Project 1: The Linear Algebra “Engine” (Classical Simulation)

Build a small linear algebra engine that simulates single- and two-qubit states.

---

Project Overview

Attribute	Value
Difficulty	Level 2: Intermediate
Time Estimate	Weekend
Main Language	Python
Alternative Languages	Julia, C++
Knowledge Area	Linear algebra
Tools	CLI or notebook
Main Book	“Quantum Computation and Quantum Information” by Nielsen & Chuang

What you’ll build: A simulator that applies basic gates to state vectors and computes measurement probabilities.

Why it teaches quantum: Quantum computing is linear algebra. This project makes states and gates concrete.

Core challenges you’ll face:

• Implementing complex vectors and matrices
• Maintaining normalization
• Modeling measurement collapse

---

Real World Outcome

You will apply gates to a qubit and verify probabilities match theory.

Example Output:

$ python sim.py --gate H --measure
State: [0.707+0j, 0.707+0j]
Measured 0: 0.50, 1: 0.50

Verification steps:

• Check normalization after each gate
• Verify Bell states for two qubits

---

The Core Question You’re Answering

“How do quantum states evolve under linear transformations?”

This is the mathematical core of quantum algorithms.

---

Concepts You Must Understand First

Stop and research these before coding:

1. State vectors
   • Why do amplitudes squared sum to 1?
   • Book Reference: Nielsen & Chuang, Ch. 2
2. Unitary gates
   • Why must quantum gates be unitary?
   • Book Reference: Nielsen & Chuang, Ch. 2
3. Measurement
   • How does measurement collapse the state?
   • Book Reference: Nielsen & Chuang, Ch. 2

---

Questions to Guide Your Design

1. State representation
   • Will you use column vectors and matrix multiplication?
   • How will you represent multi-qubit tensor products?
2. Measurement
   • Will you sample outcomes or compute probabilities only?
   • How will you handle partial measurement?

---

Thinking Exercise

Hadamard Test

Apply a Hadamard gate to

0>. What are the resulting amplitudes and probabilities?

Questions while working:

• Why is the probability split evenly?
• What happens if you apply H twice?

---

The Interview Questions They’ll Ask

Prepare to answer these:

1. “What is a quantum state vector?”
2. “Why are quantum gates unitary?”
3. “How do you simulate measurement?”
4. “What is a tensor product and why is it needed?”
5. “How do you represent a two-qubit state?”

---

Hints in Layers

Hint 1: Starting Point Implement single-qubit vectors and gates.

Hint 2: Next Level Add tensor products to build two-qubit states.

Hint 3: Technical Details Use complex numbers and normalize after operations.

Hint 4: Tools/Debugging Add checks that gate matrices are unitary.

---

Books That Will Help

Topic	Book	Chapter
State vectors	Nielsen & Chuang	Ch. 2
Unitary gates	Nielsen & Chuang	Ch. 2
Measurement	Nielsen & Chuang	Ch. 2

---

Implementation Hints

• Keep gates in a dictionary for reuse.
• Validate normalization after each step.
• Start with deterministic test circuits.

---

Learning Milestones

1. First milestone: You can apply single-qubit gates correctly.
2. Second milestone: You can model two-qubit states.
3. Final milestone: You can explain measurement outcomes quantitatively.