Azadbinrajib(abr) Quantum Gate For Quantum Computing And Quantum Information

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AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION AzadBinRajib (ABR) Quantum Gate for Quantum Computing and Quantum Information Quantum Revolution through ABR Quantum Gate Dr.A.B.Rajib Hazarika, PhD, FRAS, AES Assistant Professor, Department of Mathematics, Diphu Government College, Diphu, Assam, India-782462 ABOUT THE AUTHOR Dr. A.B.RAJIB HAZARIKA,PhD, MIAMP (GERMANY), FRAS (LONDON), MWASET, AES Born in Jammu, India on 2nd July 1970 ,did PhD in 1995 in Mathematics with specialization in Plasma Instability from Jai Narayan Vyas University, Jodhpur, Rajasthan as Junior Research Fellow, University Grants Commission(NET) and then Senior Research Fellow(UGC,NET) at the time of completion of PhD in 1995 which won the Best Thesis award from Association of Indian Universities, New Delhi in the year 1998.He was Post-Doctoral Fellow and worked as a Research Associate Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION in the Plasma Physics Division, Institute of Advanced Study in Science & Technology, Guwahati, India. He is a Fellow of Royal Astronomical Society (FRAS), London. He is a member of Scientific & Technical committee and Editorial review board on Pure and Applied Sciences of World Academy of Science, Engineering &Technology. He is a member of Advisory committee on Mathematic Education of Royal Society of London, also member of International Biographical Centre in Cambridge, UK. Currently works as Assistant Professor (Lecturer) in Mathematics in Diphu Govt. College in Diphu. He is a gazetted officer of Assam Education Service, AES-I of Government of Assam. He has published many papers in international journals and is member of 23 professional academies of international level. Under his supervision two students did MPhil in Mathematics. He has won “Leading Scientist of the world -2010” award from International Biographical Centre, UK His current interest lies in Astronomy, Astrophysics, Geophysics, Fusion Plasma, and innovation of fusion devices, design of fusion devices, simulation codes and theoretical mathematical modeling. He has innovated and applied for US patent and Trademark office for Double Tokomak collider (DTC), Magnetic confinement Tokomak collider (MCTC) hub, Duo Triad Tokomak collider (DTTC) hub .A Hall thruster as diffusion associated neoclassical indigenous system of Hall assembly (DANISHA).He is known for his theoretical research work on Gravitational instability and gravitational collapse M=23/2 M sun as a new formula for Chandrasekhar limit now known as Bhatia-Hazarika Limit , when the rotating neutron star, pulsars are formed .When the mass of the star is more than this limit a neutron star shrinks or abberates due to gravitational collapse up to a point size in space. As it is known that when the star passes limit of the size of old star more than three times that of mass of sun it passes the Schwarzchild radius and there on is a black hole from where we can receive no more information as its gravitational field is too intense to permit anything , even photons to escape. Dr. A.B.RAJIB HAZARIKA,A.E.S., ASSISTANT PROFESSOR ,DEPT. OF MATHEMATICS ,DIPHU GOVT. COLLEGE,DIPHU,KARBI ANGLONG ,ASSAM Besides being in teaching side by side research is also going on for last 14 years .I started my research work as a Junior research fellow (UGC,NET) under the guidance of Prof(Dr) P K Bhatia, DSc, FNASc, FRAS(Lond.),FIMA(UK),MIAU(Lond),Former Dean, Faculty of Science, Former Prof and Head, Dept. of Mathematics and Statistics, J N Vyas University, Jodhpur, Rajasthan on 11th May 1993 ,during that I used to teach as JRF(UGC,NET) in the K N college for Women, Jodhpur and also used to do my research work .I completed in total 7(seven) papers for my PhD thesis entitled “Some problems of plasma instabilities in partially ionized and fully ionized plasmas” which was submitted to Dept. of Mathematics and Statistics, J N Vyas University, Jodhpur, Raj. on 27th Dec 1995.From the thesis two of the papers are published in internationally reputed journal Physica Scripta published from Royal Swedish Academy of Sciences, Stockholm, Sweden viz., P K Bhatia and A B Rajib Hazarika(1995)Phys Scr 51,775 & P K Bhatia and A B Rajib Hazarika(1996) Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Phys Scr 53,57 which is a fundamental paper in the field of space plasma it gives some idea about the size when the stars will burst which came out to be true for binary stars as well as for neutron stars giving new limit known as Bhatia –Hazarika limit. Bhatia P K and Hazarika A B Rajib: J Indian Acad. Math. 29 (1), 141. And many of papers remain unpublished from the thesis. In the mean time Senior Research Fellowship (UGC, NET) was awarded on 12th May 1995.After completing PhD in Mathematics I joined Plasmas Physics Division, Institute of Advanced Study in Science & Technology, Guwahati as Research Associate (DST) in the project Development of plasma physics division being given work of developing the theoretical models for sheath instabilities which was thrusted upon by Dept. of Science and Technology (DST) , Govt. of India. I have two models for sheath i.e. Sheath Criterion for Bhom Chodura collisional magnetized sheath which is in annual report as well as was presented in 14th National Symposium of plasma science and technology, Amritsar (1999).Based on which research scholars did the experiment to check the results proved to be correct is available in annual report of IASST (1999) & in technical report IASST/98/29. One paper on BETA machine was also done and was presented in 13th National symposium of plasma science &technology (1998,2001) in oral session which was based on the stabilization process of Rayleigh –Taylor instabilities in BETA machine as theory as well as simulation code is available as technical report IASST/98/23and in annual report of IASST(1998) which contains two more paper on Rayleigh-Taylor instability in stratified polytrophic medium, another one Jhonson’s criterion of superposed fluids, Kelvin –Helmholtz instability in collisional polytrophic medium. A new type simulation code was developed for BETA machine using Fuzzy Differential Inclusion(FDI) which was presented in 15th National symposium of plasma science and technology, (2002,2003)using source ionization another with thermal diffusion. Two new type of future fusion devices were conceptualized theoretically namely Double Tokomak Collider(DTC) and Magnetic Confinement Tokomak Collider(MCTC) for which patent is applied for in US patents and trademarks office on 12th jun 2008 and paper related to above said devices were presented in 19th, 21st National Symposium of plasma science and technology,(2005,2007) and got accepted for publication in 3rd Technical meeting of IAEA on theory of plasma instabilities, York(UK).These two future fusion devices gives application wise in generation of electricity by using fusion and by hybrid technology. Application in the field of computers is making the microchip using nano-technology. In space craft thrusters so far we that different types of thrusters are used by using plasma or electric ignition similarly I am planning to make new type of thrusters using the presently working three fusion devices and using hollow cathode thrusters, Hall thrusters and so on using different gases such as xenon and argon. Most recently I have developed a new type of fusion device which may give more power than the earlier design known as Duo Triad Tokomak Collider (DTTC) which hopefully gives better thrust in space technology as well as in automobile Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION industry. Very recently I have supervised two MPhil students who have completed it. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION ABOUT THE BOOK This book deals with a new type of Quantum gate AzadBinRajib (ABR) quantum gate with application to different aspect starting from Quantum algebra, Minterms and Maxterms, Modified Karnaugh maps, Literals ,schematic design of ABR quantum gate, LAQUIT microprocessor with the application ABR quantum gate imbibed in it. Design and knowhow of the microprocessor, data compression using ABR quantum gate, quantum data signatures using ABR quantum gate, Quantum data teleportation for the word HELLO WORLD .This will be first time anyone will be able to showcasing the teleportation of more than one qubit. Dr.A.B.Rajib Hazarika, PhD, FRAS (London) , AES Author Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Table of Contents Chapter1 AzadBinRajib (ABR) Quantum Gate 6 Chapter 2 Maxterms and Minterms Literals for ABR Quantum Gate 12 Chapter 3 A Logical AzadBinRajib (ABR) Quantum Gate Unit Integrated Technology (LAQUIT) Microprocessor Architecture and Algorithm based on ABR Gate with Quantum Dots Key Distribution Encryption System 23 Chapter 4 Data Compression by AzadBinRajib (ABR) Quantum Gate 34 Chapter 5 Quantum Digital Signature using ABR Quantum Gate 38 Chapter 6 Quantum teleportation using ABR quantum gate 40 Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Chapter 1 AzadBinRajib (ABR) Quantum Gate A new type of Quantum gate is innovated as AzadBinRajib (ABR) Quantum Gate by Dr.A.B.Rajib Hazarika, PhD, FRAS (London), AES which is helpful for getting information in quantum wise it is almost like Hadamard gate with some differences with it. The reason behind the innovation this quantum gate is one must quantum gate which satisfies all conditions of logic gates and shows no decoherence, it follows all the Quantum algebra formulations. It has the capability to perform all Boolean algebra calculation with this ABR Quantum gate it justifies all steps as required. One more quantum gate Not AzadBinRajib (NABR), which is compliment of the AzadBinRajib (ABR) quantum gate. We can represent the superposition of the ABR quantum gate in matrix form as ABR = NABR 𝟏 √𝟐 [ 𝟎 𝟏 𝟏 ] 𝟏 1 1 0 ] 0 0 = √2 [ Pictorial representation schematic of Quantum logic gate is shown below Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Quantum Algebra on basis of Boolean algebra 1. Closure property: ABR quantum gate is set of quantum ABR + ABR = 2 ABR 𝟏 √𝟐 𝟎 [ 𝟏 𝟏 𝟎 𝟏 ]+ [ √𝟐 𝟏 𝟏 𝟐 𝟎 𝟏 ]= [ √𝟐 𝟏 𝟏 𝟏 ] 𝟏 For generalized case ABR + ABR= k ABR which belongs to ABR Where k is constant. Therefore, it is closure in property. 2. Commutative property Order of variables X+Y=Y+X Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 𝟏 𝟎 𝑿 = √𝟐 [ 𝟏 𝟏 𝟎 𝟏 ],𝒀 = [ √𝟐 𝟏 𝟏 𝟏 𝟏 ] 𝟏 𝟏 𝟎 𝟏 𝟐 𝟎 𝟎 𝟏 ]+ [ ]= [ √𝟐 𝟏 𝟏 √𝟐 𝟏 𝟏 √𝟐 𝟏 [ 𝟏 𝟏 𝟎 𝟏 ]+ [ 𝟏 √𝟐 𝟏 𝟎 [ √𝟐 𝟏 = 𝟏 ] 𝟏 𝟏 ] 𝟏 X*Y+Y*X 𝟏 𝟏 𝟎 𝟏 ]𝑿 [ 𝟏 √𝟐 𝟏 𝟎 [ √𝟐 𝟏 𝟏 𝟏 𝟎 𝟏 ]𝑿 [ 𝟏 √𝟐 𝟏 𝟎 [ √𝟐 𝟏 = 𝟏 𝟎 𝟏 𝟏 ]= [ ] 𝟏 𝟐 𝟏 𝟏 𝟏 ] 𝟏 3. Associative property X+(Y+Z) =(X+Y) + Z 𝟏 𝟎 √𝟐 𝟏 = 𝟑 𝟎 [ √𝟐 𝟏 [ 𝟏 𝟎 𝟏 ]+( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]=( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]+ [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]+ [ 𝟏 √𝟐 𝟏 𝟏 ]) 𝟏 𝟏 𝟎 𝟏 ]) + [ 𝟏 √𝟐 𝟏 𝟏 ] 𝟏 Again we can prove that X*(Y*Z) = (X*Y)*Z 𝟏 𝟎 [ √𝟐 𝟏 = 𝟏 𝟎 [ 𝟐√𝟐 𝟏 𝟏 𝟎 𝟏 ]𝑿( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]=( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 𝟏 ]𝑿 [ ]) 𝟏 √𝟐 𝟏 𝟏 𝟏 𝟎 𝟏 ]𝑿 [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ])𝑿 [ 𝟏 √𝟐 𝟏 𝟏 ] 𝟏 4. Identity law For addition X+0=X 𝟏 𝟎 [ √𝟐 𝟏 = 𝟏 𝟎 ]+[ 𝟏 𝟎 𝟏 𝟎 [ √𝟐 𝟏 𝟎 ] 𝟎 𝟏 ] 𝟏 For multiplication Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION X*I=X 𝟏 √𝟐 𝟎 [ 𝟏 𝟏 𝟏 ]𝑿 [ 𝟏 𝟏 𝟏 𝟎 = √𝟐 [ 𝟏 𝟏 ] 𝟏 𝟏 ] 𝟏 5. Inverse law( Complimentary law) For addition X + X’ = 0 𝟏 𝟏 𝟎 −𝟏 𝟏 𝟎 𝟏 ]+ [ ]= [ 𝟏 √𝟐 −𝟏 −𝟏 √𝟐 𝟎 𝟎 [ √𝟐 𝟏 𝟎 ] 𝟎 For multiplication X * X’ =1 𝟏 𝟎 √𝟐 𝟏 [ 𝟏 𝟏 𝟏 ]𝑿 [ 𝟏 √𝟐 𝟏 𝟏 𝟏 ]=[ 𝟎 𝟎 𝟎 ] 𝟏 6. Distributive law a. X(Y+Z) =X*Y +X*Z 𝟏 𝟎 [ √𝟐 𝟏 =[ 𝟏 𝟎 𝟏 ]𝑿( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟎 𝟏 ]=( [ 𝟏 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]+ [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]+ [ 𝟏 √𝟐 𝟏 𝟏 ]) 𝟏 𝟏 𝟎 𝟏 ]) 𝑿 [ 𝟏 √𝟐 𝟏 𝟏 ] 𝟏 b. (X + Y)*Z = (X+ Y) (X+Z) 𝟏 𝟎 [ √𝟐 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟏 ]+( [ ]𝑿 [ ]) 𝟏 √𝟐 𝟏 𝟏 √𝟐 𝟏 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 [ ]+ [ ]) 𝑿 ( [ =( √𝟐 𝟏 𝟏 √𝟐 𝟏 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 [ ]) + √𝟐 𝟏 𝟏 𝟏 ] 𝟏 7. Absorption law or Theorem of Redundancy a. X + X*Y=X 𝟏 𝟎 [ √𝟐 𝟏 𝟏 𝟎 𝟏 ]+( [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]𝑿 [ 𝟏 √𝟐 𝟏 𝟏 𝟎 𝟏 ]) = [ 𝟏 √𝟐 𝟏 𝟏 ] 𝟏 b.X (X+Y) = X Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION X (X+Y)=X*X +X*Y =X +X*Y= X(0 + Y ) = X*1=X 8. Involution law If a variable is complimented twice it retains its original state (X’)”= X If X=0 then X’=1 and X’’=0 =X 𝟏 𝟎 𝑿 = √𝟐 [ 𝟏 𝟏 X’’= √𝟐 [ 𝟏 𝟏 𝟏 ] Then 𝑿′ = [ √𝟐 𝟏 𝟎 𝟎 ] again if we do compliment we get 𝟎 𝟎 𝟏 ]=X itself again. 𝟏 𝟏 9. Idempotent law It states that X+X=X and X*X=X we have proved earlier. 10. Redundant Literals rule a. X + X’ * Y = X + Y LHS= X + X’ Y= (X+X’)(X+Y) = 1. (X+Y) =RHS b. X (X’ + Y) = XY LHS = X*(X’ + Y) = XX’ + XY = 0 + XY = XY =RHS DeMorgan’s Theorem (XY)’ = X’ + Y’ (X +Y)’ = X’ * Y’ Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Chapter 2 Maxterms and Minterms Literals for ABR Quantum Gate Title: Probability Amplitude and Measurable State of AzadBinRajib (ABR) Quantum gate A New type of Quantum Gate with Modified Karnaugh Map and Logical AzadBinRajib (ABR) Quantum gate Unit Integrated Technology (LAQUIT) Microprocessor Micro-architecture Authors: A.B. Rajib Hazarika1 Affiliations: 1 Department of Mathematics, Diphu Government College, Diphu, Assam, India-782462. *Correspondence [email protected] to: [email protected] Abstract: A new type of quantum gate namely AzadBinRajib (ABR) is studied for probability amplitude and measurable state. It justifies the probability amplitude of 50% for 0th quantum state and 50% for 1th quantum state and is measurable for up and down state as in the classical as well as for Hadamard gate. It is unitary in nature is studied. Modified Karnaugh Map is studied for ABR quantum gate with Literals. A microprocessor is designed with microarchitecture Drabrh Logical AzadBinRajib (ABR) Quantum Gate Unit Integrated Technology (LAQUIT) microprocessor microarchitecture. Key words : Quantum gate, quantum microprocessor, probability amplitude, measurable state One Sentence Summary: The probability amplitude of AzadBinRajib (ABR) quantum gate A new type of quantum gate justifies for the 0 th or up state and 1 th or down state with ½=50% probability for each state and no decoherence is observed. Literals and Modified Karnaugh Map studied Past Studies: In past several authors have studied on different quantum gates Hadamard , CNOT gate ,X,Y,Z Pauli’s gate, pi/8 gate (1,2, and 3). Modified Karnaugh map is studied by Wang et al (2) .As lot have been done on different quantum gats a new type of quantum is studied which approves all properties required for being quantum gate. As it known for the Quantum gates lot of decoherence occurs while checking for the probability amplitude of the quantum gates. Now, the question arises “what is decoherence, which people talk a lot for the calculation of quantum gates?” The answer to this question lies in the fact when we calculate the probability amplitude in digital form we get 50% and 50% for “0” and “1” respectively, without having any in-between values and probability which we call as coherence. So, when we go for the quantum gate or combination of various quantum gates the perfect value for the 0 and 1 state is not achieved, Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION rather we get some values which lies between 0 and 1, which here we call it as decoherence and probability amplitude thereon. Now, regarding the present context quantum gates i.e. the AzadBinRajib (ABR) and Not AzadBinRajib (NABR) quantum gates we will try to observe whether we achieve the milestone of getting coherence and avoiding any sort of decoherence if at all occurs in calculations of such quantum gates. For that at first we try to see that “0 “state and “1” state in terms of ket For ABR Quantum Gate 0 |1> = [ ] “1” state 1 𝛼 ABR [𝛽 ] = =𝛽 |0> √2 1 √2 1 |0> = [ ] “0” state 0 𝛽 1 1 𝛼 ] [𝛽 ] = [ ] √2 𝛼+𝛽 1 0 [ 1 + (𝛼 + 𝛽 ) |1> √2 = 𝛼 |1> √2 + 𝛽( |0>+|1> √2 ) Probability amplitude are described as 𝑃𝛼 = 𝛼𝛼 ∗ , 𝑃𝛽 = 𝛽𝛽 ∗ 𝛼𝛼 ∗ + 𝛽𝛽 ∗ = 1 Linear superposition 𝑃𝛼 + 𝑃𝛽 = 1 ̂ |0 >= 𝐴𝐵𝑅 1 √2 0 [ 1 1 1 1 ] [ ] 1 √2 0 1 0 1 |1> = 2 [ ] = √2 { 2 } √ 1 The 0th state 1 Here we find that the Eigen vector is 𝛼 = √2 ̂ |1 >= 𝐴𝐵𝑅 0 [ √2 1 1 1 1 = 2[ ] 1 1 0 ][ ] 1 1 1 |0>+|1> = √2 { √2 } The 1st state 1 Also we find that the Eigen vector 𝛽 = √2 𝛼 = 𝛼 ∗ & 𝛽 = 𝛽 ∗ as we are considering the real values the imaginary terms does not matter. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 𝛼 2 + 𝛽2 = 1 1 2 1 2 1 1 (√2) + (√2) = 2 + 2 = 1 Hence we get 50% probability for 0th state and 50% probability for 1st state which is the required measurable stat for the 0th state and the 1st state. For complex numbers Let us consider 𝛼 = 𝑥 ± 𝑖𝑦 , 𝛽 = 𝑢 ± 𝑖𝑣 Then the linear superposition is given as 𝑧 = (𝑥 ± 𝑖𝑦) |1 > √2 + (𝑢 ± 𝑖𝑣) ( |0 > +|1 > √2 ) The inner product of the terms is given < 𝛼|𝛼 ∗ > = (𝑥 + 𝑖𝑦)(𝑥 − 𝑖𝑦) = 𝑥 2 + 𝑦 2 = 1/2 < 𝛽|𝛽 ∗ > = (𝑢 + 𝑖𝑣)(𝑢 − 𝑖𝑣) = 𝑢2 + 𝑣 2 =1/2 1 1 1 < 𝛼|𝛼 ∗> = √2 (0 1) [ ] = 2 = 50% 1 1 < 𝛽|𝛽 ∗ > = √2 (1 1 0 1) [ ] = 2 = 50% 1 Therefore the ABR quantum gate is exact superposition having measure of |0> =50% and |1>=50% For the nth state of the ABR Quantum gate we derive it as 1 2𝑛−1 ̂ 𝑛 = 𝑛/2 ∑∅=1 𝐴𝐵𝑅 |∅ > 2 As for Quantum dots it rotates through an angle of 𝜃 = 𝐴𝐵𝑅 = 𝜋 ̂ cos(2. 4 ) [ 𝜋 sin( 4 ) 𝜋 4 𝜋 sin( 4 ) 𝜋 cos( 4 ) ] Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 1 0 √2 1] =[ 1 √2 √2 = 1 √2 0 [ 1 1 ] 1 For nth qubits it changes to 𝐴𝐵𝑅 𝑛 = 𝑛𝜋 𝑛𝜋 cos(2. ) sin( ) 4 4 [ 𝑛𝜋 𝑛𝜋 ] sin( ) cos( ) 4 4 Unitary Matrix U|00> = |00> And U |10>= |11> U=[ 1 0 1 𝐼 0 1 ] where 𝛼 = [ ] and 𝐼 = [ √2 0 𝛼 1 1 0 0 0 0 ] , 0=[ ] 1 0 0 Modified Karnaugh Map Karnaugh map (K-Map) is very popular technique used for the simplification of Boolean algebra, here it is used as Modified Karnaugh Map (K-Map) for Quantum Gate algebra expressions. It reduces the Boolean algebra expressions more quickly and easily as compared to algebraic reduction. It is designed in the form of a diagram having rows and columns into squares and the number of square corresponds to a row number in the truth table. Each cell in modified K-map represents one particular combination of variables product form. For a total number of n-variables modified K-map consists of 2n cells. Say for 2512 cells for n=512 variables (qubits). TWO VARIABLE MODIFIED K-MAP For two (qubits) variables A and B Modified K-Map have 22=4 cells. These are |A’B’>, | A’B >, |AB’> , |AB> viz., |00>, |01>, |10>, |11> . The two variables modified K-map must have 4 cells shown below. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Truth Table for two variables (qubits) A B |0> Fig.1 table two |0> Output (F) Quantum Logic A’B’ or m0 |00> Truth for |0> |1> |1> |1> |0> |1> A”B or m1 |01> AB’ or m2 AB or m3 |10> |11> variables (qubits) While designing the Quantum logical circuit one has to take care of two things a. Inputs to the circuits b. Desired output for each of the input To know about the both steps described above we will have to undergo certain steps which is known as Minterms and Maxterms 1. Minterms Minterms are the products because they are logical AND gate for a set of variables. These are the product terms for which the value of function is 1. A + B = A.1 +B.1 =A (B + B’) + B (A + A’) = AB + AB’ + BA + BA’ =AB + AB’ + A’B The terms AB, AB’, A’B are the minterms because all the terms contains both A and B either in direct or in compliment form. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION “1” means the variables are “not complimented” “0” means the variables are complimented. Table for Minterms for three variables A B C Minterms Shorthand Notations |0> |0> |0> A'B'C' m0 |000> |0> |0> |1> A'B'C m1 |001> |0> |1> |0> A'BC' m2 |010> |0> |1> |1> A'BC' m3 |011> |1> |0> |0> AB'C' m4 |100> |1> |0> |1> AB'C m5 |101> |1> |1> |0> ABC' m6 |110> !1> |1> |1> ABC' m7 |111> Fig. 2 . Table of Minterms for three variables (qubits) 2. Maxterms Maxterms are the sums because they are the logical OR of a set of variables. There are the sum forms for which the value of the function is “0”. A. B = (A+0) (0 + B) = (A +BB’) (AA’ + B) = (A + B) (A + B’) (A+ B) (A’ + B) = (A + B) (A + B’) (A’ + B) The terms (A+B), (A+B’), (A’+B) are known as the Maxterms because all the terms contain both the variable A and B either in direct or in compliment form. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Column1 Column2 Column3 Column4 Table for Maxterms for three variables A B |0> |0> |0> |0> |1> |1> |1> !1> C |0> |0> |1> |1> |0> |0> |1> |1> Minterms A+B+C A+B+C' A+B'+C A+B'C' A'+B+C A'+B+C' A'+B'+C A'+B'+C' |0> |1> |0> |1> |0> |1> |0> |1> Column5 Column6 Shorthand M0 M1 M2 M3 M4 M5 M6 M7 Notations |000> |001> |010> |011> |100> |101> |110> |111> Fig.3 Table of Maxterms for three variables (qubits) For Maxterms “0” means the variable is “not complicated” “1” means the variable is “complimented”. Detailed calculation of AzadBinRajib (ABR) Quantum logical algebra 1 𝐴 = √2 [ 0 1 ] 1 1 1 0 𝐵 = √2 [ 1 1 0 1 1 0 ]+ [ A + B= √2 [ 1 1 √2 1 1 = √2 [ 1 ] 1 1 0 = √2 [ 1 1 ] OR gate 1 1 0 = √2 [ 1 1 ] 1 1 0 A. B = √2 [ 1 1 ] 1 1 √2 0 [ 1 1 ] 1 1 1 0 1 ]. [ ] 1 √2 1 1 0 1 ] AND gate 1 1 1 √2 [ 1 0 0 1 ] = [ √2 1 1 1 1 ] 1 1 0 1 1 0 1 ] ] 𝐴′ = √2 [ 𝐵′ = √2 [ As it has got real terms only if 1 1 1 1 complex terms are considered case will be different. 1 0 A + A’ = √2 [ 1 1 = √2 [ 1 0 1 ]+ [ √2 1 1 1 ] 1 1 0 0 1 ] OR gate [ √2 1 1 1 1 ] 1 Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 1 = √2 [ 0 1 ] = A’ + B = A.B 1 1 AB’ = 1 = √2 [ = 1 √2 [ 1 √2 0 1 1 0 [ ]. [ 1 1 √2 1 1 ] 1 1 0 0 1 ] AND gate [ √2 1 1 1 1 ] 1 0 1 ] = A’B 1 1 G=AB+AB’+A’B 1 1 0 0 1 ]+ [ 1 1 √2 1 1 1 0 1 0 1 ] OR gate 𝐴 = [ ] OR gate √2 1 1 1 1 = √2 [ = √2 [ 1 = √2 [ 1 0 1 1 ] + [ ] √2 1 1 1 1 √2 0 [ 1 1 ] 1 0 1 ] 1 1 That means if we add three AzadBinRajib (ABR) Quantum gate together we get same value of one ABR Quantum gate which performs Quantum algebra Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Fig.4 .Drabrh Logical AzadBinRajib (ABR) Quantum Gate Unit Integrated Technology (LAQUIT) Microprocessor microarchitecture Discussion :( It has been observed in the present study that the AzadBinRajib (ABR) does not show any type of decoherence in it, always showing appropriate results of equally distributed ½-1/2 probability for each state of up or 0th state and down or 1th It is also observed that the ABR quantum satisfies if we consider complex numbers which is calculated in the present study. It follows all possible operation done in Boolean algebra is done for the quantum algebra which works properly. It follows up with the unitary matrix form which is essential for any quantum gate to be properly working quantum gate. ABR quantum gate satisfies modified Karnaugh map operation with Minterms and the Maxterms shown vividly works properly. It has been shown that the all literals and quantum algebra works. A new type quantum microprocessor microarchitecture of Drabrh Logical AzadBinRajib (ABR) quantum gate Unit Integrated Technology (LAQUIT) microprocessor. In the microprocessor Quantum dot cryptography is used. Our study is in compliment with the studies done on Hadamard gate by authors (1, 2, and 3). References and Notes: 1. “Quantum computation and Quantum Information:Nielson,M.A,Chuang,I (2000) Cambridge University press. ISBN 0521632358.OCLC 2. Modified Karnaugh Mapnfor Quantum Boolean construction.:Wang,S.A,Lu,C.U,Tsai.I-Ming,Kuo.S-Yen: IEEE conference paper Sept.(2003), doi:10.1109/NANO.2003.1230996 circuit Xplore 3. Wilde,M.M : Quantum Information Theory (2003)Cambridge University Press, ISBN 9781139525343 , doi.org/10.1017/CB09781139525343 Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Chapter 3 A Logical AzadBinRajib (ABR) Quantum Gate Unit Integrated Technology (LAQUIT) Microprocessor Architecture and Algorithm based on ABR Gate with Quantum Dots Key Distribution Encryption System A.B. Rajib Hazarika1 1 Department of Mathematics, Diphu Government College, Diphu, Assam, India-782462 E-mail: [email protected] ; [email protected] Abstract Microachitecture and Algorithm of a new Logical AzadBinRajib (ABR) quantum gate Unit Integrated Technology(LAQUIT) microprocessor is considered .The present study describes the know how of the microarcitecture and alorithm along with the Quantum dots key encryption system in detail for n qubits as well as for particlar case .Asymmetric Encryption System (AES-512) is used in microprocessor.ABR quantum gate theory, ABR Quantum gate Algorithm is studied, quantum computing and information theory is also studied.. Keywords: Quantum gate, quantum computing, quantum information, quantum algorithm, quantum microprocessor, microarchitecture Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.) ,AES Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 1. Introduction Many authors in the pas hve worked on Quantum gates and quantum computing Ying[1],Wilde[2]. Ying[1] though worked for quantum computing ,quantum theory and Arificial Intellgence .Wilde[2] worked Quantum information and decribed everything in detail in his monogram “Quantum Information.Wilde has also used Neo classicla Shannon thory to show how quantum computing can be done by using quantum dots.He has shown quantum dots based key distributin encryption system in detail.Nielson [3] has worked on his monogram “Quantum Computing different techniques how the quantum gates work and the theory behind it .All of the three authors have exclusively described the Quantum computin,quantum theory,quantum information and the Artificial Intelligence.This forms the aspect of our present study with our newly developed AzadBinRajib (ABR) quantum gate and LAQUIT microprocessor imbibing the ABR quantum gate. 1.1 Present work 1.1 The present work is based on AzadBinRajib (ABR) Quantum gate based microprocessor which is incculcated with the Logical AzadBinRajib (ABR) Quantum gate Unit Integrated Technology (LAQUIT) as the microproprocessor archictecture gets integrated with ABR quantum gate. Here we will be using the Quantum dots based cryptography with the ABR quantum gates.Since it works on Internet protocol here, Quantum dots Long Term Evolution (QDLTE)and Quantum dots Voice over Internet protocol (QDVoIP) is used..This forms the aspect of our study.In Sec.2 LAQUIT Microprocessor ,Sec.2.1. LAQUIT Architecture Sec.2.1.1. Microprocessor design Sec.2.1.2 Hardware of LAQUIT microprocessor. Sec. 2.2. ABR Quantum Gate Algorithm Sec.3.ABR Quantum Gate QDKD Encryption system Sec3.1. ABR Quantum Gate and Quantum Registers Sec.4. Discussion and Conclusion Sec.5.References 2. LAQUIT Microprocessor Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Logical AzadBinRajib (ABR) Quantum Gate Unit Integrated Technology (LAQUIT) LAQUIT 2.1. LAQUIT Architecture 2.1.1. Microprocessor design Fig.1. 2.1.2.Hardware of LAQUIT Microprocessor The hardware section have been divided into many parts as follows: i) ARM cortex M4 ( Four) A cortex M4 is based on 64 bit technology having 4nm size. ii)ARM Cortex A15 MP core it is cortex A15 with 22nm size based on 32 bits and 64 bit technology. iii)Dynamic Memory Manager It is consist of a library of subroutines for dynamic memory allocation inteded for primary and secondary memory storage.Dynamic memory means it allocate the memory in softwre development in c++ .The program is loaded into system into stack ,heap,and code codeon Quantum dots ABR Quantum gate cryptography.. iv)Shared Memory controller/Direct Memory Access(DMA) It is a method of transforming data from computer’s random acess memory (RAM) to another allocation without processing the central processing unit (CPU). Many hardware use DMA such as disk drive controller ,graphics cards,network cards and sound cards. Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION v)Graphical processing Unit (GPU) vi)2D Graphics processor vii)DSP ( Digital Signal Processor) It is microprocessor chip with optimisation of digital signal processing .It is used for audio signal processing telecomunications ,digital image processing,radar,sonarand speech recognition system.It uses the data compression with dicrete cosine transform(DCT) which uses matrix operation of convolution for filtering,dot product,polynomial evaluation.It is used for streaming data. viii)Video Accelerator ix)Multi pipe Display Sub System(DSS) It is a device or a et of devices which controls input or output process of the display. x)AzadBinRajib (ABR) quantum gate A new type of Quantum gate works on quantum dots.Detailes are provided in Sec. 2. xi)Image signal processor xii)Audio processor xiii)Cache memory (L2,L3,L4) xiv)L3 cache Boot secure ROM It is input/output based booting cache memory wich works on read only memory (ROM). xv)M Shield security tech SHA-1/SHA-2 It is secure Hash algorithm , which is discussed in the Sec.1.2. xvi)MDS/3 (MicroarchitecturalData Sampling ).It uses hyper-threading and leak data acess protection boundaries.It reads the data buffer filter to operating systems,web browsers and microcode are necessary. xvii)Digita Encryption System(DES) It is digtal encryption based system which uses RSA encryption. xviii)RNG (Random Number Generator) It is adevice that generates random numbers or symbols that be predicted.It uses Hash algorithm or in creating a mortized searching and sorting algorithm. xix)Quantum dots asymmetric Encryption(QDAES) It deals with Asymmetric Encryption System-512 for 512 qubits with Qantum dots based AzadBinRajib (ABR) Quantum gate.The detailed study is given in Sec.2 . Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION xx)Quantum dots key distribution system.It is also based on AzadBinRajib (ABR) Quantum gate cryptographic key distribution theory as public and private key distribution which is discussed in Sec.2 xxi)Crpto DMA- Along with SHA-2 hash and DMA capability designed for fast integration,low gate count and fulltransforms.It provides a reliableand cost effectiveembedded IP solutionthat is easy to integrate high speed processing pipeline.It boosts the performance ofn host processor.Secondly ability to store keysinan integrated RAM via DMA andkeeps it inaccessible to others but usuable to for the host/application. It provides hardwre cryptographic algorithm implimentation for optimal performance,good user experience,battery lifetime and robust security.Due to flexibilitynof its configuaration such as 3G/4G/LTE/VoLTE/QDLTE/ QDVoLTE specific version are avialable on request inside secure basic IP modules. xxii)Secure WDT( Watch Dog Timeer) or COP (Computer operating properly)It works when any malfunction occurs it detects and recover,rests the timer to prevent from elapsing or “time out”.If it fails to recover or time lapse and show “time out” signal ,to intiate corrective action/actions ,thereby keeping the timer at normal process of restoring to normal system operations. xxiii)PKA ( Public Key Address It is using the Public key cryptography for generation of Address by using the cryptography. 2.2. ABR Quantum Gate Algorithm I) DSP algorithm depends on Far Infra-Red (FIR) filters, Fast Fourier Transform (FFT), specialized instruction for modulo addressing in ring buffers and bit-reversed addressing mode for FFT cross-referencing. Data instructions is saturation arithematics for operation with Minterms and Maxterms. Fixed point operations to speed up arithematics processing. Single cycle operations to increase the benefits of pipe lining. XMOS produces a multi-core multi-threaded line of processor well suited for DSP operations. II) Hash algorithm (Hash function) : It is used to checksums,check digits,finger prints,lossy compression,randomization function error correcting codes and ciphers.Hashing is computationally and acess to storage space efficiencyfrom data which avoids the non-linear access of time of ordered and unordered lists and structured trees.Sometimes may need exponential storage space or data access having large or variable length keys. III) Shor algorithm It is quantum algorithm to know about the superposition as well as the up or down state of quantum computer.It solves the integer Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION factorization specically it quantum gates for fast calculation uses quantum Forier transformand modular exponetial=tion by repeated squaring without succumbing to quantum noice or other quantum decoherence.. IV) Grover’s algorithm It helps in searching the combination of gates to be used for quantum computation. V) ABR Quantam Gate Algorithm It is designed to find the eigen values for its higher state by itself using the superposition state aas a) Initialize the sytem to the state b) | 𝑨𝑩𝑹 > = 𝟏 𝒏 𝟐𝟐 𝒏−𝟏 𝟐 ∑𝒃=𝟏 |𝒃 > c) Perform following “ABR iteration” r(n) .The function r(n) which is asymtotically O(2𝑛/2 ) which is much faster then the Grover’s algorithm. d) Then Grover’ algorithm is observed for quantum oracle operator finds the 𝑰 𝟎 ] ,where alpha = ABR gate. Unitary matrix 𝑼 = [ 𝟎 𝜶 e) The unitary operator works on |𝑨𝑩𝑹 > . 𝑼 |𝒃 > → |𝑨𝑩𝑹 > |𝒃 + 𝒇(𝑨𝑩𝑹) > Where |ABR> is index qubit here we see that 0th state flips to 1 th state 𝑼 |𝑨𝑩𝑹 > → (−𝟏)𝒇(𝑨𝑩𝑹) |𝑨𝑩𝑹 > . 3. ABR Quantum Gate QDKD Encryption ABR Quantum gate based quantum dots key distribution encryption deals with two type of keys a) Public key distribution b) Private key distribution The public key is given and known universally to whom so ever a message is sent whereas it is not possible to decrypt the message as he is not the intended person who have to receive the message. Only private key holder will be in position to decrypt the message. For knowing these we must have the Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION knowledge of the qubits and quantum registers, entanglement, addresses, memory allocation etc. 3.1. Qubits and quantum registers of ABR Quantum Gate here we are considering ABR quantum gate as basic element used for QKD to get register we have |𝐀𝐁𝐑 >= 𝛂 ( and 𝜷 ( |𝟏> √𝟐 |𝟎> +|𝟏> √𝟐 ) + 𝛃( |𝟎> +|𝟏> √𝟐 ) where |𝟏> √𝟐 ) are two basis state 𝛼 , 𝛽 are complex numbers. A state of a quantum registers consists of n qubits is 𝟏 𝟐𝒏−𝟏 𝒏 ∑𝒕=𝟏 𝜶𝒕 |𝒕 > | 𝑨𝑩𝑹 > = 𝟐𝟐 = 𝟏 𝒏 𝟐𝟐 𝒏−𝟏 ∑𝟐𝒕𝟏 𝒕𝟐………..𝒕𝒏∈(𝟎,𝟏) 𝜶𝒕𝟏 𝒕𝟐 ……..𝒕𝒏 |𝒕𝟏 𝒕𝟐 … … … 𝒕𝒏 > Where the complex values 𝜶𝒕𝟏 𝒕𝟐……..𝒕𝒏 are required for normalization condition. ∑ |𝜶 𝒕 |𝟐 = 𝒕∈(𝟎,𝟏)𝒏 𝟏 𝟐𝒏−𝟏 ∑ |𝜶𝒕𝟏𝒕𝟐 ……..𝒕𝒏 | 𝒏 𝟐𝟐 𝒕𝟏 𝒕𝟐 ………..𝒕𝒏 ∈(𝟎,𝟏) 𝟐 =𝟏 The state |ABR> is a superpositionof computational basis sate |𝒕𝟏 𝒕𝟐 … … … 𝒕𝒏 > 𝒕𝟏 𝒕𝟐 … … … . . 𝒕𝒏 ∈ (𝟎, 𝟏) of the quantum register. The number 𝜶𝒕𝟏𝒕𝟐 ……..𝒕𝒏 ’s are the probability of |ABR> we get | 𝑨𝑩𝑹 > = 𝟏 𝒏 𝟐𝟐 𝒏−𝟏 𝟐 ∑𝒃=𝟎 𝜶𝒕 |𝒕 > fo the integers 𝒕𝟏 𝟐𝒏−𝟏 + 𝒕𝟐 𝟐𝒏−𝟐 + ⋯ … … . . +𝒕𝒏 𝟐𝟎 Identified with its binary representation 𝒕 = 𝒕𝟏 𝒕𝟐 … … … . . 𝒕𝒏−𝟏 Or can be written as column vector Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION 𝜶𝟏 |𝑨𝑩𝑹 > = [ … . ] 𝜶𝟐𝒏−𝟏 Several registers can be put to get the generalized register for n qubits state. | 𝑨𝑩𝑹𝒊 > = 𝟏 𝟐𝒏−𝟏 𝒏 ∑ 𝟐𝟐 𝒃=𝟎 𝜶𝒊,𝒕 |𝒕(𝒊) > Be an 𝒏𝒊 qubit state for each 𝟏 ≤ 𝒊 ≤ 𝒌 Then their tensor product is given by 𝒌 |𝑨𝑩𝑹𝟏 > ⋯ … … . |𝑨𝑩𝑹𝒌 >= ∏ |𝑨𝑩𝑹𝒊 > 𝒊=𝟏 = 𝟏 𝒏 𝟐𝟐 𝒏−𝟏 ∑𝟐𝒕𝟏 𝒕𝟐………..𝒕𝒏∈(𝟎,𝟏) 𝜶𝒕𝟏 𝒕𝟐 ……..𝒕𝒏 |𝒕𝟏 𝒕𝟐 … … … 𝒕𝒏 > For k times |𝑨𝑩𝑹 >∗𝒌 for |𝑨𝑩𝑹 > ⋯ … … . |𝑨𝑩𝑹 > 𝒌 𝒕𝒊𝒎𝒆𝒔 Entanglement is crucial feature of multiple qubits, say two qubits entanglement is |𝑨𝑩𝑹 >= 𝜶 ( |𝟏𝟏 > √𝟐 ) + 𝜷( |𝟎𝟎 > +|𝟏𝟏 > ) √𝟐 2.2 ABR Quantum Gate To realize the quantum computer by quantum circuits consisting of quantum gates n qubits can be described as 𝟐𝒏 𝑿 𝟐𝒏 unitary matrix, so that 𝑼 𝑼+ where 𝑼+ is transpose of Unitary matrix is known as Hermitian matrix here 𝑼𝒊𝒋 changes to 𝑼𝒋𝒊 . 𝒏−𝟏 𝑼 = 𝑼𝒊𝒋 𝟐𝒊,𝒋=𝟎 Is a quantum gate then the outcome of performing 𝑼 on |ABR> is the nth state Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION |𝑨𝑩𝑹 > = [ Where 𝑨𝑩𝑹𝟏 … . ]= U|ABR> 𝑨𝑩𝑹𝟐𝒏−𝟏 U|ABR> is given according to the usual matrix multiplication | 𝑨𝑩𝑹𝒊 > = 𝟏 𝟐𝒏−𝟏 𝒏 ∑ 𝟐𝟐 𝒃=𝟎 𝑼𝒊𝒋 |𝑨𝑩𝑹𝒊 > For I =0, 1, 2,3,…… 𝟐𝒏−𝟏 we get 𝟏 |𝑨𝑩𝑹 >= √𝟐 ( Let 𝟎 𝟏 𝟏 ) for a single qubits . 𝟏 be a gate acting on the register for each 𝟏 ≤ 𝒊 ≤ 𝒌. Then the tensor of the big register formed by the k registers is defined by 𝒌 𝒌 𝒌 ∏ 𝑼𝒊 ∏ |𝑨𝑩𝑹𝒊 > . = ∏ 𝑼𝒊 |𝑨𝑩𝑹𝒊 > 𝒊=𝟏 𝒊=𝟏 𝒊=𝟏 Together with linearity, where | 𝑨𝑩𝑹𝒊 > Is a state of the ith register for each i. we can write 𝑼∗𝒌 for 𝑼 ∗ 𝑼 ∗ 𝑼 … … … . 𝑼 𝒌 𝒕𝒊𝒎𝒆𝒔 2.3. ABR Quantum Gate Algorithm It is discussed in Sec.2.2 (v) 4. Discussion and Conclusion It has been observed in the present study that the ABR quantum gate follows all necessary requirements needed for the microprocessor to work is discussed in Sec1. The details of microprocessor LAQUIT microarchitecture is discussed in Sec1.1 whereas the algorithm involving the LAQUIT microprocessor is discussed in Sec .1.2 .In this section we have discussed that ABR Quantum Gate algorithm a novel one is discussed which involves certain steps of Grover’s algorithm. The nth quantum state of ABR quantum gate is shown how it operates. In Sec.2 ABR Quantum Gate Quantum dot based Key distribution encryption system involving Asymmetric Encryption (AES-512) is used to study the entanglement for n quantum states, initially shown for 2 qubits state and so on. Then it is shown for k times qubits .It works as it is desired for our calculations in quantum computing. Unitary Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION matrix and Hermitian matrix is how it will work on nth or kth state of qubits. Providing us the Big quantum register with n qubits. Thereby showing memory allocation .The results will be same as for Yin [1] , Wilde[2] and Neilson [3] References [1]Ying,M 20010 Artificial Intelligence 174 162 [2] Wilde,M.M 20003 Quantum Information Theory , Cambridge University Press, Cambridge (UK) ISBN 9781139525343 , doi.org/10.1017/CB09781139525343 [3] Nielson,M.A 2000 “Quantum computation and Quantum Information: Cambridge University press,Cambridge (UK ). ISBN 0521632358.OCLC Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Chapter 4 Data Compression by AzadBinRajib (ABR) Quantum Gate A.B.Rajib Hazarika* Department of Mathematics,Diphu Government College,Diphu Assam,India782462 [email protected] , [email protected] Abstract AzadBinRajib (ABR) Quantum Gate is studied for data compression and loosy compression with Quantum Shannon theory, von-Neumann theory and with Schur -Weyl transform. The study is done for n length string or word by finding the density matrix as well for nth qubits state with fidelity. Keywords : Quantum gate, data compression,spectral density, fidelity 1. Introduction Data compression for classical computers with 1’s and 0’s are easy enough but now the challenge comes when we are considering the Quantum computers.Dataa compression means for a string of a data the frequency of 1’s and 0’s are checked which might require just a certain bits (binary values) .Registering the information about the order of those bits (1 and 0) would need slightly larger string, but still will be shorter than the original sequence of message (data). For Quantum data which involves qubits (||0>, |1>) or the 0th state and 1th state becomes cumbersome to calculate the superposition state of the quantum data (message).Here instead of 1’s and 0’s we will be using the probability amplitude of the quantum data. So far many authors have used the data compression to study with neo-classical Shannon theory Wilde [1]. While some of the authors tried to study the use of von-Neumann theory by Nielson [2], which uses he Eigen values for the compression of quantum data. Literals such as SOP (sum of products), POS (product of sums) are also used to study the present context. In past the data compression for classical computers with internet protocol was studied by Hazarika [3], Zubair et. al {4-7], similarly we will try to obtain results for quantum data compression also. Netflix uses a new trick to analyze each shot in a video and compress it without losing the image quality thus reducing the data it uses the encoding method known as “Dynamic Optimizer “.It was tried on a 555kbps stream of video to compress it to run on 100kbps stream without affecting the image quality rather improved the image quality by applying dynamic optimizer. Netflix uses data compression for large scale streaming .It stored 1 Petabyte of data on Amazon. WhatsApp messenger use cross platform messages and video over IP is another method of data compression, for voice codec they used Opus [12-14] * Corresponding author. Tel.:+0-000-000-0000 ; fax:+0-000-000-0000 ; e-mail: [email protected] . Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION which uses the modified discrete cosine transform (MDCT) and linear predictive coding (LPC) audio compression algorithm .Instagram use both photo and video for data compression [15].Twitter uses data compression for messages, photo, video. Nishida [16] has given that tweet classified by data compression can happen. Zoom video communication developed a coding that uses multiple video-over-Internet Protocol with data compression and multiple conferencing up to 9-10 people at a time. Lisovoki [17] have very recently tried for data compression in sonar and radar systems for remote sensing purposes. Rozema et al [18] have shown the Quantum data compression with qubits encoding. Reskill [19] have published Quantum computation theory describing data compression also in it. The objective of this study encompasses on the vivid use of spectral density with wavelet theory on AzadBinRajib (ABR) Quantum Gate for data compression. 2. Data compression by ABR Quantum gate Present study is with the spectral density using Wavelet theory on AzadBinRajib (ABR) Quantum Gate. 2.1. ABR data compression The density matrix 𝒏−𝟏 |𝑨𝑩𝑹𝒏 > = ∑𝟐𝒏=𝟏 𝑷𝒏 |𝑨𝑩𝑹𝒏 > (1) Literal for 2 qubits |𝑨𝑩𝑹 > = 𝜶|𝟎𝟎 > + 𝜷𝟏 |𝟎𝟏 > +𝜷𝟐 |𝟏𝟎 > (2) Spectral density matrix is given by |𝑨𝑩𝑹𝒏 > = 𝜶𝒏 ( |𝟏> √𝟐 ) + 𝜷𝒏 ( |𝟎> + |𝟏> √𝟐 ) (3) 𝟐𝒏−𝟏 𝝆 = ∑ 𝑷𝒏 𝝆𝒏 𝒏=𝟏 ∑ 𝝆𝒏 |𝜶𝒏 |𝟐 𝝆=[ ∑ 𝝆𝒏 |𝜶𝒏 𝜷′𝒏 | ∑ 𝝆𝒏 |𝜶𝒏 ′𝜷𝒏 | ] ∑ 𝝆𝒏 |𝜷𝒏 |𝟐 (4) Letter written in Quantum Shannon theory for Quantum data compression is 𝑳𝒄𝒐𝒑𝒎𝒓𝒆𝒔𝒔𝒊𝒐𝒏 = 𝑳𝟎 . . ∑𝒏 𝑷𝒏 𝒍𝒐𝒈𝟐 𝑷(𝒏) (5) Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Letters written in Quantum von –Neumann Theory for Quantum data compression 𝑲𝒄𝒐𝒑𝒎𝒓𝒆𝒔𝒔𝒊𝒐𝒏 = −𝑲𝟎 . . ∑𝒏 ℵ𝒏 𝒍𝒐𝒈𝟐 ℵ(𝒏) ℵ Is Eigen value (6) For 2 qubits 𝑺(𝝆)𝒄𝒐𝒑𝒎𝒓𝒆𝒔𝒔𝒊𝒐𝒏 = − ∑𝒏 𝑷𝒏 𝒍𝒐𝒈𝟐 𝑷(𝒏) Is Shannon entropy The quantum states will be |00>, |01 >, |10 >, |11 > |𝑨𝑩𝑹 > = 𝜶𝜶|𝟎𝟎 > + 𝜶𝜷𝟏 |𝟎𝟏 > +𝜶𝜷𝟐 |𝟏𝟎 > (7) As the last term with |11> is zero. For every system decode we need Fidelity 𝑭= 𝟏 𝟒 𝟐 ∑ 𝑷𝒏 |< 𝑳𝒋 | ∗ < 𝑳𝒋 |𝑪 >| + 𝑷𝒏 |< 𝑳𝒋 | ∗ < 𝑳𝒋 |𝟎𝟎 >| 𝟐 𝒋=𝟏,𝟐….𝒏 (8) Where represents compression of data. 2.2. We can derive for n qubits for ABR Quantum Gate it changes to |𝑨𝑩𝑹𝒏 > = 𝜶𝒏 ( |𝟏> √𝟐 ) + 𝜷𝒏 ( |𝟎> + |𝟏> √𝟐 |𝑨𝑩𝑹 > = ∑ ∏𝒏 𝜶𝒏 | ∏𝒏 𝒙𝒏 ∏𝒏 𝒙′𝒏 > ) (9) (10) Represents Sum of products (SOP) and the system with nth term of Fidelity we get 𝟏 𝑭= 𝒏 𝑷𝒏 |< 𝑳𝒋 | ∗ < 𝑳𝒋 |𝑪 >| ∑ 𝒋=𝟏,𝟐….𝒏 𝟐 𝟐 +𝑷𝒏 |< 𝑳𝒋 | ∗ < 𝑳𝒋 | ∏ 𝒙𝒏 >| 𝒏 (11) Dr.A.B.Rajib Hazarika,PhD,FRAS (Lond.),AES ,Assistant Professor, Diphu Govt. College,Diphu,Assam,India-782462 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION ∏𝒏 𝒙𝒏 Represents product of (0’s ) zero’s and ∏𝒏 𝒙′𝒏 represents product of 1’s (ones ) 𝑳𝒋 ,represents number of letters in word, 𝑷𝒏 represents probability amplitude of nth degree of freedoms whereas the sum all n probability amplitudes should be equal to 1. 𝒏−𝟏 𝟐 ∑𝒏=𝟏 𝑷𝒏 = 𝟏 (12) C represents compression data. Now we can compress data of n qubits having 𝟐𝒏 length string of message. 2.3. Quantum data compression by Schur- Weyl Transform for ABR Quantum Gate 𝒏−𝟏 𝝅 𝝅 𝒊=𝒋=𝟐 |𝑨𝑩𝑹𝒊𝒋 > = ∑𝒊=𝟏,𝒋=𝟏 | 𝐜𝐨𝐬 (𝒊) ( 𝟐 + 𝒋 𝟒 ) > (13) 𝒏−𝟏 𝝅 𝝅 𝝅 𝝅 𝟐 𝟒 𝟐 𝟒 𝟐 ∑𝒊=𝒋=𝒌=𝟎 | 𝐜𝐨𝐬 ( + 𝒊 ) 𝐜𝐨𝐬 ( + 𝒋 ) … … … … 𝒌 𝒕𝒊𝒎𝒆𝒔⟩ (14) which gives us the matrix for nth qubits. 3. Discussion and Conclusion In our study of the data compression with AzadBinRajib (ABR) Quantum Gate we found that it is behaving in similar fashion as in the classical case as well as that with Quantum data compression of a qubits ensemble by Rozema et. al [18] and in compliment of Wilde [1], Neumann [2] and Preskill [19] if considered with Hadamard gate instead of ABR quantum gate. Though other authors have not worked for n qubits we have tried to get the calculations done for n qubits on ABR quantum gate. References [1] Wilde,M.M (2003) “Quantum Inforation “,Cambridge University Press,Cambridge,UK [2] Nielson = 1 √2 (|0 > +|1 >) Ancilla qubit is used as control on the targets |fk> and |f’k>in Fredkin gate. ABR Quantum Gate applied on Ancilla qubit and finally the first qubit gets measured .If both states are same |0> result is measured. If both state are nearly orthogonal, the result can be either |0> or |1>. |𝜑0 >= |𝑎 > |𝑓𝑘 > |𝑓𝑘, > |𝜑0 >= |𝜑0 >= 1 √2 1 √2 (|0 > +|1 >) |𝑓𝑘 > |𝑓 ′ 𝑘 > (|0 > |𝑓𝑘 > |𝑓 ′ 𝑘 > +|1 > |𝑓𝑘 > |𝑓 ′ 𝑘 > ) After the Fredkin gate is applied we use ABR Quantum Gate to first qubit 1 |𝜑0 >= 2 (|1 > |𝑓𝑘 > |𝑓 ′ 𝑘 > +(|0 > +|1 > )|𝑓′𝑘 > |𝑓𝑘 > ) After sorting for |0> and |1> respectively 1 |𝜑0 >= 2 (|0 > |𝑓 ′ 𝑘 > |𝑓𝑘 > + |1 > (|𝑓𝑘 > |𝑓 ′ 𝑘 > +|𝑓 ′ 𝑘 > |𝑓𝑘 >) ) Now it is easy to see the state on |𝑓𝑘 > = |𝑓 ′ 𝑘 > Which are the desired measured states. ABR= a |1> + b (|0> + |1 >) DR .A.B.RAJIB HAZARIKA , PHD, FRAS (LONDON) ,AES ASSISTANT PROFESSOR , DIPHU GOVT. COLLEGE, DIPHU ASSAM,INDIA -782462 36 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION Chapter 6 Quantum Teleportation using ABR Quantum Gate If it story of two persons Alice and Bob who wants to send a message not digitally rather quantum wise .Alice tries to send ultimately “HELLO WORLD”. At first, Alice starts with sending a ‘0’ which in matrix form is 1 |0 > = [ ] 0 On using ABR Quantum Gate which passed through digital ‘0’ turns out to be as 𝟏 √𝟐 𝟎 [ 𝟏 1 0 𝟏 1 ][ ] = [ ] √2 𝟏 0 1 |1> 0 [ ] which is |0> state or and he understands that √2 1 √2 Alice sent to him ‘0’th state or 0 digitally. Then Bob receives 1 0 Similarly, when Alice sends ‘1’ digitally, quantum wise |1 > = [ ] using 1 ABR Quantum gate it gets converted to 1 1 𝟎 𝟏 0 [ ] [ ] = [ ] noted down state value is as received by Bob, then he √𝟐 𝟏 𝟏 1 √2 1 understands that Alice sent him 𝟏 1 “1th” state √2 |1 > + 1 √2 (|0 > +|1 >) = 1 √2 1 [ ] 1 Sending HELLO WORLD Each and every alphabet as well as word is at first converted to ASCII code to binary code .Then every each alphabet is taken as the string of 8qubits as ket |L> ,say. ASCII code for the words H=072=01001000 E=069=01000101 L=076=01001100 L=076=01001100 O=079=01001111 W=087=01010111 O=079=01001111 R=082=01010010 L=076=01001100 DR .A.B.RAJIB HAZARIKA , PHD, FRAS (LONDON) ,AES ASSISTANT PROFESSOR , DIPHU GOVT. COLLEGE, DIPHU ASSAM,INDIA -782462 37 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION D=068=01000100 As we considering 1 Qubyte=8 qubits= 23qubits 8 qubits make one alphabet here five alphabets are making a word “HELLO” and “WORLD”. |L0> = Represents the cursor |L1> = “H” =|01001000> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 6 |1> =8 √2 + 2 8 (|0 > +|1 >) |L2> = “E” =|01000101> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 5 |1> =8 √2 + 3 8 (|0 > +|1 >) |L3> = “L” =|01001100> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 5 |1> =8 √2 + 3 8 (|0 > +|1 >) |L4> = “L” =|01001100> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 5 |1> =8 √2 + 3 8 (|0 > +|1 >) |L5> = “O” =|01001110> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 4 |1> =8 √2 + 4 8 (|0 > +|1 >) |L6> = “Blank space” =|11111111> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 0 |1> =8 √2 + 8 8 (|0 > +|1 >) |L7> = “W” =|01010111> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 3 |1> =8 √2 + 5 8 (|0 > +|1 >) |L8> = “O” =|01001110> DR .A.B.RAJIB HAZARIKA , PHD, FRAS (LONDON) ,AES ASSISTANT PROFESSOR , DIPHU GOVT. COLLEGE, DIPHU ASSAM,INDIA -782462 38 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 4 |1> =8 √2 + 4 8 (|0 > +|1 >) |L9> = “R” =|01010010> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 5 |1> =8 √2 + 3 8 (|0 > +|1 >) |L10> = “L” =|01001100> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 5 |1> =8 √2 + 3 8 (|0 > +|1 >) |L11> = “D” =|01000100> = 𝛼1 |0𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > + 𝛽1 |1𝑡ℎ 𝑠𝑡𝑎𝑡𝑒 > 6 |1> =8 √2 + 2 8 (|0 > +|1 >) Now to represent the string or a alphabets into words then to sentence. At first the word “HELLO” is represented by alphabets taken together as all strings of messages. “HELLO” = |L1>*|L2>*|L3>*|L4>*|L5> “Blank space” |L6> “WORLD”= |L7>*|L8>*|L9>*|L10>*|L11> As we have described the words along with the blank space now we will form the full sentence as combination of words. “HELLO” “Blank space” “WORLD” |L> = |L1>*|L2>*|L3>*|L4>*|L5>*|L6>*|L7>*|L8>*|L9>*|L10>*|L11> This |L> represents the string of messages as “HELLO” blank space “WORLD” which will be teleported and we can read it as “HELLO WORLD” Which is the teleported message. DR .A.B.RAJIB HAZARIKA , PHD, FRAS (LONDON) ,AES ASSISTANT PROFESSOR , DIPHU GOVT. COLLEGE, DIPHU ASSAM,INDIA -782462 39 AZADBINRAJIB (ABR) QUANTUM GATE FOR QUANTUM COMPUTING AND QUANTUM INFORMATION THANK YOU DR .A.B.RAJIB HAZARIKA , PHD, FRAS (LONDON) ,AES ASSISTANT PROFESSOR , DIPHU GOVT. COLLEGE, DIPHU ASSAM,INDIA -782462 40