Introduction to Quantum Computing Assignment 1 MULT20015
MULT20015 – Introduction to Quantum Computing
Due: 2025-08-28 (Thursday) at 5pm
Total points: 20.
Weight for subject mark: 20%.
Number of problems: 5.
Submit your completed work as a PDF only via Canvas/LMS (no other formats accepted). You may use apps
like CamScanner to convert handwritten work to PDF. Keep files under 10MB. Include your name and student
number on the front page. Show all working.
You are encouraged to discuss the assignment with classmates (e.g. in informal discussion sessions), but you must
write your own solution based on your understanding. Copying full or partial solutions from other students may
invalidate your submission, please remind yourself of UniMelb’s Academic Integrity Policy and Penalties.
1 Single Qubit States [0.5+0.5+1.0 = 2.0 points]
This question aims to solidify our knowledge of single qubit states. In class, we said that the data or “quan-
tum state” of any single qubit register can be expressed as |ψ⟩ = a0 |0⟩ + a1 |1⟩ where a0, a1 are complex
numbers.
(a) Consider the state |ψ⟩ = a0 |0⟩+ a1 |1⟩ where a0 =
√
150/251 and a1 = i
√
101/251.
Plot the amplitudes a0 and a1 in the complex plane. [Hint: draw both the real and imaginary axis and
mark the point co-ordinates.]
(b) Convert the state to polar notation form: |ψ⟩ = |a0|eiθ0 |0⟩+ |a1|eiθ1 |1⟩with the basis state phase angles
θ0 and θ1 in radians (expressed as multiples of π).
(c) Now convert the state to “Bloch sphere” form |ψ⟩ = eiθglobal (cos θB2 |0⟩+ eiϕB sin θB2 |1⟩), specifying the
global phase θglobal and the Bloch angles θB and ϕB . Then plot the state on the Bloch sphere.
2 Single Qubit Gates [1.5+3.0+2.0 = 6.5 points]
This question aims to solidify our knowledge of single qubit gates. In class, we said that any single qubit logic
gate (or single qubit program) can be represented as a 2× 2 matrix.
(a) Consider the following single-qubit operators in matrix form, corresponding to rotations by angle θR about
some axis with the global phase θglobal explicitly included
RY (θR) = e
iθglobal
[
cos θR2 − sin θR2
sin θR2 cos
θR
2
]
RX+Y (θR) = e
iθglobal
[
cos θR2 − i√2 sin
θR
2 − 1√2 sin
θR
2
− i√
2
sin θR2 +
1√
2
sin θR2 cos
θR
2
]
Both gates are π rotations around some axis. The first gate is a rotation about the X axis and corresponds
to the gate Y =
[
0 −i
i 0
]
. The second gate corresponds to a rotation about the axis in between the X and
Y axis. We call this gate the Hadamard-like gate HL =
1√
2
[
0 1− i
1 + i 0
]
. Find the value of the global
phase for both gates so that RY (π) = Y and RX+Y (π) = HL.
(b) In matrix form a rotation by θR about the −Y − Z axis, ie. R−Y−Z(θR), is given by the 2× 2 matrix
R−Y−Z(θR) =
[
cos θR2 +
i√
2
sin θR2
1√
2
sin θR2
− 1√
2
sin θR2 cos
θR
2 − i√2 sin
θR
2
]
Where we have set the global phase θglobal to zero. This can be programmed into the QUI with a particular
angle θR to produce the following state from the initial state |0⟩: (i) Find the angle θR that produced
the state |ψ⟩ = R−Y−Z(θR) |0⟩, in both radians (multiples of π) and degrees. Show your working. (ii)
Explain qualitatively (using diagrams) how the operation moves the state around the Bloch sphere. (iii)
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Introduction to Quantum Computing Assignment 1 MULT20015
Figure 1: probabilities and amplitudes.
Construct this gate in QUI and then apply the Hadamard gate. Record the new state in polar form and
take a screenshot of the QUI amplitudes like in Figure 1.
(c) Consider a general single qubit program U . In matrix notation, we are given U parameterised by α, β ∈ C:
U =
[
α β
−β∗ α∗
]
where α∗ denotes the complex conjugate of α. It is easy to show that U is unitary, i.e. U†U = UU† = I
where † denotes the Hermitian conjugate (transpose then complex conjugate) and I is the 2× 2 identity
matrix, if |α|2 + |β|2 = 1. For the matrices below explain why they are not unitaries and therefore not
valid quantum programs
M1 =
[
1 i
−i 1
]
, M2 =
[
0 0
0 1
]
, M3 =
[
1 0
1 1
]
.
3 Multi-qubit states [0.5+2.0+1.0 = 3.5 points]
This question aims to solidify our knowledge of multiqubit states.
(a) Consider a three-qubit state for which the QUI state information cards are:
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Introduction to Quantum Computing Assignment 1 MULT20015
Write out the three-qubit state in ket form in a decimal representation with phases in polar notation
(b) Consider the following circuit over 3 qubits:
(i) At each time step, write the state in ket form and show your working. (ii) At the end of the circuit,
add an X gate to one of the wires to change the probabilities. Verify with QUI and include a screenshot
the circuit and resulting probabilities.
(c) For the state you have calculated in part (b) compute the probability of measuring each of the basis states
using the state at the end of the circuit. That is compute the probability of measuring any of the eight
basis states |000⟩ , |001⟩ , |010⟩ , ..., |111⟩ e.g. Pr(000) =?.
4 Entanglement [0.5+(0.5+2.0+0.5)+2.0+1.5 = 7.0 points]
This question aims to solidify our knowledge of entanglement in multiqubit systems. The following screenshot
form QUI will be useful for the below question.
(a) From the information provided, reproduce this program in the QUI to determine the final state produced
in the ket form, listing the amplitudes
|ψ⟩ = a00 |00⟩+ a01 |01⟩+ a10 |10⟩+ a11 |11⟩ . (3)
(b) The amount of entanglement entropy in the state |ψ⟩ is reported in the QUI by hovering over the time
slider between qubit lines – in this case it is 0.601 bits. From your answer in (a), calculate the entanglement
entropy between two qubits by the following procedure.
i Construct the matrix
A =
[
a00 a01
a10 a11
]
.
ii Calculate the eigenvalues λ1 and λ2 of the matrix product AA
† where A† is the Hermitian conjugate
of the matrix A.
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Introduction to Quantum Computing Assignment 1 MULT20015
iii The entanglement entropy between the qubits (in units of bits) is given by:
H = −λ1 log2(λ1)− λ2 log2(λ2) . (4)
For the state in question, |ψ⟩, find the eigenvalues λ1 and λ2 and calculate the entanglement entropy
between the two quibts in units of bits. Confirm that this value matches that reported by the QUI.
(c) Does global phase affect the entanglement value? Justify your answer.
(d) In the QUI, modify the rotation angle of the R-gate in the initial circuit such that the entanglement
entropy is about 0.700π +±0.01π. Report the angle in R-gate and your entanglement entropy.
5 Ed, Complex numbers & Matrices Quiz [0.5+0.5 = 1.0 point]
(a) Complete the complex numbers and matrices quiz on Canvas / LMS. Look under module 0.
(b) Ask a question or answer a question on Ed Discussions. It can be about the material or fall in the general
scope of quantum computing. It is totally acceptable if there already is an answer if you believe your
answer will contribute meaningfully. You can even answer your own question if you made some full or
partial progress in answering it. Please include a screenshot of your question or answer in your solution
submission.
You can use Ed Discussions and the lab sessions to organize other meetups or discussions, such as study meetups
to discuss the assignments (not copy each other’s solution!) or to prepare for the final exam.
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