---
title: "Quantum Study Guide"
author: "Jessica McPhaul"
date: "2025-02-06"
format: 
  html:
    theme: cosmo
    code-fold: true
    html-math-method: katex
server: shiny
execute:
  echo: true
---

## Shiny Interactive Component

::: {.cell}

```{.r .cell-code}
library(shiny)

:::

fluidPage(
  titlePanel("Old Faithful Geyser Data"),
  sidebarLayout(
    sidebarPanel(
      sliderInput("bins",
                  "Number of bins:",
                  min = 1,
                  max = 50,
                  value = 30)
    ),
    mainPanel(
      plotOutput("distPlot")
    )
  )
)

Old Faithful Geyser Data

function(input, output, session) {
  output$distPlot <- renderPlot({
    x <- faithful[, 2]
    bins <- seq(min(x), max(x), length.out = input$bins + 1)
    
    hist(x, breaks = bins, 
         col = 'darkgray', 
         border = 'white',
         xlab = 'Waiting time to next eruption (in mins)',
         main = 'Histogram of waiting times')
  })
}
function (input, output, session) 
{
    output$distPlot <- renderPlot({
        x <- faithful[, 2]
        bins <- seq(min(x), max(x), length.out = input$bins + 
            1)
        hist(x, breaks = bins, col = "darkgray", border = "white", 
            xlab = "Waiting time to next eruption (in mins)", 
            main = "Histogram of waiting times")
    })
}

Quantum Computing: Math & Theory Study Guide

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): \(|\psi\rangle = \begin{pmatrix} a \\ b \end{pmatrix}\)
    • Row (Bra): \(\langle\psi| = \begin{pmatrix} a & b \end{pmatrix}\)
    • Inner Product: \(\langle\phi | \psi \rangle\) = dot product
    • Outer Product: \(|\psi\rangle \langle\phi|\) = matrix multiplication
  • Matrices:
    • Identity: \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
    • 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: \(|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

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) \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) Flips \(|0\rangle \leftrightarrow |1\rangle\)
Pauli-Y \(\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\) Phase + Flip
Pauli-Z \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) Phase shift
Hadamard (H) \(\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\) Superposition
Phase Gate (S) \(\begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}\) Phase shift
T Gate \(\begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}\) Phase shift

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 

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