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CS 166: Quantum Computation

Homework 4

2022

1. Let lu）be a one qbubit state such that lu）has real amplitudes in the standard basis. That is, lu）= al0）+ Xl1）, where a and X are real.

(a) Find a one qubit state lul）that is perpendiculate to lu）and has real amplitudes. Express lul）in the standard basis.

(b)  Suppose that Alice and Bob share an entangled pair lΦ）= 1/.2(l01）_ l10）) and Alice measures her qubit in the {lu）, lul）} basis. What is the probability of each outcome and what is the state of both qubits after each possible outcome?

2. In class we showed the no cloning theorem, which says that there can be no cricuit with input lo）l0） and outputs lo）lo） for every qubit state  lo）.  The proof of the no cloning theorem allows for the possibility that there is a circuit which clones a qubit state if the state is known to be one of two orthogonal states. Design a circuit that will clone a state from the {li）, l _ i）} basis. That is, on input li）l0）, the output should be li）li）, and on input l _ i）l0）, the output should be l _ i）l _ i）. Give a brief justification that your circuit works as expected.

3.  Consider the game played by Alice and Bob in the CHSH protocol. In class, we worked out the probabilities of the outcomes for the cases where z = y = 0 and z = y = 1. In this problem, you should work out the probabilities for the other two cases (z = 0, y = 1 and z = 1, y = 0).  Give the probabilities for each of combination for Alice and Bob’s output bits (a and X) then give the probability in each case that Alice and Bob win the game.

4. Work out a version of the quantum teleportation protocol if Bob and Alice are given the entangled pair 1/.2(l01）_l10）) instead of 1/.2(l00）+l11）). Recall that an overall global phase doesn’t matter, so al0）+ 8l1）is effectively the same as _al0）_ 8l1）.

5.  Consider the small circuit shown below.

Suppose that the 2-qubit operation U is specified by the matrix:

┌ a0     80     30     60  -

' a3     83     33     63  '

(a) If the third qubit is measued at the end of the circuit, what are the probabilities of each outcome and the resulting state after each outcome?

(b)  Suppose that the third qubit is measured and then the first two qubits are mea- sured.  What are the probabilities of the outcomes of the measurement of the first two qubits.  (You will need to consider the probabilities of the outcome of measuring the third qubit and then determine the probabilities for the first two qubits conditioned on the outcome from the measurement of the third qubit.)

(c)  Suppose that at the end of the circuit, just the first two qubits are measured. What is the probabilityof each outcome?