
Quantum Cost of Toffoli Gate

                             Input       Outpu
                                           t
                         A           B    R
                         0           0    C
                         0           1    C
                         1   Have anything wr
                                0   C
                         1           1    C’
Quantum Cost of Toffoli Gate

                               Input       Outpu
                                             t
                           A           B    R
                           0           0    C
                           0           1    C
                           1           0    C
                           1           1    C’




                      Ok
Quantum Cost of Toffoli Gate       (cont…)

Alternate representation of Quantum circuit of TG…
Quantum Cost of Fredkin Gate




                       But How?
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Fredkin Gate   (cont…)
Quantum Cost of Peres Gate
Quantum Cost of NFT Gate
Quantum Cost of NFT Gate
Quantum Cost of MIG Gate
Assignment




             Find out cost
About Author
               Sajib Kumar Mitra is an MS student of Dept.
               of Computer Science and Engineering,
               University of Dhaka, Dhaka, Bangladesh. His
               research interests include Electronics, Digital
               Circuit Design, Logic Design, and Reversible
               Logic Synthesis.
THANKS TO ALL

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Quantum Cost Calculation of Reversible Circuit

  • 1. Quantum Cost Calculation of Reversible Circuit Sajib Mitra Department of Computer Science and Engineering University of Dhaka sajibmitra.csedu@yahoo.com
  • 2. OVERVIEW  Reversible Logic  Quantum Computing  Quantum Gates  Realization of Quantum NOT  Quantum wire and Special Cases  Quantum Cost Calculation of RC  Conclusion  Assignment  References
  • 3. Reversible Logic  Equal number of input and output vectors  Preserves an unique mapping between input and output vectors of the particular circuit  One or more operation can implement in a single unit called Reversible Gate  (N x N) Reversible Gate has N number of inputs and N number of outputs where N= {1, 2, 3, …}
  • 4. Reversible Logic (cont…)  Advantage  Recovers bit-loss as well as production of heat  Adaptable for Quantum Computing  Multiple operations in a single cycle  Uses low power CMOS technology
  • 5. Reversible Logic (cont…)  Limitation  Feedback is strictly restricted  Maximum and minimum Fan-out is always one
  • 6. Reversible Logic (cont…) Most Popular reversible gates are as follows: Fig. 3x3 Dimensional Reversible gates
  • 7. Reversible Logic (cont…) Most Popular reversible gates are as follows: Fig. 4x4 Dimensional Reversible gates
  • 8. Quantum Computing  First proposed in the 1970s, quantum computing relies on quantum physics by taking advantage of certain quantum physics properties of atoms or nuclei that allow them to work together as quantum bits, or qubits, to be the computer's processor and memory.  Qubits can perform certain calculations exponentially faster than conventional computers.  Quantum computers encode information as a series of quantum-mechanical states such as spin directions of electrons or polarization orientations of a photon that might represent as 0 or 1 or might represent a superposition of the two values. q =α 0 + β 1
  • 9. Quantum Computing (cont…)  Quantum Computation uses matrix multiplication rather than conventional Boolean operations and the information measurement is realized using qubits rather than bits The matrix operations over qubits are simply specifies by using quantum primitives as follows:
  • 10. Quantum Computing (cont…) Input Output Input/output Symbol A B P Q Pattern 0 0 0 0 00 a 0 1 0 1 01 b 1 0 1 1 10 c 1 1 1 0 11 d
  • 11. Quantum Computing (cont…)
  • 12. Quantum Computing (cont…) Input Output A B P Q 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0
  • 13. Quantum Gates Fig: Quantum Gates are used for realizing Reversible Circuit
  • 14. Quantum Gates (cont…)  What is SRN? But
  • 15. Quantum Gates (cont…)  What is SRN? But NOT But How?
  • 16. Realization of Quantum NOT Basic operator for single input line: 1. NOT 2. Coin Flip 3. Quantum Coin Flip
  • 17. Realization of Quantum NOT (cont…)
  • 18. Realization of Quantum NOT (cont…) Probability of 0 or 1 based on Coin Flip: 1 1/2 1/2 0 1 1/2 1/2 1/2 1/2 0 1 0 1 1/4 1/4 1/4 1/4
  • 19. Realization of Quantum NOT (cont…) Probability of 0 or 1 based on Coin Flip: 1 1/2 1/2 So the Probability of P(0)=1/2 0 1 P(1)=1/2 1/2 1/2 1/2 1/2 0 1 0 1 1/4 1/4 1/4 1/4
  • 20. Realization of Quantum NOT (cont…) Probability of |0> or |1> based on Quantum Coin Flip: | 1> 1 1 2 2 | | 0> 1> 1 − 1 1 1 2 2 2 2 | | | | 0> 1> 0> 1> 1 −1 1 1 2 2 2 2
  • 21. Realization of Quantum NOT (cont…) Probability of |0> or |1> based on Quantum Coin Flip: | 1> 1 1 2 2 So the Probability of | | P(|0>)=1 0> 1> P(|1>)=0 1 − 1 1 1 2 2 2 2 | | | | 0> 1> 0> 1> 1 −1 1 1 2 2 2 2
  • 22. Realization of Quantum NOT (cont…)  NOT operation can be divided into to SRN matrix production 1 NO 0 T
  • 23. Quantum Cost (QC) of any reversible circuit is the total number of 2x2 quantum primitives which are used to form equivalent quantum circuit.
  • 24. Quantum Wire and Special Cases (cont…) Quantum XOR gate, cost is 1
  • 25. Quantum Wire and Special Cases (cont…) Two Quantum XOR gates, but cost is 0
  • 26. Quantum Wire and Special Cases (cont…) Quantum Wire
  • 27. Quantum Wire and Special Cases (cont…) Quantum Cost of V and V+ are same , equal to one. SRN and its Hermitian Matrix on same line. VV+= Identity and the total cost = 0
  • 28. Quantum Wire and Special Cases (cont…) SRN and its Hermitian Matrix on same line. VV+= Identity and the total cost = 0
  • 29. Quantum Wire and Special Cases (cont…) The attachment of SRN (Hermitian Matrix of SRN) and EX-OR gate on the same line generates symmetric gate pattern has a cost of 1. Here T= V or V+
  • 30. Quantum Wire and Special Cases (cont…) The cost of all 4x4 Unitary Matrices (b, c, d) and the symmetric gate pattern (e, f, g, h) are unit.
  • 32. Quantum Cost of Toffoli Gate But How?
  • 33. Quantum Cost of Toffoli Gate INPUT OUTPUT a b r 0 0 c 0 1 c 1 0 c 1 1 c’
  • 34. Quantum Cost of Toffoli Gate INPUT OUTPUT a b r 0 0 c 0 1 c 1 0 c 1 1 c’
  • 35. Quantum Cost of Toffoli Gate INPUT OUTPUT a b r 0 0 c 0 1 c 1 0 c 1 1 c’ INPUT OUTPUT a b r 0 0 c 0 1 c 1 0 c 1 1 c’
  • 36. Now
  • 37. Quantum Cost of Toffoli Gate Input Outpu t A B R 0 0 C 0 1 C 1 Have anything wr 0 C 1 1 C’
  • 38. Quantum Cost of Toffoli Gate Input Outpu t A B R 0 0 C 0 1 C 1 0 C 1 1 C’ Ok
  • 39. Quantum Cost of Toffoli Gate (cont…) Alternate representation of Quantum circuit of TG…
  • 40. Quantum Cost of Fredkin Gate But How?
  • 41. Quantum Cost of Fredkin Gate (cont…)
  • 42. Quantum Cost of Fredkin Gate (cont…)
  • 43. Quantum Cost of Fredkin Gate (cont…)
  • 44. Quantum Cost of Fredkin Gate (cont…)
  • 45. Quantum Cost of Fredkin Gate (cont…)
  • 46. Quantum Cost of Fredkin Gate (cont…)
  • 47. Quantum Cost of Peres Gate
  • 48. Quantum Cost of NFT Gate
  • 49. Quantum Cost of NFT Gate
  • 50. Quantum Cost of MIG Gate
  • 51. Assignment Find out cost
  • 52. About Author Sajib Kumar Mitra is an MS student of Dept. of Computer Science and Engineering, University of Dhaka, Dhaka, Bangladesh. His research interests include Electronics, Digital Circuit Design, Logic Design, and Reversible Logic Synthesis.