Q Jeff Kinne

Under direction of Dr. Gary Lewandowski. Quantum computing Jeff Kinne Under direction of Dr. Gary Lewandowski.

Church-Turing thesis What is a Turing Machine? Nothing more powerful than a TM? Then maybe we can’t deal with NP-hard problems…

Information theory Computation deals with information. Ultimately, information must be stored physically. Physics will determine the limits of computation. Quantum mechanics?

Quantum computer bits and gates are the basis for our computers. A qubit has the following form: |a|(1,0)+|b|(0,1). a and b are complex numbers. (1,0) is a basis vector and corresponds to the logical state of 0. (0,1) corresponds to 1.

Quantum computer The modulus of a is the probability that the qubit will be measured as (1,0) = 0. The qubit can exist in a superposition of the states 0 and 1 but will always become 0 or 1 when it is measured. Put many qubits together to form a register of quantum bits. A quantum gate maps a superposition of qubits to another superposition.

Deutsch’s algorithm Function mapping {0,1} into itself. There are 4 such possible functions. Seek to determine whether the function is one to one. This could be done by evaluating f(0) XOR f(1).

Fowee’s algorithm Prepare a superposition of all possible two bit states by sending 0,1 through the H gate: H(0) = (1/2)0+(1/2)1, H(1) = (1/2)0-(1/2)1. Put that superposition through the following gate: G(i,j) = (i, j XOR f(i)) Then put it through H again. In the end, we’ll get 1 for the first bit if the function is one to one and 0 otherwise. Measure a global property of the system by evaluating f only once.

Cryptography Method for key distribution that is possibly unbreakable. Pair of properties that cannot simultaneously be measured. Alice sends particles to Bob and picks which property to measure for each. Bob also randomly picks which property to measure.

Cryptography Afterwards, they figure out over a normal channel for which bits they measured the same property. Those bits are the key that they can use for encrypting. They will detect any listening by Eve. If she was listening, they start over.

Other quantum algorithms Factoring integers efficiently. Searching an unsorted list in square root time. Not all that many yet.

Feasibility - hardware 7 qubit register. But need at least 419 qubits to be like a 42 bit computer. NMR technology for qubits. Also need to have the system be isolated long enough to perform many of calculations. Many advances necessary.

Feasibility – software There aren’t that many algorithms yet. Perhaps a compiler-like program could translate a circuit into an algorithm. Systems programs? General purpose quantum paradigm?

Conclusion Results at http://www.cs.xu.edu/~heile/quantum/ soon, also links to more information. TM may die a horrible death but not for a while (at least 30 years). Much potential for QC, but not realized yet.