# McKay Graybill.  They already exist  Different models and ideas  Quantum Parallelism  Measurement is tricky, inherently imprecise.

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McKay Graybill

 They already exist  Different models and ideas  Quantum Parallelism  Measurement is tricky, inherently imprecise

 Represented by a unitary matrix ◦ Controlled-Not ◦ Controlled-Controlled-Not (Toffoli) ◦ Hadamard

 Can simulate any combination of gates  Example (Steane, 1998): C NOT =

 “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

 M, a, m ∈ ℤ : M 2 < 2 m < 2M 2  Qubits required: m + ⌈log 2 (M)⌉  f(x) = a x mod M, for all 0 < x < 2 m – 1 ◦ Measure the state to align amplitude function  Fourier transform, Euclidean verification, repeat as necessary

 O(√n) search on unsorted dataset of size n  Requires N + 1 qubits, where 2 N ≥ n ◦ Extra qubit is Boolean result of test function P(x)  Custom transformation  Repeat transformations to increase reading state where P(x) is true

 Truly Random Number Generation  Data Key Distribution and Truly Secure Communication  Constraint Satisfaction Problems  Simulation of actual physical environments  Probably more…

 Error threshold  Scalability  Specialized ◦ Math and Physics is still the API  Shor’s 2003 Paper

 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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