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Quantum Computing and Qbit Cryptography Patrick Lii 5 May 2009 Physics 138

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Outline Motivation for Quantum Computing A Review of Classical Computers Qbits and Quantum Algorithms Quantum Cryptography Conclusion

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What is a Quantum Computer? A quantum computer (QC) is a computational device which operates on data using quantum algorithms QC in proof-of-concept stage Current motivations: Cryptography Factorization Database searching http://www.acceleratingfuture.com/michael/blog/category/random/page/2/

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Classical versus Quantum Computers Example: Large number factorization QCs ->advantage of parallelism qbits are in superpositions of states backwards compatible w/ classical algorithms

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Performance Advantage of QCs classical: ~1-10 gflops of computing power quantum: ~10 tflops Factorization speed: for an integer N with size: the factorization time of a classical comp is: For a QC ORNLs Jaguar Supercomputer

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Speed Comparison Classical Computer Quantum Computer 1 hr 4.11 days 7.47 years ~73000 years 1 hr 8 hrs 2.76 days 21.3 days Assume CC and QC can factor a 78 digit number (n = 256) in 1 hour QC easily defeats RSA encryption! n = 256 (AES) n = 512 n = 1024 n = 2048 (RSA)

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Outline Motivation for Quantum Computing A Review of Classical Computers Qbits and Quantum Algorithms Quantum Cryptography Conclusion

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The Classical Computer Classical bits (cbits): 0 or 1 2 cbits 4 states, 3 cbits 8 states n cbits 2 n states data is represented in binary 138 10001010 q 01110001

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Classical Operations Based on logic gates Example: 1-bit gate NOT gate X:0 1 X:1 0 2-bit gates: AND/NAND OR/NOR XOR/XNOR AND Gate Bit 1Bit 2Output 111 100 010 000 XOR Gate Bit 1Bit 2Output 110 101 011 000

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Classical Algorithms All 1, 2, 3-cbit gates together form universal set classical algorithm: a complex operation that uses a sequence of classical gates http://www.inetdaemon.com/img/gates.gif

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Outline Motivation for Quantum Computing A Review of Classical Computers Qbits and Quantum Algorithms Quantum Cryptography Conclusion

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The Quantum Bit (Qbit) Unlike cbits, state of a qbit is a superposition of 1 and 0: w/ normalization condition: In matrix form: n qbits are in superposition of 2 n states qbits can be any two-level quantum system

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The Quantum Bit In general, for n bits: w/ normalization:

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Purely quantum property of qbit Two qbits are entangled if wavefunction cannot be written as product of 1 qbit states Qbit Entanglement

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Quantum Logic Gates All quantum operations are unitary UU = U U = 1 Gate can be any unitary quantum operator Ex: quantum NOT gate 2-bit gates can operate on entangled pairs Quantum logic gate using lasers

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Important Quantum Gates Conditional Not Hadamard Transformation CNOT Gate Bit 1 InBit 2 InBit 1 OutBit 2 Out 1110 1011 0101 0000 π/8 Phase Gate These 3 gates form a universal set

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The Measurement Gate Borns Law Given qbit: Probability of measuring state = amplitude squared M gate Most important gate in QC Collapses qbit wavefunction Result based on probability We may not always get correct answer Irreversible!

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Quantum Algorithms Similarly to classical algorithms, quantum algorithms are sequences of quantum gates In general, QCs have a simple processing structure: Complex processing lies in the U Gates

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Shors Algorithm Developed by Peter Shor in 1994 Efficient factorization of large numbers RSA Encryption Based on multiplying 2 very large prime numbers (~200 digits each) CCs cannot factor this in a reasonable time However, using Shors algorithm, a QC can Lots of interest from government

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Physical Implementations of QCs In 2001, a group at IBM led by Vandersypen created a 7-qbit QC NMR implementation Used it to demonstrate Shors algorithm by factoring 15 into 3 and 5 Other possibilities Optical lattices Polarized light Diamond based Superconductor (SQUIDs) Trapped ion any two level system w/ orthogonal bases Biggest problem in implementation of QC: controlling decoherence of qbits

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Outline Motivation for Quantum Computing A Review of Classical Computers Qbits and Quantum Algorithms Quantum Cryptography Conclusion

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Quantum Cryptography (BB84) Called BB84: Bennett and Brassard 1984 Method of secure key distribution Created using only 1-qbit gates Can be implemented using current tech (transmission w/ polarized light) interception can be detected

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Message Security Say we want to transmit the number 83 In binary: 1010011 (7-bits) We securely (and randomly) generate a key w/ equal bit-length take: 1011011 We then use this key to encode the message flip message bits everywhere the key equals 1 Message becomes 0001000 impossible for someone w/o a key to unencrypt this Cryptography comes down to: Random key generation Secure key distribution

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Key Generation I Alice sends Bob a long stream of photons (qbits) She randomly assigns each a type: circular or linear pol Then, randomly assigns a polarization sub-state based on the type LH or RH for circ X or Y for linear Example: Alice sends 8 qbits |<<|< Legend: X, 0 bit | Y, 1 bit >RHCP, 0 bit

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Key Generation II Bob randomly decides on a linear or circular measurement of each incoming photon For measurement, Bob chooses: O+O++OOO Legend: +Linear OCircular And he measures: <> for reference, Alice sent: |<<|<

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Key Generation III Bob calls Alice and tells her his choice of measurement (circ or lin) for each photon Alice then tells Bob which of his types agree with her transmission types NYYYNNYN They then use the agreeing values as a key In example, A&B have 4 agreeing qbits: |<-< Their key is: 1101

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Eavesdropping for Eve to eavesdrop on A&Bs transmission, she must also randomly make circ or lin measurements of each photon This changes polarization of about half the qbits 1/4 th of Bobs result will not agree w/ Alices prep A&B can compare some check bits over the phone to see if anyone is eavesdropping

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Qbit Interception Suppose Eve uses a more sophisticated attack: intercepts the transmission processes it in a QC restores it to original state and sends it back off to Bob This is defeated by the no-cloning theorem Forbids creation of identical copy of an arbitrary state Eve gets no useful information from her interception

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Outline Motivation for Quantum Computing A Review of Classical Computers Qbits and Quantum Algorithms Quantum Cryptography Conclusion

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Future Developments in QC Largely in proof-of-concept stage formidable technological obstacles Still need to: discover more algorithms overcome decoherence of qbits Deeper understanding of QM may make it easier to do this We are decades away from truly powerful QC (~2050?)

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Conclusion 1.Quantum computers are based on enacting quantum operations on qbits 2.Quantum operations are simply unitary operators in the Hilbert space of the system 3.QCs have the potential to vastly outperform classical computers because of the QM nature of their operations 4.QCs are still many years off; however, they will fundamentally change computation as we know it 5.Qbits can also be employed in generating an undefeatable cryptography scheme which may prove useful once RSA encryption is defeated by QCs

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References Quantum Computing (General) Kaye, Phillip, Raymond Laflamme, and Michele Mosca. An Introduction to Quantum Computing. 1st ed. Oxford: Oxford University Press, 2007. Print. Lieven M.K. VandersypenLieven M.K. Vandersypen et al. (1999). "Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing". Phys. Rev. Lett 83: 1054–1057. Mermin, David. Quantum Computer Science. 1st Ed. Cambridge: Cambridge University Press, 2007. Print. [great introductory resource for Quantum Computers from a professor at Cornell, not rigorous however] Classical Computing http://joshblog.net/projects/logic-gate-simulator/Logicly.htmlhttp://joshblog.net/projects/logic-gate-simulator/Logicly.html [cool logic gate simulator] Quantum Cryptography C. H. Bennet and G. Brassard, Quantum Cryptography: Public key distribution and coin tossing, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984) http://fredhenle.net/bb84/http://fredhenle.net/bb84/ [BB84 transmission simulator] Shors Algorithm Shor, P. (1994) Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20-22, 1994.

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Let: |Ф μ >, μ = 0, …, 3 = four states of Alices qbits (X, Y, RH, LH) |ψ> = initial state of qbits on Eves QC Since qbit must emerge in original state: Eve must find a U that yields four distinct |ψ μ >

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