HW5 - Problem 1

Problem 1: The \(n\)-Qubit Hadamard

Let’s practice applying the \(n\)-qubit Hadamard we encountered in lecture (Proposition 11.1).

  1. Consider the case where \(n = 3\), and \(x = 101\).
  • Write down what \(H^{\otimes n} \left|x\right>\) would be in line (82) for this instance.
  • Distribute the tensor product you have, and verify that the amplitude of each term is generated by the bitwise dot product \(x \cdot y\) as stated in the proposition.
  1. Consider the case where \(n = 4\), and \(x = 0000\).
  • What is equation (82) in this instance? You don’t have to distribute this one.
  • What is the probability that we measure 0010 if we measure the four qubits in the standard basis?
  • More generally, what can we say about the distribution of outputs when we measure this state in the standard basis?
  1. Suppose after running an algorithm for some number of steps, we get the following state:

\(\frac{1}{\sqrt{2^2}} (\left|00\right> - \left|01\right> + \left|10\right> - \left|11\right>)\)

The next step of the algorithm is to apply a Hadamard gate to each of the two qubits. What is the state after applying the Hadamard gates?

  1. Suppose after running an algorithm for some number of steps, we get the following state:

\(\frac{1}{\sqrt{2^3}} (\left|000\right> - \left|001\right> -\left|010\right> +\left|011\right> +\left|100\right> -\left|101\right> -\left|110\right> + \left|111\right>)\)

The next step of the algorithm is to apply a Hadamard gate to each of the two qubits. What is the state after applying the Hadamard gates?