Assignment 4

Exercise 1

Create a single-qubit circuit using the IBM quantum composer. Apply a Hadamard gate on the qubit.

What possible states, in Dirac Braket notation, exist? (1 pt)

Provide a screenshot containing the circuit, probabilities, and q-sphere. (2 pts)

Exercise 2 (3 pts)

Create a 2-qubit circuit using the IBM quantum composer. Apply any two single-qubit gates of your choice (one per qubit) such that both qubits are in superposition.

What possible states, in Dirac Braket notation, exist? (1 pt)

Provide a screenshot containing the circuit, probabilities, and q-sphere. (2 pts)

Exercise 3 (3 pts)

Create a 2-qubit circuit using the IBM quantum composer. Apply exactly two gates on the circuit to maximally entangle both qubits. This should produce the Bell state.

What possible states, in Dirac Braket notation, exist? (1 pt)

Provide a screenshot containing the circuit, probabilities, and q-sphere. (2 pts)

Exercise 4 (3 pts)

Create a 3-qubit circuit using the IBM quantum composer. Apply a 180-degree (or radians) rotation around the X, Y, and Z axis for qubits 0, 1, and 2, respectively. You can do this with only one gate per qubit.

What possible states, in Dirac Braket notation, exist? (1 pt)

Provide a screenshot containing the circuit, probabilities, and q-sphere. (2 pts)

Exercise 5 (3 pts) Consider the quantum circuit below.

What is a Dirac Braket notation for this qubits state after the gates? (1 pt)

What gate(s) needs to be appended to transform the qubits state such that its Dirac Braket notation is |1? (1 pt)

What gate(s) needs to be appended to transform the qubits state such that its Bloch sphere representation is pointing at the north pole? (1 pt)

Exercise 6 (2 pts)

Consider this scenario. Youre trying to break a lock that uses a case sensitive 16character alphanumeric passphrase (characters may be a-z, A-Z, and 0-9). How many qubits at a minimum would your quantum computer need to simultaneously represent all possibilities for this passphrase? Show your work.

Exercise 7 (2 pts)

Consider the quantum circuit below.

There are two qubits each with a single-qubit gate applied to them. These two gates can be combined into a single 2-qubit gate using a math operation. State the math expression that yields the matrix for the 2-qubit gate. Show your work.

Exercise 8

(1 pt) Assume a Hadamard gate is applied to a qubit in ground state. Which of these gates when inserted before the Hadamard gate changes final the state vector? Circle all that apply.

X-gate

Y-gate

Z-gate

Identity

Hadamard