Quantum Phase Estimation and Implementing and Testing the Deutsch-Jozsa Algorithm

  1. (20 pts.) Using the quantum phase estimation (QPE) algorithm we created in class, with the controlled U gate being the controlled phase gate with \(\phi=\frac{\pi}{3}\), determine how many counting qubits are needed to match the eigenvalue to one decimal place, two decimal places, and three decimal places. From these results, comment on the expected number of qubits you would need to achieve the computational precision. Here we will define this as the precision held by the standard float (about 10 decimal places).

  2. (15 pts.) Rewrite the QPE algorithm from class to use any other gate for U (that is not the phase gate). Set up your algorithm and prove that you can get (at least approximately) both eigenvalues of your new U gate.

  3. (15 pts.) Using the Deutsch-Jozsa algorithm developed in class, set up the circuit to work for input strings of length 3, length 5, and length 7. Test each of the circuits and show the output and comment on the results.