Quantum Computing: Math & Theory Study Guide

This guide provides key concepts, equations, and mnemonics to help me grasp ( err try. shit’s hard) quantum computing fundamentals. It includes handy references, shortcuts, and visualization techniques to make complex topics more intuitive.

1. Complex Numbers & Linear Algebra

Complex Numbers:

  • Form: \(z = a + bi\), where \(i^2 = -1\).
  • Magnitude: \(|z| = \sqrt{a^2 + b^2}\).
  • Conjugate: \(\overline{z} = a - bi\).
  • Multiplication Rule: \((a+bi)(c+di) = (ac - bd) + (ad + bc)i\).

Linear Algebra Basics:

  • Vector Notation (Dirac bra-ket):
    • Column (Ket): \(|\psi\rangle = \begin{bmatrix} a \\ b \end{bmatrix}\).
    • Row (Bra): \(\langle\psi| = \begin{bmatrix} a & b \end{bmatrix}\).
    • Inner Product: \(\langle\phi | \psi \rangle\) = dot product.
    • Outer Product: \(|\psi\rangle \langle\phi|\) = matrix multiplication.
  • Matrices:
    • Identity: \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\).
    • Transpose: Switch rows & columns.
    • Conjugate Transpose: Conjugate each entry and transpose.
    • Matrix Multiplication: Row-column multiplication.

Mnemonic for Matrix Multiplication:

“Row times Column gives the Sum.”

2. Qubits & Quantum States

A qubit is a 2D vector in a complex Hilbert space:
\[ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad \text{where } |\alpha|^2 + |\beta|^2 = 1. \] - Computational Basis: \(|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).

Bloch Sphere Representation:

\[ |\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle \] Mnemonic:
- Latitude (θ) controls probability distribution.
- Longitude (φ) controls phase difference.
- “Theta tilts, Phi spins.”

3. Quantum Gates & Transformations

Basic Quantum Gates:

Gate Matrix Representation Effect
Pauli-X (NOT) \(X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) Flips \(|0\rangle \leftrightarrow |1\rangle\)
Pauli-Y \(Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\) Phase + Flip
Pauli-Z \(Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) Phase shift
Hadamard (H) \(H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\) Superposition
Phase Gate (S, T) \(S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}\), \(T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix}\) Phase shifts

Mnemonic for Hadamard Gate:

“Halfway between 0 and 1.” (Splits \(|0\rangle\) and \(|1\rangle\) into an equal mix.)

4. Superposition & Entanglement

Superposition:

\[ H |0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \]

Bell Pairs (Entanglement):

\[ |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle) \]

Mnemonic for Bell States:
- “Entangled twins” (They behave as a single unit.)

5. Quantum Measurement & Probability

  • Measurement collapses a qubit into \(|0\rangle\) or \(|1\rangle\).
  • Probability of getting \(|0\rangle\): \(P(0) = |\alpha|^2\).
  • Probability of getting \(|1\rangle\): \(P(1) = |\beta|^2\).

6. Quantum Algorithms

Deutsch’s Algorithm:

  • Determines if a function is constant or balanced in one query.
  • Uses Hadamard & X gates.
from qiskit import QuantumCircuit, Aer, execute

qc = QuantumCircuit(2, 1)
qc.h([0,1])  # Create superposition
qc.cx(0,1)   # Apply function
qc.h(0)      # Interference step
qc.measure(0,0)

sim = Aer.get_backend('qasm_simulator')
result = execute(qc, sim, shots=1000).result()
print(result.get_counts())

Quantum Fourier Transform (QFT):

  • Generalization of the Fourier Transform to quantum states.
  • Key in Shor’s Algorithm for factorization.
import numpy as np
from qiskit import QuantumCircuit

def qft(n):
    qc = QuantumCircuit(n)
    for i in range(n):
        for j in range(i):
            qc.cp(np.pi / 2**(i-j), j, i)  # Controlled phase gates
        qc.h(i)
    return qc

qc = qft(3)
qc.draw('mpl')  # Visualize QFT circuit

Mnemonic for QFT:

“Interference creates frequency magic.”

7. Shortcuts & Tips

  • Use Hadamard gates to create superposition.
  • Apply controlled gates (CNOT) for entanglement.
  • Use phase gates (S, T) for controlled rotations.
  • QFT is a quantum version of the FFT—useful for factoring numbers.
  • Shor’s Algorithm breaks RSA encryption using QFT.

Final Cheat Sheet

Concept Formula / Mnemonic
Qubit State \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\)
Hadamard Gate \(H = \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)\)
Entanglement \(|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)\)
Measurement Probability \(P(0) = |\alpha|^2, P(1) = |\beta|^2\)
Deutsch’s Algorithm Uses Hadamard, CNOT, Hadamard
QFT Phase shift + Hadamard