MIDTERM EXAM IN SPRING 2004.

1. Using Maslov/Miller/Dueck algorithm synthesize Kerntopf gate from Toffoli and Feynman gates. Explain your procedure step-by-step. Discuss its strength and weaknesses on this an possibly other example.
2. Illustrate De Morgan's Laws using Quantum Array notation. What is the use of this transformation?
3. Feynman Gate is sometimes called a Quantum NAND. If Feynman is a Quantum NAND what would be a Quantum NOR? Prove that this new gate is reversible. How to realize it in some well-known quantum cell library?
4. Realize from quantum gates a circuit that converts a binary function to a Reed-Muller representation of this function. Function is given originally as a binary vector of its coefficient. For instance function A AND B is given as a vector [0,0,0,1] and function A OR B is given as a vector [0,1,1,1]. The result is a vector of PPRM coefficients. For instance [1,1,0,1] corresponds to 1 XOR A XOR AB, or to 1 XOR B XOR AB, depending on term ordering. You can assume arbitrary term ordering. The circuit should be for functions of 4 variables.
5. Design a quantum circuit for a two-by-three multiplier. No garbage bits allowed, all inputs must be repeated on output. You can have arbitrary width of the quantum cascade and arbitrary number of ancilla bits that have constant values on both primary inputs and primary outputs.
6. Using only Feynman, Fredkin and Toffoli gates realize a function with 3 inputs (A,B,C) and 3 outputs (X,Y,Z) that realizes symmetric functions. S^{1,2} (A,B,C), S^{0,3} (A,B,C), S^{1,3} (A,B,C). Minimize garbage. Minimize total number of inputs to gates. Do not try to minimize the width of the circuit.
7. Assuming that the bit C is a feedback bit and bits A and B are input/output, calculate and draw a diagram of a probabilistic finite state machine corresponding to this function.