HW3 - Problem 2

Problem 2: Cloning States

Recall the basis \(\{\left|i\right>, \left|-i\right>\}\), which were defined as

  • \(\left|i\right> := \frac{1}{\sqrt{2}}(\left|0\right> + i \left|1\right>)\)
  • \(\left|-i\right> := \frac{1}{\sqrt{2}}(\left|0\right> - i \left|1\right>).\)
  1. Determine whether a state \(\left|\psi\right>\) that we know to be one of the above two states is clonable or not. Briefly justify your answer.
  2. Design a quantum circuit that clones a state if we know it is in the above basis. That is, a quantum circuit \(U\) such that
  • \(U(\left|i\right>\left|0\right>) = \left|i\right>\left|i\right>\)
  • \(U(\left|-i\right>\left|0\right>) = \left|-i\right>\left|-i\right>\)
  1. Design a 4 qubit quantum circuit that clones 2 qubit standard basis states. That is, a quantum circuit \(U\) such that
  • \(U(\left|00\right>\left|00\right>) = \left|00\right>\left|00\right>\)
  • \(U(\left|01\right>\left|00\right>) = \left|01\right>\left|01\right>\)
  • \(U(\left|10\right>\left|00\right>) = \left|10\right>\left|10\right>\)
  • \(U(\left|11\right>\left|00\right>) = \left|11\right>\left|11\right>\)
  1. Recall the Bell basis listed below.
  • \(\left|\Phi^+\right> := \frac{1}{\sqrt{2}}(\left|00\right> + \left|11\right>)\)
  • \(\left|\Phi^-\right> := \frac{1}{\sqrt{2}}(\left|00\right> - \left|11\right>)\)
  • \(\left|\Psi^+\right> := \frac{1}{\sqrt{2}}(\left|01\right> + \left|10\right>)\)
  • \(\left|\Psi^-\right> := \frac{1}{\sqrt{2}}(\left|01\right> - \left|10\right>)\)

Design a 4 qubit quantum circuit that clones the Bell basis states. That is, a quantum circuit \(U\) such that

  • \(U(\left|\Phi^+\right>\left|00\right>) = \left|\Phi^+\right>\left|\Phi^+\right>\)
  • \(U(\left|\Phi^-\right>\left|00\right>) = \left|\Phi^-\right>\left|\Phi^-\right>\)
  • \(U(\left|\Psi^+\right>\left|00\right>) = \left|\Psi^+\right>\left|\Psi^+\right>\)
  • \(U(\left|\Psi^-\right>\left|00\right>) = \left|\Psi^-\right>\left|\Psi^-\right>\)

You may want to reference your solution from problem 1.