[Solved] Quantum Homework 13

1. Let P represent the portion of the eigenvalue approximation circuit shown below.

.

.

.

|ui

We consider the circuit for arbitrary unitary m-dimensional U, |ui B^m, and x {0,1}ⁿ (the eigenvalue estimation circuit took x = 0ⁿ and |ui to be an eigenvector).

Show that P |x,ui = |xi U^[x] |ui, where [x] is the number with binary representation x and U^[x] is matrix exponentiation.

2. Let U be a unitary operator and suppose that |i and |i are eigenvectors with respective eigenvalues , C. Prove that if 6= then h | i = 0 (i.e. |i and |i are orthogonal).
3. Let : {0,1} {0,1} be the identity function, defined by (x) = x. The function (x,y) = (x,x y) has the property that

(x,0) = (x,x),

meaning that it clones the bit x in the first register to the second register.

Let |i B be an arbitrary 1-qubit quantum state. Show that

.

That is, the quantum operator corresponding to the classical 1-bit cloning operator fails to clone |i unless |i is in a state corresponding to a classical bit.

4. Let U be a (2n)-qubit operator that clones two n-qubit quantum states, |i,|i Bⁿ, meaning

U(|i |0ⁿi) = |i |i and U(|i |0ⁿi) = |i |i.

Prove that U clones |i and |i if and only if |i = |i or h | i = 0. [Hint: take the inner product of the two equations.]

5. Use the previous question to prove that a there are no quantum cloning operators that work for all pairs of states.