HW6 - Problem 2

Problem 2: More Bombs

Recall the bombs we talked about in the first module of the course. These bombs would detect photons passing through, and if they work, they would explode. If they were defective, nothing would happen. We used superposition to come up with a strategy to detect working bombs with nonzero probability, but let’s see if we can do better using the geometric intuition we got from Grover’s algorithm.

As a refresher, we model an experiment using the bombs in the following way:

We have a laser initially pointing in the \(\left|0\right>\) direction, and then we use a beam splitter to create a superposition state of

\(\cos\theta \left|0\right> + \sin\theta\left|1\right>.\)

Given this state, the “Bomb gate” does the following:

  • If the bomb is defective:
    • The gate acts as an identity gate.
  • If the bomb is not defective:
    • The gate acts as a standard basis measurement. In other words, with probability \(\cos^2\theta\) you observe \(\left|0\right>\) and with probability \(\sin^2\theta\) you observe \(\left|1\right>\). If we measure \(\left|1\right>\), the bomb explodes.
  1. Let’s set \(\theta = \frac{1}{100}\). Draw the state \(R_\theta \left|0\right>\) in the \(\left|0\right>\), \(\left|1\right>\) plane.
  2. Draw the state after applying \(R_\theta\) and \(B\) once each in the case that the bomb is defective. If there are multiple outcomes, label each outcome and its probability.
  3. Draw the state after applying \(R_\theta\) and \(B\) once each in the case that the bomb is not defective. If there are multiple outcomes, label each outcome and its probability.
  4. Now suppose we applied the \(R_\theta\) gate and \(B\) gate consecutively for a total of 50 times. Draw the state in the case that the bomb is defective. If there are multiple outcomes, label each outcome and its probability. In another picture, draw the state in the case that the bomb is not defective. If there are multiple outcomes, label each outcome and its probability.
  5. How many applications of the pair of \(R_\theta\) gate and \(B\) gate do we need to apply until we can maximally be certain about which case we are in? Call this number \(T\).