Quantum Algorithms and Software Engineering

quantum

course work

pennylane

Some personal reflections on Quantum

Author

Zak Toews

Published

March 8, 2026

Quantum Algorithms and Software Engineering: Term Portfolio Project

By Zak Toews

4th Year Software Engineering Student

Part 1: Documenting my Quantum Journey
Part 2: Documenting my Quantum Journey

Part 1: Documenting my Quantum Journey

1. How did you get up to speed in basic linear algebra (e.g., Euler formula, complex plane, linear transformations or operators, eigenvectors and eigenvalues), including which resources you consulted in this process?

In this course, it was brought to our attention that the foundations of Quantum Computing relies on a understanding of linear algerbra, trigonometry, complex numbers, and matrix operations. We have yet to work with eigenvalues and eigenvectors.

For the matrix algebra, I had recalled some of my knowledge from my first year course which is almost 5 years ago now. But, I was not too concerned as if I had learned it fine the first time, I could learn it again without much of a hitch. I looked up the basic Matrix Operations 1, practice was real time during Quizzes and Assignments, and I would verify my answers with some tools [2][3]. Although, I am still pretty weary about trusting the math that AI pumps out, but Gemini is pretty good at it now.

For trigonometry and Euler’s formula, I understand the basics, but I have been consistently going back to the slides that contain Euler’s identity then trying to imagine what how the phase change affects the qubit on the Bloch sphere. The complex plane as it is has been drilled home for me from classes such as ECE 360 Control Systems (T_T).

I included some code to show the different basis states below.

Show code

import matplotlib.pyplot as plt
from qiskit.visualization import plot_bloch_vector

# Coordinates for the 6 basis states [x, y, z]
states = {
    "|0>": [0, 0, 1],
    "|1>": [0, 0, -1],
    "|+>": [1, 0, 0],
    "|->": [-1, 0, 0],
    "|i>": [0, 1, 0],
    "|-i>": [0, -1, 0]
}

# Create subplots with 3D projection
fig = plt.figure(figsize=(20, 5))

for i, (label, coord) in enumerate(states.items()):
    # Add a 3D subplot for each basis vector
    ax = fig.add_subplot(1, 6, i + 1, projection='3d')
    plot_bloch_vector(coord, title=label, ax=ax)

plt.show()

2. How did you get started in documenting linear algebra formulas (e.g., Euler formula or matrices) using LaTeX Markdown in Jupyter Notebooks? Develop your own cheat sheet of the quantum computing formulas and Dirac notation to ease assignment typesetting.

To get started with the formulas and matrices that I would need, I would pull the relevant information from the slides. I had actually done my assignment by hand to avoid the typesetting, but for the next assignment I will be able to copy and paste from the following cheat sheet:

Matrix Symbols and Multiplication Properties

1. Definitions * Complex Conjugate: A^* or \bar{A} * Transpose: A^T * Inverse: A^{-1} * Conjugate Transpose (Adjoint): A^\dagger

2. Multiplication Properties \begin{aligned} &\text{Distributive:} & A(B + C) &= AB + AC \\ &\text{Associative:} & (AB)C &= A(BC) \\ &\text{Non-Commutative:} & AB &\neq BA \quad \text{(Generally)} \\ \\ &\text{Transpose of Product:} & (AB)^T &= B^T A^T \\ &\text{Inverse of Product:} & (AB)^{-1} &= B^{-1} A^{-1} \\ &\text{Adjoint of Product:} & (AB)^\dagger &= B^\dagger A^\dagger \end{aligned}

3. Properties of Unitary Matrices (U) A matrix U is unitary if its conjugate transpose is also its inverse. \begin{aligned} &\text{Defining Condition:} & U^\dagger U &= UU^\dagger = I \\ &\text{Inverse Relationship:} & U^{-1} &= U^\dagger \\ &\text{Product of Unitaries:} & (UV)^\dagger &= V^\dagger U^\dagger \quad \text{(The product } UV \text{ is also unitary)} \\ &\text{Norm Preservation:} & \langle U\psi | U\psi \rangle &= \langle \psi | \psi \rangle \\ &\text{Determinant:} & |\det(U)| &= 1 \end{aligned}

4. Properties of Tensor Products (\otimes) \begin{aligned} &\text{Distributive:} & A \otimes (B + C) &= (A \otimes B) + (A \otimes C) \\ &\text{Scalar Multiplication:} & c(A \otimes B) &= (cA) \otimes B = A \otimes (cB) \\ &\text{Mixed-Product Property:} & (A \otimes B)(C \otimes D) &= (AC) \otimes (BD) \\ \\ &\text{Transpose:} & (A \otimes B)^T &= A^T \otimes B^T \\ &\text{Conjugate:} & (A \otimes B)^* &= A^* \otimes B^* \\ &\text{Adjoint (Dagger):} & (A \otimes B)^\dagger &= A^\dagger \otimes B^\dagger \\ &\text{Inverse:} & (A \otimes B)^{-1} &= A^{-1} \otimes B^{-1} \end{aligned}

5. Dirac Notation & Computational Basis The computational basis for a single qubit consists of the states |0\rangle and |1\rangle.

Column Vectors (Kets): |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
Row Vectors (Bras): \langle 0| = \begin{pmatrix} 1 & 0 \end{pmatrix}, \quad \langle 1| = \begin{pmatrix} 0 & 1 \end{pmatrix}
Linear Combination of Unit Kets: A general quantum state |\psi\rangle is a superposition: |\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} where \alpha, \beta \in \mathbb{C} and |\alpha|^2 + |\beta|^2 = 1.

Inner Product and Orthnormality The inner product of two vectors |\phi\rangle and |\psi\rangle is written as \langle \phi | \psi \rangle. For the computational basis: \langle i | j \rangle = \delta_{ij} \implies \langle 0|0\rangle = 1, \quad \langle 0|1\rangle = 0

Measurement and the Born Rule When measuring a state |\psi\rangle = \alpha|0\rangle + \beta|1\rangle in the computational basis:

Probability of outcome |0\rangle: P(0) = |\langle 0|\psi\rangle|^2 = |\alpha|^2
Probability of outcome |1\rangle: P(1) = |\langle 1|\psi\rangle|^2 = |\beta|^2
General Born Rule: For a state |\psi\rangle, the probability of measuring an eigenvalue associated with eigenvector |x\rangle is: P(x) = |\langle x|\psi\rangle|^2

6. Quantum Logic Gates (Single-Qubit)

\begin{aligned} &\mathbf{X \text{ (NOT):}} & X &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ &\mathbf{Z \text{ (Phase Flip):}} & Z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ &\mathbf{Y \text{ (Bit \& Phase Flip):}} & Y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ &\mathbf{H \text{ (Hadamard):}} & H &= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \\ &\mathbf{S \text{ (Phase):}} & S &= \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \\ &\mathbf{T \text{ (}\pi/8\text{ Gate):}} & T &= \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} \\ &\mathbf{P(\phi) \text{ (Phase Shift):}} & P(\phi) &= \begin{pmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{pmatrix} \\ &\mathbf{R_z(\theta) \text{ (Rotation):}} & R_z(\theta) &= \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix} \\ &\mathbf{\sqrt{X} \text{ (SX Gate):}} & \sqrt{X} &= \frac{1}{2}\begin{pmatrix} 1+i & 1-i \\ 1-i & 1+i \end{pmatrix} \end{aligned}

7. Common Gate Identities Note: Identities are often given “up to a global phase” (e^{i\theta}). \begin{aligned} &\text{Phase/Rotation Relations:} & S &= P(\pi/2) \equiv R_z(\pi/2) \\ & & T &= P(\pi/4) \equiv R_z(\pi/4) \\ &\text{Hadamard Basis Change:} & H X H &= Z \\ & & H Z H &= X \\ &\text{Square Root Relations:} & (\sqrt{X})^2 &= X \\ & & S^2 &= Z \\ &\text{Self-Inverse Gates (Hermitian):} & H^2 = X^2 &= Y^2 = Z^2 = I \\ &\text{Pauli Multiplication:} & XY &= iZ, \quad YZ = iX, \quad ZX = iY \end{aligned}

9. Tensor Product of Matrices (Kronecker Product) For two matrices A (size m \times n) and B (size p \times q), the tensor product A \otimes B is a matrix of size mp \times nq: A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B & \dots \\ a_{21}B & a_{22}B & \dots \\ \vdots & \vdots & \ddots \end{pmatrix}

10. 2-Qubit Computational Basis The basis for a 2-qubit system is formed by the tensor product of two 1-qubit basis vectors. There are 2^2 = 4 possible states:

\begin{aligned} |00\rangle &= |0\rangle \otimes |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 \\ 1 \cdot 0 \\ 0 \cdot 1 \\ 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \\ \\ |01\rangle &= |0\rangle \otimes |1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 0 \\ 1 \cdot 1 \\ 0 \cdot 0 \\ 0 \cdot 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \\ \\ |10\rangle &= |1\rangle \otimes |0\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 \\ 0 \cdot 0 \\ 1 \cdot 1 \\ 1 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \\ \\ |11\rangle &= |1\rangle \otimes |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 \\ 0 \cdot 1 \\ 1 \cdot 0 \\ 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \end{aligned}

General Rule for n-qubits For an n-qubit state |x\rangle where x is the decimal representation of the binary string: * The vector has a length of 2^n. * The vector has a 1 at the x-th index (starting from 0) and 0 elsewhere.

Mixed Tensor Products (Superposition) If |\psi\rangle = \alpha|0\rangle + \beta|1\rangle and |\phi\rangle = \gamma|0\rangle + \delta|1\rangle, then: |\psi\rangle \otimes |\phi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \otimes \begin{pmatrix} \gamma \\ \delta \end{pmatrix} = \begin{pmatrix} \alpha\gamma \\ \alpha\delta \\ \beta\gamma \\ \beta\delta \end{pmatrix}

Basis Definitions

Hadamard Basis (X-basis): |+\rangle = H|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad |-\rangle = H|1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}

Tensor Products of Basis Vectors \begin{aligned} \\ &\text{Hadamard ($X$-basis):} & |++\rangle &= |+\rangle \otimes |+\rangle = \frac{1}{2}\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} \\ & & |+-\rangle &= |+\rangle \otimes |-\rangle = \frac{1}{2}\begin{pmatrix} 1 \\ -1 \\ 1 \\ -1 \end{pmatrix} \\ & & |--\rangle &= |-\rangle \otimes |-\rangle = \frac{1}{2}\begin{pmatrix} 1 \\ -1 \\ -1 \\ 1 \end{pmatrix} \\ & & |-+\rangle &= |-\rangle \otimes |+\rangle = \frac{1}{2}\begin{pmatrix} 1 \\ 1 \\ -1 \\ -1 \end{pmatrix} \end{aligned}

Mixed Basis States |+\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}

11. 2-Qubit Gate Tensor Products The following matrices represent the operations when gates are applied in parallel to two separate qubits.

\begin{aligned} &I \otimes I = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} &I \otimes X = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \\ \\ &X \otimes I = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} &X \otimes X = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} \\ \\ &Z \otimes Z = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} &X \otimes Z = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} \\ \\ &Z \otimes X = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{pmatrix} &H \otimes H = \frac{1}{2} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{pmatrix} \end{aligned}

12. 3-Qubit Hadamard Tensor Product (H^{\otimes 3}) Applying a Hadamard gate to three qubits simultaneously creates a uniform superposition of all 8 possible basis states.

H \otimes H \otimes H = \frac{1}{2\sqrt{2}} \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{pmatrix}

13. General n-qubit Identity and Parity * I^{\otimes n}: An identity matrix of size 2^n \times 2^n. * Z^{\otimes n}: A diagonal matrix where the entry is +1 if the basis state has an even number of 1s and -1 if it has an odd number of 1s.

14. Entanglement: The 4 Bell States

The Bell states (or EPR pairs) are the four maximally entangled two-qubit states. They form an orthonormal basis for the 4-dimensional Hilbert space of two qubits.

Generating Bell States via Circuit The general recipe to create a Bell state from a basis state |xy\rangle is: \text{Bell}(x,y) = \text{CNOT}_{0 \to 1} (H \otimes I) |xy\rangle

Matrix of the CNOT Gate The CNOT gate flips the target (second) qubit if the control (first) qubit is |1\rangle: \text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}

The Four Bell States

Initial State	Bell State Notation	Vector / Formula
\|00\rangle	\|\Phi^+\rangle	\frac{1}{\sqrt{2}}(\|00\rangle + \|11\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}
\|10\rangle	\|\Phi^-\rangle	\frac{1}{\sqrt{2}}(\|00\rangle - \|11\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}
\|01\rangle	\|\Psi^+\rangle	\frac{1}{\sqrt{2}}(\|01\rangle + \|10\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}
\|11\rangle	\|\Psi^-\rangle	\frac{1}{\sqrt{2}}(\|01\rangle - \|10\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}

*Of most of which was generated by Gemini [3]

It is also possible to visualize the equations in LaTeX using the draw method of Statevector object in Qiskit. For example:

Show code

from qiskit.quantum_info import Statevector
from qiskit import QuantumCircuit

qc = QuantumCircuit(2)
qc.x(1)
qc.h(1)
qc.z(1)
psi = Statevector(qc)
psi.draw("latex")

\frac{\sqrt{2}}{2} |00\rangle+\frac{\sqrt{2}}{2} |10\rangle

3. How did you get started running your first quantum circuits using IBM Qiskit and Xanadu Pennylane Jupyter Notebook platforms?

I first started checking out Qiskit in the labs and seeing how it generates the circuits in the IBM Quantum Composer. As for PennyLane, one of the other students said that the PennyLane Codebook was helpful in learning PennyLane and comprehension off the material. So, I have since completed the Introduction segment, but I plan to continue to follow along [4].

4. How would you motivate other students to join the journey into quantum computing given the motivational materials presented in class and found in the references? Your answers to this question will likely evolve during this course. Revisit regularly this quesion.

Well, I have already suggested to my friend, who is interested in the material but couldn’t join the course, that he should check out the content on PennyLane and go through the CodeBook. There are also the tutorials available on the IBM Qiskit website [5].

Also, suggesting the Bloch Sphere is sometimes a good tool for visualizing phase changes and the relative locations of the basis states; however, it is not an accurate tool, as the basis kets are intended to orthogonal, but instead are viewed as opposing in the Bloch Sphere.

The short article about Quantum and Computational Chemistry was insightful and approachable [6]. I would recommend that.

From my understanding, Quantum Computing is inevitable and will become a common way for sophisticated problem in the not-too-distant future. Therefore it would be advised that everyone in Software Engineering and Computer Science start to understand at least the basics of Quantum Computing. Take this article “Quantum Computing Moves from Theoretical to Inevitable” that explains the upcoming markets that will benefit the most from Quantum Computing [7].

5. Which algorithm is your favourite quantum algorithm so far?

I think that all the algorithms, so far, are interesting. The most difficult, yet intriguing, part is trying to grasp the significance of how gates modify the quantum state and how the math verifies the expected results in the end.

From a curiosity perspective, I would say that the Teleportation algorithm is the most important because it proves the ability to move quatum states around, and it also kills the misconception that many people have, that Quantum states can somehow transfer information faster than the speed of light.

I have included a snippiet of code that simulates transportation. It was completed with the aid of Gemini [3].

Show code

import pennylane as qml

# We need 3 wires for teleportation
dev = qml.device("default.qubit", wires=3)

@qml.qnode(dev)
def teleportation_circuit():
    
    # 1. PREPARE THE STATE TO TELEPORT (on wire 0)
    # Let's just create a random state using a rotation
    qml.RX(1.23, wires=0)
    qml.RY(0.45, wires=0)

    # 2. CREATE ENTANGLEMENT (between Alice/wire 1 and Bob/wire 2)
    qml.Hadamard(wires=1)
    qml.CNOT(wires=[1, 2])

    # 3. ALICE'S OPERATIONS (Bell State Measurement)
    qml.CNOT(wires=[0, 1])
    qml.Hadamard(wires=0)

    # 4. MEASURE AND CONDITIONALLY APPLY CORRECTIONS
    # In PennyLane, we use m_0 and m_1 to represent Alice's measurement results
    m_0 = qml.measure(0)
    m_1 = qml.measure(1)

    # Bob applies corrections based on Alice's classical bits
    # If m_1 is 1, apply PauliX; if m_0 is 1, apply PauliZ
    qml.cond(m_1, qml.PauliX)(wires=2)
    qml.cond(m_0, qml.PauliZ)(wires=2)

    # Return the probabilites in the state
    return qml.probs(wires=2)

# Execute
result = teleportation_circuit()
print(result)

[0.6504817 0.3495183]

6. What was the most challenging part in understanding Grover’s algorithm?

I don’t think we have covered Grover’s algorithm yet in class, but I looked it up and this PennyLane article covers it pretty well [8].

I suppose one of the most challenging parts of using Grover’s algorithm or any other Oracle based algorithm is being able to imagine and then describe how the oracle needed for the specific problem.

7. Required: What are your personal insights, aha moments, and epiphanies you experienced in the first part of this course?

Some things that ocurred to me were:

The matrix math becomes unwieldy very quickly, and that Dirac notation is much preferred.
Quantum computing in inevitable
The algorithms needed for solving a modern problem must be huge (many quibits, many gates) with large portions being progromatically generated or transpiled.
Modern Quantum Computing is the culmination of many very clever people.
Quantum computing will not likely be of much use alone but will be integrated wtih standard computing; something that can be called upon when the certain cases arise (such as with simulation or searching).

8. Required: How did you experience Generative AI as a learning tool for this course?

Gen AI, especially Google Gemini, has been extremely helpful in this course. I would say that Gen AI is helpful in a many scenarios, but academically, it has been essential for easing the initial learning curve involved with such things as Dirac notation and explaining phase rotation. The optimist in me says that AI is a super useful tool that has come along at a time when it is most in need where we can all benefit from the culmination of our efforts. The pessimist in me says that all the most useful AI models will be behind paywalls soon enough and we may find ourselves reliant on them, especially if we don’t actually understand the information that it gives back to us. So, ultimately, GenAI is amazing for boiler-plate work, but for knowledge work (exploratory solutions and the like), the results should still be critisized and the technicalites understood.

9. Record your Generative AI prompts and contexts for Basic Quantum Terms, Complex Linear Algebra, Quantum Algorithms, Python, Qiskit, and PennyLane inquiries for easy recall. Use the same line item or project in your genAI engine (e.g., ChatGPT, Google Gemini, Perplexity, or others) to build up the “quantum context”.

“Hey can you explain superposition, entanglement, and the no cloning theorem? keep it simple and just tell me why each one actually matters for making a quantum computer faster than a regular one.”

“Quantum states use a lot of linear algebra. Can you explain how bra ket notation works with unitary matrices and hermitian operators. Also, why does a unitary matrix have to keep the total probability at 1 when it acts on a qubit state?”

“Can you explain to me how Grover’s algorithm works? I would like the Dirac and Matrix representation.”

“I need to make a bell state in qiskit. Can you give me the python code to set up 2 qubits, apply them with a hadamard and a cnot, and then run it to see the results. keep the code simple and just use comments to say which qubit is which.”

“I want to try a teleportation circuit in pennylane. Can you show me a template for that? Please explain the different steps.”

Part 1 Bibliography

[1] GeeksforGeeks, “Matrix Operations,” GeeksforGeeks, Nov. 23, 2020. https://www.geeksforgeeks.org/maths/matrix-operations/
[2] “Matrix Calculator - Reshish,” matrix.reshish.com. https://matrix.reshish.com/
[3] Google, “Gemini,” gemini.google.com, 2025. https://gemini.google.com/app
[4] “Map | PennyLane Codebook,” Pennylane.ai, 2025. https://pennylane.ai/codebook/learning-paths
[5] “IBM Quantum Computing | Qiskit,” Ibm.com, 2026. https://www.ibm.com/quantum/qiskit#tutorials (accessed Feb. 12, 2026).
[6] “Fertilizer and other quantum computer chemistry,” Quantum Flagship. https://qt.eu/applications/fertilizer-and-other-quantum-computer-chemistry
[7] G. Dunn, V. Sinha, Laurent-Pierre Baculard, S. Ali, and W. Chang, “Quantum Computing Moves from Theoretical to Inevitable,” Bain, Sep. 23, 2025. https://www.bain.com/insights/quantum-computing-moves-from-theoretical-to-inevitable-technology-report-2025/
[8] L. Botelho, “Grover’s Algorithm,” PennyLane Demos, Jul. 03, 2023. https://pennylane.ai/qml/demos/tutorial_grovers_algorithm

Part 2: Documenting my Quantum Journey

1. In your own words, describe the Quantum Fourier Transform (QFT) algorithm. How is QFT used in quantum computing?

The Quantum Fourier Transform is the quantum analog for the traditional Fourier Transform algorithm. A generalization of the the Fourier transform can utilize the Euclidean space to allow for easy manipulations of qubits. Superposition, entanglement, and amplitude amplification working together in this algorithm allows for exponential speed-up. It effectivley maps a quantum state’s amplitude distribution from the computational basis to the phase basis. Because of this, a QFT is usually followed by the inverse to get useful measurements.[1]

QFT usually consists of a Hadamard transform, to create equal superposition, then phase shifts are applied to encode the frequency components. [1]

It is prodominantly used Shor’s Algorithm, Quantum Phase Estimation, and Quantum Machine Learning. [1]

2. In your own words, describe the Quantum Phase Estimation (QPE) algorithm. How is QPE used in quantum computing?

Quantum Phase Estimation tackles a problem that is at the basis of quantum computing. How can we know the phase of a quantum system if when we measure a qubit directly, the superposition and phase immediately collapses. So, the QPE is able to transfer the phase using “phase kickback” to different, counting qubits, that can be applied to the inverse QFT to then be measured. [2]

QPE usually consists of putting the counting qubits in superposition and the using Controlled-U gates to kick back the phase. Then the inverse QFT is applied to the counting register. The measurement is a bitstring representation of the phase in binary. [2]

QPE is used in Shor’s algorithm, quantum simulation, and solving linear systems. [2]

QPE is primarily theoretical at the moment as current implimentations are hardware intensive. [2]

3. For you, what was the most challenging part of understanding Grover’s algorithm and why?

For me, the most difficult part of understanding Grover’s algorithm, as with most of the algorithms is understanding how the “phase kickback” works and implementing it. Otherwise, I understand pretty well how the Oracle and the Diffusion Operator work in tandem to isolate and highlight the “hidden” result.

4. Which is your favourite quantum algorithm and why?

I still don’t know if I am educated enough to make this statement, yet, but my ideAL algorithm would be one where I could solve a real world problem with the existing technology and real qubits in a way that shows temporal or computational complexity impro vements over a classical calculation.

5. What is the role of eigenvalues and eigenstates (eigenvectors) in quantum computing?

They remove a lot of complexities in the computations. Instead of dealing with entire matrices, you can instead create the results with eigen values as scalars, and eigen vectors together as matrices.

All observable quantities, in quantum computing, can be represented by Hermitian operators and the results that come from measurements of these quantities are the eigenvalues of the operators. The eigenstates are then the “stable” configurations of the operators.

A superposition can be seen as a linear combination of several eigenstates.

6. Summarize the decomposition strategies presented in class for developing hybrid quantum-classical systems.

Phase Oracles

The phase oracle is a black-box function that recognize the solution in the solution space and changes the phase of this solution. It goes -> Define the logic, initialize the Ancilla, Flip the solution phase in the solution space, Reverse initialization logic. [3]

QUBO/HUBO

Creating a QUBO is translating Constraints into Energy where the “correct” answer is the lowest possible value in a quadratic equation. It goes -> Start with constraints, turn them into penalties, simplify to binary variables, map result to Quantum gates Z and I and scalars.

7. Which one is your favourite decomposition strategy? Describe this strategy in detail.

Well, they are all quite difficult to derive, but I think that the QUBO strategy of problem decomposition provides the most promising results especially with the hardware focus that the local D-Wave lab provides. The algorithm for mapping the problems to a Quadratic Unconstrained Binary Optimization problem open the door to a bunch of possiblities when designing quantum algorthims for existing and novel software problems.

8. Write a summary of the different quantum applications presented by the graduate students.

Protein Folding
Naomi Phan - Protein Structure Folding

Naomi explores the world of protein folding and how amazingly complex (with regard to amino acid rotations and permutations). “Brute Force for 1 protein takes longer than the age of the universe”. Naomi uses a Variational Quantum Eigensolver mixed with a classical optimizer to employ the quantum advantage.

Route Planning (2 different students)
Aden Knelson-Dobson - Quantum Traffic Flow Optimization

Aden showed that a traffic flow optimization problem can be formulated as QUBO and thus could be aplicable to be solved with quantum annealing. Papers referenced are Lucas (2014) and Neukart et al. (2017).

Joyce can de Vegte - Quantum Traffic Optimization

Joyce explored the trafic optimization problem while giving a clear explanation of how to create the cost Hamiltionian and using QAOA with 1 and 2 layers. Namely, A. Giovagnoli’s “An Introduction to the Quantum Approximate Optimization Algorithm” was referenced.

graph colouring
Chuan Zhang - Graph Coloring Using QAOA

Chuan explains how Grover’s algorithm is not a great choice because of the large amount of hardware resources need. He instead goes onto suggest that a QUBO and QAOA is a better approach. After comparison between algorithm types, it is shown that RQAOA is the best contender for solving this type of problem.

quantum encryption
Juliiana Unesikhina - A Study of Encrypted Quantum State Distribution

Juliiana discusses the challenges of distributing quantum tasks, and how these limitations actually become opportunities. She highlights why we should strive for Quantum State Distribution with several references.

[4]

9. Which applications inspired you the most? Describe this application in more detail. Please note graduate students who have presented an application cannot write about the application they present for their grad project.

I definitely think that the presentations on route finding were the most intriguing as I am working on a Trip Planner application at the moment to demonstrate on my personal website. I will also have a section dedicated to discussion of quantum applcations and how they can be used in my software. So, I find these presentations, not only easy to understand, because I feel that everyone is more or less domain familiar, but it can also be potentially rolled out in my application in the near future.

10. Did you use Generative AI tools (e.g., ChatGPT, Gemini, Copilot, or Perplexity) in the learning journey of this course? If so, what was your experience?

Yes, I used them plenty. Mostly Gemini for this kind of content. I thought that it did a pretty good job and dumbing difficult topics down for this layperson.

I find that you have to use the AI conscientiously. You have to consider resources, ask for citations, compare against other LLMs and even word of mouth to ensure that what you are propogating is correct and not a hallucination of the AI.

I think generally, AI will excelerate all facets of research, exploration, and study, but like not always clicking the first link on Google, or assuming the float representing a rational number from a program is going to be correct, you should not assume that the first response from AI is correct.

11. What topics, methods, and tools of this course did you add already or are you going to add to your résumé?

I am going to add many things to my protfolio website including this document here. For my resume, I think I will add that I understand the basics, but I plan to continue to look into Quantum Algorithms.

12. What have been your personal insights, aha moments, or epiphanies you experienced in the second part of your quantum learning journey?

We are here. This is it. The Quantum revolution is coming, but it will not be easy, and as it stands, each algorithm is complex and specifc to each problem, but it has room to emply the quantum advantage. I forsee that there will be libraries (like there are now) and a level of generalization that will make it easy it for quantum software engineers and compilers to create the highest-level algorithms to ping (I think local quantum computers will be reserved for the utmost secure and wealthy) for solutions over the quantum web that solve problems that traditional computers simply cannot solve.

Part 2 Bibliography

[1] https://www.quera.com/glossary/quantum-fourier-transform [2] https://www.quera.com/glossary/quantum-phase-estimation [3] https://gemini.google.com/app [4] Relevant, grad-student made presentations from the SEng 457 course BrightSpace page.

Quantum Algorithms and Software Engineering: Term Portfolio Project

By Zak Toews

4th Year Software Engineering Student

Table of Contents

Part 1: Documenting my Quantum Journey

1. How did you get up to speed in basic linear algebra (e.g., Euler formula, complex plane, linear transformations or operators, eigenvectors and eigenvalues), including which resources you consulted in this process?

2. How did you get started in documenting linear algebra formulas (e.g., Euler formula or matrices) using LaTeX Markdown in Jupyter Notebooks? Develop your own cheat sheet of the quantum computing formulas and Dirac notation to ease assignment typesetting.

Matrix Symbols and Multiplication Properties

3. How did you get started running your first quantum circuits using IBM Qiskit and Xanadu Pennylane Jupyter Notebook platforms?

4. How would you motivate other students to join the journey into quantum computing given the motivational materials presented in class and found in the references? Your answers to this question will likely evolve during this course. Revisit regularly this quesion.

5. Which algorithm is your favourite quantum algorithm so far?

6. What was the most challenging part in understanding Grover’s algorithm?

7. Required: What are your personal insights, aha moments, and epiphanies you experienced in the first part of this course?

8. Required: How did you experience Generative AI as a learning tool for this course?

Part 1 Bibliography

Part 2: Documenting my Quantum Journey

1. In your own words, describe the Quantum Fourier Transform (QFT) algorithm. How is QFT used in quantum computing?

2. In your own words, describe the Quantum Phase Estimation (QPE) algorithm. How is QPE used in quantum computing?

3. For you, what was the most challenging part of understanding Grover’s algorithm and why?

4. Which is your favourite quantum algorithm and why?

5. What is the role of eigenvalues and eigenstates (eigenvectors) in quantum computing?

6. Summarize the decomposition strategies presented in class for developing hybrid quantum-classical systems.

7. Which one is your favourite decomposition strategy? Describe this strategy in detail.

8. Write a summary of the different quantum applications presented by the graduate students.

9. Which applications inspired you the most? Describe this application in more detail. Please note graduate students who have presented an application cannot write about the application they present for their grad project.

10. Did you use Generative AI tools (e.g., ChatGPT, Gemini, Copilot, or Perplexity) in the learning journey of this course? If so, what was your experience?

11. What topics, methods, and tools of this course did you add already or are you going to add to your résumé?

12. What have been your personal insights, aha moments, or epiphanies you experienced in the second part of your quantum learning journey?

Part 2 Bibliography