Day 2 : Complex numbers, Probability Theory, and Calculus in Quantum Computing

Self Introduction¶

Hey there! I’m Rohan Sai, aka Aiknight.

Here’s a joke for you:

Why did the quantum computer refuse to share its snacks? Because other person couldn’t find where they are exactly! 😄

In Day 1, we kicked off with the foundations of quantum computing.

Today in Day 2, we’re getting mathematical with complex numbers, probability theory, and calculus—the building blocks of quantum mechanics.

Stick around, and let’s decode the quantum universe together! 🚀

Did you know?¶

Quantum particles can "tunnel" through barriers, literally defying classical physics. Mind-blowing, isnt it ?

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1. Quantum Computing¶

Concept:¶

Quantum computing is a revolutionary area of computing that harnesses the principles of quantum mechanics to process information. Unlike classical computing, where bits represent either a 0 or 1, quantum computing uses quantum bits or qubits that can represent both 0 and 1 simultaneously. This property is known as superposition. In addition to superposition, quantum computing also leverages phenomena like entanglement and quantum interference.

Key Principles:¶

1. Superposition:

   • A qubit can exist in multiple states (0, 1, or any quantum superposition) at once. This allows quantum computers to handle a vast amount of data at once.

2. Entanglement:

   • When qubits become entangled, the state of one qubit is directly related to the state of another, no matter the distance between them. This allows faster and more complex computations.

3. Quantum Interference:

   • Quantum algorithms utilize interference to amplify the probability of correct answers and suppress incorrect ones.

Types of Quantum Computing:¶

1. Gate-based Quantum Computing:

   • This model uses quantum gates to manipulate qubits. The quantum gates are analogous to classical logic gates but operate on quantum bits.

2. Quantum Annealing:

   • Quantum annealing is used to find the minimum energy state of a system. It’s particularly useful for optimization problems.

3. Topological Quantum Computing:

   • This is an advanced model that relies on qubits that are more resistant to error due to their topological properties.

Example:¶

A simple quantum algorithm like Grover's algorithm can be used to search an unsorted database more efficiently than classical algorithms. In Grover's algorithm, the quantum computer finds the marked item in √N steps, where N is the number of items in the database, instead of N steps.

Formulae:¶

1. Superposition: $ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle $ where $ \alpha $ and $ \beta $ are complex numbers representing the probability amplitudes, and $ |\psi\rangle $ is the state of the qubit.

2. Quantum Gate Operations:

   • Hadamard gate (H) to create superposition: $ H = \frac{1}{\sqrt{2}} $

     \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

Code Implementation:¶

Using Python and Qiskit, a popular quantum computing library:

from qiskit import QuantumCircuit, Aer, execute

# Create a quantum circuit with 1 qubit
qc = QuantumCircuit(1, 1)

# Apply Hadamard gate to the qubit
qc.h(0)

# Measure the qubit
qc.measure(0, 0)

# Execute the circuit on a simulator
simulator = Aer.get_backend('qasm_simulator')
result = execute(qc, simulator, shots=1024).result()

# Get the result
counts = result.get_counts()
print("Result: ", counts)

Benefits:¶

• Can solve problems faster than classical computers (e.g., factoring large numbers, simulating quantum systems).
• Useful for optimization, cryptography, and machine learning.

Demerits:¶

• Requires stable qubits, which are hard to maintain.
• Quantum computers are still in the early stages of development, and there are issues like error correction and decoherence.

Sorry for getting out of context , its just exciting to write about it thats it

So getting into the topics of the day... Lets start with complex numbers

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2. Complex Numbers¶

Concept:¶

A complex number is a number that has both a real and an imaginary component.

It is written in the form:

$ z = a + bi $

where:

• $ a $ is the real part,
• $ b $ is the imaginary part, and
• $ i $ is the imaginary unit, defined as $ i^2 = -1 $.

Operations with Complex Numbers:¶

1. Addition/Subtraction:

   $ (a + bi) + (c + di) = (a+c) + (b+d)i $

2. Multiplication:

   $ (a + bi)(c + di) = (ac - bd) + (ad + bc)i $

3. Division:

   $ \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} $

4. Modulus (Magnitude):

   $ |z| = \sqrt{a^2 + b^2} $

5. Argument (Angle):

   $ \theta = \text{atan2}(b, a) $

6. Conjugate:

   $ \text{Conjugate}(a + bi) = a - bi $

Example:¶

For $ z = 3 + 4i $:

• Modulus $ |z| = \sqrt{3^2 + 4^2} = 5 $
• Conjugate of $ z = 3 - 4i $

Code Implementation:¶

In Python, complex numbers are supported directly:

# Create complex numbers
z1 = complex(3, 4)
z2 = complex(1, 2)

# Addition
z3 = z1 + z2

# Modulus
modulus = abs(z1)

# Conjugate
conjugate = z1.conjugate()

print("Sum:", z3)
print("Modulus:", modulus)
print("Conjugate:", conjugate)

Benefits:¶

• Useful in many fields, such as electrical engineering, fluid dynamics, quantum mechanics, and signal processing.

Demerits:¶

• Not applicable in all scenarios (only in contexts where imaginary numbers make sense).

I know.. stay with me for more , next up is vector spaces

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3. Vector Spaces¶

Concept:¶

A vector space is a collection of vectors, which are objects that can be added together and scaled by numbers, known as scalars. The set of scalars is typically a field, such as real numbers, complex numbers, or rational numbers.

Axioms of Vector Space:¶

1. Closure under addition and scalar multiplication.
2. Commutativity of addition.
3. Associativity of addition.
4. Existence of a zero vector (additive identity).
5. Existence of additive inverses.
6. Distributive properties.

Types of Vector Spaces:¶

1. Euclidean Space:

   • Real-numbered vectors.

2. Function Space:

   • Vectors are functions, and the operations are defined for these functions.

3. Polynomial Space:

   • Vectors are polynomials of degree less than or equal to n.

Example:¶

Consider the vector space $ \mathbb{R}^2 $, consisting of 2-dimensional vectors like:

$ v = (x, y) $

where $ x, y \in \mathbb{R} $.

Operations such as vector addition and scalar multiplication can be performed:

• Addition:

  $ (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) $

• Scalar Multiplication:

  $ k(x, y) = (kx, ky) $

Code Implementation (Using NumPy):¶

import numpy as np

# Define two vectors
v1 = np.array([2, 3])
v2 = np.array([1, 4])

# Vector addition
v_add = v1 + v2

# Scalar multiplication
scalar = 3
v_scalar = scalar * v1

print("Vector addition:", v_add)
print("Scalar multiplication:", v_scalar)

Benefits:¶

• Vectors are central in physics and engineering, especially in the fields of linear transformations, machine learning, and computer graphics.
• Provides the foundation for linear algebra, essential for data science and optimization.

Demerits:¶

• For higher-dimensional spaces, it becomes harder to visualize.
• Complex operations can be computationally intensive.

Below is a basic project for complex number calculation do check it out

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class ComplexNumber:
    """A class to represent a complex number and perform advanced operations."""

    def __init__(self, real: float, imag: float):
        self.real = real
        self.imag = imag

    def __repr__(self):
        return f"{self.real} + {self.imag}i" if self.imag >= 0 else f"{self.real} - {-self.imag}i"

    def __add__(self, other):
        """Addition of two complex numbers."""
        return ComplexNumber(self.real + other.real, self.imag + other.imag)

    def __sub__(self, other):
        """Subtraction of two complex numbers."""
        return ComplexNumber(self.real - other.real, self.imag - other.imag)

    def __mul__(self, other):
        """Multiplication of two complex numbers."""
        real_part = self.real * other.real - self.imag * other.imag
        imag_part = self.real * other.imag + self.imag * other.real
        return ComplexNumber(real_part, imag_part)

    def __truediv__(self, other):
        """Division of two complex numbers."""
        denominator = other.real**2 + other.imag**2
        if denominator == 0:
            raise ValueError("Cannot divide by zero in complex numbers.")
        real_part = (self.real * other.real + self.imag * other.imag) / denominator
        imag_part = (self.imag * other.real - self.real * other.imag) / denominator
        return ComplexNumber(real_part, imag_part)

    def magnitude(self):
        """Returns the magnitude of the complex number."""
        return (self.real**2 + self.imag**2)**0.5

    def conjugate(self):
        """Returns the conjugate of the complex number."""
        return ComplexNumber(self.real, -self.imag)

    def polar_coordinates(self):
        """Returns the polar representation (magnitude, angle in radians)."""
        import math
        magnitude = self.magnitude()
        angle = math.atan2(self.imag, self.real)
        return magnitude, angle

    @staticmethod
    def from_polar(magnitude, angle):
        """Creates a complex number from polar coordinates."""
        import math
        real = magnitude * math.cos(angle)
        imag = magnitude * math.sin(angle)
        return ComplexNumber(real, imag)


def display_operations():
    """Demonstrates operations on complex numbers with explanations."""
    print("-- Complex Number Calculator Demonstration --\n")

    # Creating two complex numbers
    c1 = ComplexNumber(3, 4)
    c2 = ComplexNumber(1, -2)
    print(f"Complex Number 1: {c1}")
    print(f"Complex Number 2: {c2}\n")

    # Addition
    print(f"Addition: {c1} + {c2} = {c1 + c2}")

    # Subtraction
    print(f"Subtraction: {c1} - {c2} = {c1 - c2}")

    # Multiplication
    print(f"Multiplication: {c1} * {c2} = {c1 * c2}")

    # Division
    try:
        print(f"Division: {c1} / {c2} = {c1 / c2}")
    except ValueError as e:
        print(f"Division Error: {e}")

    # Magnitude
    print(f"Magnitude of {c1}: {c1.magnitude()}")
    print(f"Magnitude of {c2}: {c2.magnitude()}")

    # Conjugate
    print(f"Conjugate of {c1}: {c1.conjugate()}")
    print(f"Conjugate of {c2}: {c2.conjugate()}")

    # Polar Coordinates
    mag, angle = c1.polar_coordinates()
    print(f"Polar Coordinates of {c1}: (Magnitude: {mag}, Angle: {angle} radians)")

    # From Polar
    print(f"Complex Number from Polar (Magnitude: {mag}, Angle: {angle} radians): {ComplexNumber.from_polar(mag, angle)}")


if __name__ == "__main__":
    display_operations()

Now let's dive into Probability Theory and Statistics for Quantum Computing

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Probability Theory & Statistics in Quantum Computing¶

Introduction¶

In quantum computing, probability and statistics are central because quantum systems are probabilistic in nature. The wavefunction, which describes the state of a quantum system, provides probabilities for different outcomes upon measurement. Understanding probability and statistical concepts helps us interpret these outcomes and analyze the behavior of quantum algorithms.

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1. Probability Theory in Quantum Computing¶

1.1 Basics of Quantum Probability¶

In classical systems, probabilities arise due to lack of information, but in quantum systems, probabilities are fundamental. They emerge from the Born rule, which connects the wavefunction to observable probabilities.

1.2 Concepts in Quantum Probability¶

1. State Vector and Amplitude:
   A quantum state is represented as a vector in a Hilbert space:

   $ |\psi\rangle = \sum_{i} c_i |i\rangle $

   Here, $ c_i $ are complex amplitudes. The probability of observing state $ |i\rangle $ is:

   $ P(i) = |c_i|^2 $

   Ex : State vector is like a list of possibilities or possible states of a qubit. Consider flipping a coin, which is single qubit it represents the state of both head and tail simultaneously

   and

   Amplitude is like the likelihood of a state. It represents the probability of measuring the qubit in a specific state (heads or tails) when the superposition is observed.

   Like peekaboo and there is your amplitude...JK 😄

2. Born Rule:
   The probability of measuring a quantum system in a particular state is the squared magnitude of the amplitude.

Example:
For a qubit in the state: $ |\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle $ The probabilities are:

• $ P(0) = \left|\frac{1}{\sqrt{2}}\right|^2 = 0.5 $

• $ P(1) = \left|\frac{1}{\sqrt{2}}\right|^2 = 0.5 $

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1.3 Quantum Superposition and Probabilities¶

In superposition:

$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle $

The probabilities are:

$ P(0) = |\alpha|^2, \quad P(1) = |\beta|^2 $

Such probabilities must satisfy normalization: $ |\alpha|^2 + |\beta|^2 = 1 $

Example:
For $ \alpha = \frac{1}{2}, \beta = \frac{\sqrt{3}}{2} $ :

$ P(0) = \frac{1}{4}, \quad P(1) = \frac{3}{4} $

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1.4 Types of Probabilities in Quantum Computing¶

1. Measurement Probability: Probabilities of outcomes after measuring a quantum state.

2. Transition Probability: Probability of transitioning between states.

   $ P_{\text{transition}} = |\langle \phi | \psi \rangle|^2 $

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1.5 Quantum Entanglement and Joint Probabilities¶

For entangled states: $ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) $ The joint probabilities: $ P(00) = P(11) = \frac{1}{2}, \quad P(01) = P(10) = 0 $

If that was more to fit in , take a break and come back ... 😄

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2. Statistics in Quantum Computing¶

2.1 Statistical Description of Quantum Systems¶

1. Expectation Value:

   For an observable $ \hat{A} $: $ \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle $

   Example:
   For a qubit in $ |\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) $, the expectation of $\hat{\sigma}_z $ (Pauli-Z matrix) is:

   $ \langle \psi | \hat{\sigma}_z | \psi \rangle = 0 $

2. Variance:

   $ \text{Var}(\hat{A}) = \langle \hat{A}^2 \rangle - (\langle \hat{A} \rangle)^2 $

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2.2 Density Matrices and Mixed States¶

1. Pure States:
   A quantum state $ |\psi\rangle $ can be represented as:

   $ \rho = |\psi\rangle\langle\psi| $

2. Mixed States:
   For probabilities $ p_i $:

   $ \rho = \sum_i p_i |\psi_i\rangle\langle\psi_i| $

Example:
If $ |\psi_1\rangle = |0\rangle $, $ |\psi_2\rangle = |1\rangle $, and $ p_1 = p_2 = 0.5 $:

$ \rho = 0.5|0\rangle\langle0| + 0.5|1\rangle\langle1| $

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2.3 Statistical Tools in Quantum Computing¶

1. Fidelity: Measures similarity between quantum states.

   $ F(\rho, \sigma) = \left(\text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2 $

2. Entropy:

   Von Neumann Entropy: $ S(\rho) = -\text{Tr}(\rho \log \rho) $

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3. Code Implementations¶

3.1 Probability of Measurement¶

import numpy as np

# Quantum state amplitudes
alpha = 1 / np.sqrt(2)
beta = 1 / np.sqrt(2)

# Probabilities
P_0 = abs(alpha)**2
P_1 = abs(beta)**2

print(f"Probability of |0>: {P_0}")
print(f"Probability of |1>: {P_1}")

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3.2 Expectation Value¶

import numpy as np

# Pauli-Z operator
sigma_z = np.array([[1, 0], [0, -1]])

# Quantum state
psi = np.array([1 / np.sqrt(2), 1 / np.sqrt(2)])

# Expectation value
expectation_value = np.dot(psi.conj().T, np.dot(sigma_z, psi))
print(f"Expectation value: {expectation_value}")

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3.3 Density Matrix¶

import numpy as np

# Pure state
psi = np.array([1 / np.sqrt(2), 1 / np.sqrt(2)])

# Density matrix
rho = np.outer(psi, psi.conj())
print("Density Matrix:")
print(rho)

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3.4 Fidelity Calculation¶

from scipy.linalg import sqrtm

# Density matrices
rho = np.array([[0.5, 0.5], [0.5, 0.5]])
sigma = np.array([[1, 0], [0, 0]])

# Fidelity
sqrt_rho = sqrtm(rho)
fidelity = np.trace(sqrtm(np.dot(sqrt_rho, np.dot(sigma, sqrt_rho))))**2
print(f"Fidelity: {fidelity}")

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Benefits and Demerits¶

Benefits¶

1. Enables accurate modeling of quantum measurements.
2. Essential for designing and analyzing quantum algorithms.
3. Provides tools for characterizing quantum systems (e.g., entanglement).

Demerits¶

1. Computationally intensive for large systems.
2. Probabilistic nature can be counterintuitive.
3. Requires advanced mathematical tools.

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Time for a project

Probability Simulator¶

Concept Explanation¶

Overview¶

A Probability Simulator is a tool designed to simulate, visualize, and analyze various probability distributions and statistical scenarios. It allows users to understand complex probabilistic concepts through interactive and computational methods. The simulator can handle multiple probability distributions (e.g., Binomial, Normal, Poisson), perform Monte Carlo simulations, and visualize outcomes through graphs and statistical metrics.

Objectives¶

1. Simulate different types of probability distributions.
2. Perform custom experiments with varying parameters.
3. Visualize results using interactive plots.
4. Provide statistical insights and metrics for better understanding.

Key Components¶

1. Distributions: Implementation of common distributions like Uniform, Normal, Binomial, Poisson, etc.
2. Random Experiments: Simulate coin flips, dice rolls, and other probabilistic scenarios.
3. Visualization: Generate histograms, line plots, and scatter plots to understand distributions and outcomes.
4. Monte Carlo Simulations: Estimate probabilities and statistical metrics through repeated random sampling.
5. User Interface: Provide a simple interface for users to input parameters and view results.

Code Implementation¶

For code implementation do try it out in my colab Colab Link

OUTPUT :

Estimated Pi: 3.1408¶

Calculus for Quantum Computing¶

Introduction¶

Calculus forms a foundational element in understanding quantum computing. It provides tools for analyzing continuous change, optimizing systems, and modeling quantum behaviors like wavefunctions, probabilities, and amplitudes.

Quantum mechanics, the backbone of quantum computing, heavily relies on calculus concepts like differentiation and integration. These concepts help describe the time evolution of quantum states, probabilities of measurements, and optimization in quantum algorithms.

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1. Basic Concepts of Calculus¶

1.1 Differentiation¶

Differentiation involves finding the rate of change of a function. In quantum computing, this is useful for:

• Describing the time evolution of quantum states.
• Optimizing quantum algorithms using gradient-based methods.

Formula:¶

$ \frac{d}{dx} f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $

Example:¶

Given a wavefunction $ \psi(x) = e^{-x^2} $,

find $ \frac{d\psi}{dx} $:

$ \frac{d\psi}{dx} = \frac{d}{dx} e^{-x^2} = -2x e^{-x^2} $

Code Implementation:¶

import sympy as sp

x = sp.symbols('x')
psi = sp.exp(-x**2)
derivative = sp.diff(psi, x)
print(f"Derivative: {derivative}")

1.2 Integration¶

Integration calculates the area under a curve and is vital for:

• Normalizing quantum wavefunctions.
• Computing probabilities.

Formula:¶

$ \int_a^b f(x) dx = F(b) - F(a) $

where $ F(x) $ is the antiderivative of $f(x)$.

Example:¶

Normalize the wavefunction $ \psi(x) = e^{-x^2} $:

$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} $

Code Implementation:¶

import sympy as sp

integral = sp.integrate(sp.exp(-x**2), (x, -sp.oo, sp.oo))
print(f"Integral: {integral}")

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2. Types of Calculus in Quantum Computing¶

2.1 Differential Calculus¶

Used for analyzing instantaneous rates of change in quantum systems.

Example: Time Evolution¶

The Schrödinger equation describes time evolution:

$ i\hbar \frac{\partial}{\partial t} \psi(x, t) = \hat{H} \psi(x, t) $

where $ \hat{H} $ is the Hamiltonian operator, and $ \psi(x, t) $ is the wavefunction.

Code Implementation:¶

# Solving Schrödinger's Equation
from sympy.physics.quantum import Operator

t = sp.symbols('t')
psi_t = sp.Function('psi')(t)
H = Operator('H')
schrodinger_eq = sp.I * sp.Symbol('hbar') * sp.diff(psi_t, t) - H * psi_t
print(f"Schrödinger Equation: {schrodinger_eq}")

2.2 Integral Calculus¶

Integral calculus is central to probability computations and system analysis.

Example: Probability Density¶

Calculate the probability of a particle in a region $ [a, b] $:

$ P(a \leq x \leq b) = \int_a^b |\psi(x)|^2 dx $

Code Implementation:¶

psi_squared = sp.Abs(psi)**2
probability = sp.integrate(psi_squared, (x, -1, 1))
print(f"Probability: {probability}")

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3. Advanced Calculus in Quantum Computing¶

3.1 Variational Calculus¶

Used in variational quantum algorithms like VQE (Variational Quantum Eigensolver).

Principle:¶

Find $ \psi $ that minimizes $ E = \langle \psi | \hat{H} | \psi \rangle $.

Example:¶

Variational Ansatz

Choose a trial wavefunction $ \psi(\theta) $ and optimize $ \theta $:

$ E(\theta) = \langle \psi(\theta) | \hat{H} | \psi(\theta) \rangle $

Code Implementation:¶

from scipy.optimize import minimize

# Example Hamiltonian and trial function
def energy(theta):
    H = 1  # Example Hamiltonian
    psi_theta = sp.cos(theta)**2
    return H * psi_theta

result = minimize(energy, x0=0.5)
print(f"Optimal theta: {result.x}")

3.2 Functional Derivatives¶

In quantum field theory and advanced quantum algorithms, functional derivatives are used to handle systems with infinitely many degrees of freedom.

Formula:¶

$ \frac{\delta F[\phi]}{\delta \phi(x)} $

Example: Euler-Lagrange Equation¶

$ \frac{\partial L}{\partial \phi} - \frac{d}{dx} \left( \frac{\partial L}{\partial \phi'} \right) = 0 $

3.3 Advanced Optimization Techniques¶

Optimization is critical in quantum computing for training quantum machine learning models, finding ground states, and tuning algorithm parameters.

Gradient Descent in Quantum Algorithms¶

In Quantum Approximate Optimization Algorithm (QAOA), gradient descent is used to optimize parameters $ \gamma $ and $ \beta $:

$ \nabla_{\theta} C(\theta) = \frac{\partial C}{\partial \theta} $

3.4 Quantum Natural Gradient¶

Natural gradients consider the geometry of the parameter space:

$ \theta_{t+1} = \theta_t - \eta F^{-1} \nabla C(\theta) $

where $ F $ is the Fisher information matrix.

Example: Quantum Variational Algorithms¶

Natural gradients improve convergence in training quantum models.

I will cover more of it in furthur chapters ahead and adding more as course moves on

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4. Applications and Implications¶

4.1 Benefits¶

• Precise modeling of quantum systems.
• Optimization of quantum algorithms.
• Robust analysis of quantum probabilities.

4.2 Limitations¶

• Computational complexity.
• Sensitivity to errors in numerical implementations.

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Don't forget to checkout the projects which are Probability Simulator and Function Visualizer at Colab Link

Stay Tuned For Day 3 wherein i will come with more of Quantum Stuff.

If you like my work you can follow me on Linkedin and X

Happy Learning...