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.