It’s exam time!!

Well, all two of you who actually read this blog may have wondered at my lack of posts this week (or maybe not).  In any case, I’m in the throes of giving exams, grading, putting together the next issue of The Quantum Times, doing some research, etc. etc. but should be a bit less busier soon.

However, I thought it might be interesting to post the final exam I gave in my Quantum Cryptography class (I always write my exams from scratch so I don’t care if future students see this and this is an exam I already gave).  It might stimulate some interesting discussion.

Here is a link to a PDF file of the exam, but I’ll post the questions below in order to try to stimulate a little discussion.

Question 1

Given the following circuit, what is the output state in column vector format?  What is the state (really a sub-state) of the first (top) qubit?  Suppose the input of the first (top) qubit was |1> instead.  What would the output state be now?  Describe how this demonstrates the controlled natured of the CONTROLLED–Z gate.  Are either of these states entangled?  Briefly explain why or why not.

Question 2

Suppose we have a quantum computer whose y-register has 6 qubits.  This computer is executing Shor’s factoring algorithm and that we are trying to factor the number 51.  Further suppose that measurement of the x-register has yielded the result c = 768 and take a to be 7.  Find the number of qubits, m, in the x-register.  Compute c/2^m and estimate how close to this a fraction j/r must be if c is to be a reasonably likely result.  By trial and error, find a fraction with a relatively small denominator (≤ 25) that satisfies this such that r is the correct period of our function (check this!).  What is the probability that a measurement on the x-register will yield a value of c = 768?  What are the prime factors of 51, then, given by Shor’s algorithm?

Question 3

Consider a quantum circuit with two qubits as its input.  Both qubits pass through individual Hadamard gates before then passing through a CONTROLLED–T² gate where the bottom qubit is the control qubit.  They then pass through a SWAP gate.  Draw this circuit.  Physically, a T² gate is the same thing as a quarter-wave plate.  A regular quarter-wave plate acts on single qubits and its effect is to leave horizontal and vertical polarizations alone but significantly changing the 45º-polarized states.  How do you think a CONTROLLED–T² gate, which acts on two qubits, might be physically implemented?