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Quantum Algorithms

Quantum Algorithms. McKay Graybill. Quantum Computers. They already exist Different models and ideas Quantum Parallelism Measurement is tricky, inherently imprecise. Quantum Gates. Represented by a unitary matrix Controlled-Not Controlled-Controlled-Not ( Toffoli ) Hadamard.

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Quantum Algorithms

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  1. Quantum Algorithms McKay Graybill

  2. Quantum Computers • They already exist • Different models and ideas • Quantum Parallelism • Measurement is tricky, inherently imprecise

  3. Quantum Gates • Represented by a unitary matrix • Controlled-Not • Controlled-Controlled-Not (Toffoli) • Hadamard

  4. Universal Quantum Computer • Can simulate any combination of gates • Example (Steane, 1998): CNOT =

  5. Shor’s Big Idea • “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer”, 1994 • Inspired by the work of Dan Simon, 1993-1994 • Computations are dependent upon physics • Finding the period of a function

  6. Shor’s Algorithm • M, a, m ∈ ℤ : M2 < 2m < 2M2 • Qubits required: m + ⌈log2(M)⌉ • f(x) = ax mod M, for all 0 < x < 2m – 1 • Measure the state to align amplitude function • Fourier transform, Euclidean verification, repeat as necessary

  7. Grover’s Search Algorithm • O(√n) search on unsorted dataset of size n • Requires N + 1 qubits, where 2N ≥n • Extra qubit is Boolean result of test function P(x) • Custom transformation • Repeat transformations to increase reading state where P(x) is true

  8. Other Ideas • Truly Random Number Generation • Data Key Distribution and Truly Secure Communication • Constraint Satisfaction Problems • Simulation of actual physical environments • Probably more…

  9. Current State of Research • Error threshold • Scalability • Specialized • Math and Physics is still the API • Shor’s 2003 Paper

  10. References • Doug Applegate • BACON, D., AND VAN DAM, W. 2010. Recent progress in quantum algorithms. Communications of the ACM, Vol. 53, No. 2, 84-93. • CHANG, K. 2012. I.B.M. researchers inch toward quantum computer. The New York Times. http://www.nytimes.com/2012/02/28/technology/ibm-inch-closer-on-quantum-computer.html • CORLEY, A. 2009. Quantum chip helps crack code. IEEE Spectrum. http://spectrum.ieee.org/computing/hardware/chip-does-part-of-codecracking-quantum-algorithm • FEYNMANN, R. P. 1986. Quantum mechanical computers. Foundations of Physics, Vol. 16, No. 6. • GROVER, L. K. 1996. A fast, quantum mechanical algorithm for database search. http://arXiv.org/quant-ph/9605043v3. • GUIZZO, E. 2010. Loser: D-Wave does not quantum compute. IEEE Spectrum. http://spectrum.ieee.org/computing/hardware/loser-dwave-does-not-quantum-compute • RIEFFEL, E., AND POLAK, W. 2000. An introduction to quantum computing for non-physicists. ACM Computing Surveys Volume 32 Issue 3, 300-335.SHOR, P. W. 1995. Polynomial-time Algorithms for Prime-factorization and Discrete Logarithms on a Quantum Computer. http://arXiv.org/quant-ph/9508027v2. • SHOR, P. 1996. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. http://arXiv.org/quant-ph/9508027v2. • SHOR, P. 2003. Why haven’t more quantum algorithms been found? Journal of the ACM Vol. 50 Issue 1, 87-90. • STEANE, A. 1998. Quantum computing. Rep. Prog. Phys.Vol. 61, No. 2, 171. • VAN METER, R., AND OSKIN, M. 2006. Architectural implications on quantum computing technologies. ACM Journal on Emerging Technology in Computing Systems Vol. 2 No. 1. 31-63.

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