MULT90063 Introduction to Quantum Computing
Assignment 1
Due: 5pm Friday, April 8th, Week 6, 2022
Assignment 1 for MULT90063 Introduction to Quantum Computing.
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Work on your own, attempt all questions, and hand in your completed written work on or before the due date as per instructions above. The circuits you create for this assignment should be created in QUI and images along with all the relevant parameters (such as rotation angles and global phases) clearly indicated in your assignment submissions. Submit this assignment online as a single PDF document via LMS.
Total marks = 64
Question 1 [6 marks]
(a) Starting from the |0⟩ state, what state do you end up in if you apply a Hadamard gate?
(b) Starting from the |0⟩ state, find an operation you can apply resulting in the state:
|0⟩ − 𝑖 |1⟩ √2
(c) Find the Cartesian co-ordinates and plot the state you found in part (b) on the Bloch sphere.
(d) Starting from the |0⟩ state, find an operation you can apply to result in the state,
!”# |0⟩+√3𝑒 $|1⟩
(e) Find the Cartesian co-ordinates, and plot the state found in part (d) on the Bloch sphere.
(f) Starting from the state
find an operation which results in the state,
!”# |0⟩+√3𝑒 $|1⟩
|0⟩ + 𝑖 |1⟩ √2
MULT90063 Assignment 1, © C. Hill et al 2022
Question 2 [6 marks]
Consider the following entangled two-qubit state: |ψ⟩ = 𝑐𝑜𝑠(θ) |00⟩ + 𝑠𝑖𝑛 (θ) |11⟩.
(a) Write out the circuit containing one R-gate around a single cartesian axis (indicating the axis, rotation angle, and global phase of the gate which you use), and one two-qubit gate that produces this state.
(b) Using the definition of entanglement entropy (described in an aside in the lectures), 𝑆 = − 7 𝑝” 𝑙𝑜𝑔 𝑝”
calculate the entanglement entropy in this state as a function of 𝜃,
(c) Plot the entropy of entanglement for angles between 𝜃 = 0 and 𝜋.
Question 3 [10 marks]
Below is shown the decomposition of the 3-qubit Toffoli gate into an equivalent circuit comprising only 1-
qubit and 2-qubit gates.
(a) Write down the action of 𝐻, 𝑇 and 𝑇% on an arbitrary single qubit state in ket notation.
(b) Starting from the initial 3-qubit state |111⟩ and using ket notation, trace through each step of the circuit
to verify the output produced.
(c) Explain why such a decomposed circuit might be important in physical quantum computers.
MULT90063 Assignment 1, © C. Hill et al 2022
Question 4 [8 marks]
A three-qubit GHZ state is given by:
|𝐺𝐻𝑍⟩ = |000⟩ + |111⟩ √2
(a) Demonstrate a circuit which constructs the GHZ-state, using only single qubit and two-qubit operations. Optimize the circuit as much as possible.
A three-qubit W-state is given by:
|𝑊⟩ = |001⟩ + |010⟩ + |100⟩ √3
(b) Find a circuit which constructs the three qubit W-state, using only single qubit and two-qubit operations. Optimise this circuit as much as possible.
Question 5 [10 marks]
(a) Construct an oracle with five qubits, which identifies (ie. applies a phase to) numbers which are cubes
(0, 1, 8 and 27) less than 32. You may use multiply controlled operations.
Optimize your circuit to use as few operations as possible. Briefly describe your implementation of this circuit including any optimizations which you have made.
(b) Use the oracle which you constructed in part (a) to implement a single iteration of Grover’s algorithm, starting from an equal superposition, and calculate the probability of measuring a number which is a cube at the output. Show your working.
(c) How many iterations of Grover’s algorithm are required to give a high probability of success using your oracle? Show your working.
(d) The question is changed to find all cubes (strictly) less than 220 (using 20 qubits). How many iterations of Grover’s algorithm would now be required?
MULT90063 Assignment 1, © C. Hill et al 2022
Question 6 [6 marks]
Alice, Bob and Charlie have been captured by quantum bandits who do not believe they know quantum computing. The bandits tell them, “We are going to separate you so that you cannot communicate, and ask you each for a value of either x, or y. You must respond with a value of +1, or -1. We will either ask for two y and one x value, when your answers must multiply to give -1, or we will ask you all for a value of x when your answers must multiply to give +1. If your answers don’t multiply to the correct answer, then a horrible fate awaits you all.”
Alice, Bob and Charlie correctly write down the following equations, required to fulfil the bandit’s demands:
𝑋 𝑌 𝑌 = −1 &'(
𝑌 𝑋 𝑌 = −1 &'(
𝑌𝑌𝑋 =−1 &'(
𝑋 𝑋 𝑋 = +1 &'(
(a) It is not possible for Alice, Bob and Charlie to assign values of +1 or -1 to each of 𝑋 ,𝑋 ,𝑋 ,𝑌 ,𝑌 &'(&’
and 𝑌 , so that they always answer the bandit’s questions correctly, and avoid their horrible fate. (
(b) What (classical) strategy can they use to have the highest probability of success? What is that probability?
Alice, Bob and Charlie realize that if they share a GHZ state that they can always answer correctly. Their strategy is as follows:
They each obtain one qubit of the GHZ state. If they are asked for value of x, they measure in the x-basis by applying a Hadamard gate first, and measuring their qubit. If they asked for the value of y, they measure their qubit in the y-basis by applying an 𝑆% gate followed by a Hadamard gate. They then measure their qubit and if they measure the state “0” they reply +1, and if they obtain the result “1”, they reply -1.
(c) Construct the circuits for each of the four possible choice of questions.
(d) With reference to the states obtained from these circuits, show that Alice, Bob and Charlie can escape
100% of the time if they follow this quantum strategy.
Question 7 [8 marks]
(a) Implement a function which takes three qubits as input and counts the number of qubits set to “1” in that input. This function should have a two-qubit output register, which outputs the number of bits which are set to 1, assuming that the initial input to this register is |00⟩. You may use CNOT, Toffoli and single qubit gates. Show your resulting circuit and comment on your construction.
(b) How would you generalise this circuit to larger inputs? Give a method for constructing a circuit which counts the number of 1’s in an arbitrary, n-qubit, input.
MULT90063 Assignment 1, © C. Hill et al 2022
Question 8 [10 marks]
Qutrits are quantum states with three basis states, . An operation on a qubit is given by
|0i,|1i,|2i U = e x p ⇣ i ⇡3 4 ⌘
24 0 0 1 35 4= 0 0 0 100
(a) Write out the matrix representation for U. Show your working.
(b) Using two qubits you can implement the qutrit operation using the states |00i , |01i , |10i? Implement this circuit in the QUI. Show your circuit in your assignment and comment briefly on the construction of your circuit.
A particular two-qutrit operation is given by:
V = e x p ⇣ i ⇡6 4 ⌦ 4 ⌘
(c) Write out the 9×9 matrix representation for V. Show your working.
(d) Using two qubits for each qutrit, you can implement the two-qutrit operation using four qubits. Implement this circuit in the QUI. Show your circuit and comment briefly on the construction of your circuit.
MULT90063 Assignment 1, © C. Hill et al 2022
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