Midterm Exam for Quantum Computing Class Marek Perkowski

Problem 1 – Analysis of quantum permutative circuits 1.Write equation for every output of the above reversible circuit 2.Draw a Karnaugh Map of this circuit 3.Write a truth table of this circuit 4.Represent the truth table as a single permutation vector.

Problem 2 – Analysis of quantum circuits 1.Design a quantum circuit1 with 3 inputs (from top), a, b, c and three corresponding outputs (from top) – A, B, C. 2.The circuit should have entanglement when a=0 and have no entanglement when a=1. 3.Analyze the circuit using two methods to prove that the circuit operates as in point 2. 1.By operating on matrices using Heisenberg’s Notation. 2.By operating on formulas using Dirac’s Notation. (you should comment also in English) 3.Be very specific what is entanglement and where the circuit represents entanglement. 4.(bonus) Draw a 4-input 4-output circuit2 with maximum entanglement in which all qubits are involved. abcabc ABCABC abcdabcd ABCDABCD Circuit 1 Circuit 2

Problem 3 – Design of an oracle. 1.Given is a vector V of four numbers, each number has 2 bits. 2.The vector V can be sorted (in non-decreasing order) or be not sorted like this. Ex. A vector [0,0,0,0] is sorted in non-decreasing order so we treat it as “sorted” for the oracle. 3.If the vector is sorted, the output of the oracle is “1”. Otherwise the output of the oracle is “0”. 4.Using classical reversible gates draw the oracle for this problem. 5.Using a concept of a counter to exercise all possible binary states, explain how this circuit distinguishes the sorted from the not-sorted vectors. Explain in English. 6.(This is a bonus question). If you put Hadamard gates on every input variable, and initialize all inputs to |0>, what would be the quantum state on the output.