Introduction to Quantum Computing

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February 5, 2025

Introduction

This lecture serves as an introductory course to quantum mechanics, specifically tailored for quantum computing and quantum communication. As these fields are under very active development, we will focus on the fundamental principles. This course aims to provide a comprehensive overview, establishing a foundation for more advanced topics. We will begin with a general introduction to the subject, covering essential formalism and key concepts. This will be followed by a formal introduction to linear algebra and the principles of quantum mechanics. The course will then progress through quantum statistical mechanics, density matrices, quantum circuits, quantum complexity, quantum algorithms, and quantum communication, concluding with an overview of quantum computer implementations.

Course Textbook

The primary textbook for this course is:

Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang, Cambridge University Press.

This is a standard reference in the field, offering a comprehensive resource for further study.

Course Program

The program will cover the following topics:

General Introduction: An overview of quantum computing to provide context and motivation.
Linear Algebra: A formal introduction to the necessary mathematical tools.
Quantum Mechanics Principles: Fundamental principles of quantum mechanics, focusing on systems relevant to quantum computing, particularly qubits.
Quantum Statistical Mechanics and Density Matrix: Introduction to quantum statistical mechanics and the density matrix formalism for systems with incomplete information.
Quantum Circuits: Representation and operation of quantum circuits, and their differences from classical circuits.
Quantum Complexity: An analysis of quantum computational complexity compared to classical complexity.
Quantum Algorithms: Exploration of key quantum algorithms.
Quantum Communication: Introduction to the principles of quantum communication.
Quantum Computer Realizations: A brief overview of the physical implementations of quantum computers.

Mathematical Preliminaries: Complex Numbers

Motivation for Complex Numbers

Complex numbers extend the real number system to provide solutions for equations like $x^2 = -1$. This extension is analogous to how number systems evolved from natural numbers to integers, rationals, and reals to overcome limitations in mathematical operations. Complex numbers are thus introduced to solve equations that have no solutions within the real numbers.

Definition and Components

A complex number $z$ is expressed as $z = a + ib$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined by $i^2 = -1$.

Real and Imaginary Parts

For $z = a + ib$, the real part of $z$, denoted as $\operatorname{Re}(z)$, is $a$. We write $\operatorname{Re}(z) = a$.

For $z = a + ib$, the imaginary part of $z$, denoted as $\operatorname{Im}(z)$, is $b$. We write $\operatorname{Im}(z) = b$.

Remark. Remark 1. Note: the imaginary part is the real number $b$, not $ib$.

Operations with Complex Numbers

Multiplication

For two complex numbers $z_1 = a + ib$ and $z_2 = c + id$, their product is: \[\begin{aligned} z_1 \cdot z_2 &= (a + ib)(c + id) \\ &= ac + iad + ibc + i^2bd \\ &= (ac - bd) + i(ad + bc) \end{aligned}\] The real part of the product is $ac - bd$, and the imaginary part is $ad + bc$.

Complex Conjugation

The complex conjugate of $z = a + ib$, denoted as $z^*$, is obtained by reversing the sign of the imaginary part:

Modulus of a Complex Number

The modulus (or absolute value) of $z = a + ib$, denoted as $|z|$, is a non-negative real number given by: \[\begin{aligned} |z|^2 &= z \cdot z^* = (a + ib)(a - ib) = a^2 + b^2 \\ |z| &= \sqrt{a^2 + b^2} = \sqrt{z \cdot z^*} \end{aligned}\] The modulus represents the distance from the origin to $z$ in the complex plane.

Geometric Representation: Complex Plane

The complex plane represents complex numbers geometrically:

The horizontal axis is the real axis.
The vertical axis is the imaginary axis.

A complex number $z = a + ib$ corresponds to a point $(a, b)$ or a vector from the origin to $(a,b)$. The conjugate $z^* = a - ib$ is the reflection of $z$ across the real axis, and $|z|$ is the vector’s length.

In polar coordinates $(r, \phi)$, where $r = |z|$ and $\phi$ is the argument, we have: \[\begin{aligned} a &= r \cos(\phi) \\ b &= r \sin(\phi) \end{aligned}\] Thus, $z = r(\cos(\phi) + i\sin(\phi))$.

Exponential Form: Euler’s Formula

Euler’s formula states that for any real $\phi$: This is derived from Taylor series expansions. Using Euler’s formula, the polar form becomes the exponential form: \[z = re^{i\phi}\]

Remark. Remark 2. Note that $|e^{i\phi}| = \sqrt{\cos^2(\phi) + \sin^2(\phi)} = 1$.

For multiplication of $z_1 = r_1e^{i\phi_1}$ and $z_2 = r_2e^{i\phi_2}$: The argument $\phi$ can be found using: \[\phi = \arctan\left(\frac{b}{a}\right) = \arctan\left(\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right)\]

Introduction to Quantum Computing and Cubits

Quantum Mechanics for Quantum Computing

Quantum computing utilizes principles of quantum mechanics to perform computations, offering advantages over classical computing for specific problems. Unlike classical bits, which are either 0 or 1, quantum computers use qubits.

Cubits: Superposition and Quantum Information

Superposition

Qubits can exist in a superposition of states, representing 0 and 1 simultaneously. This is a fundamental departure from classical bits and allows qubits to hold more information and perform computations in fundamentally different ways. A qubit state is described as a linear combination of the basis states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$.

Wave Function Representation

The state of a qubit is represented by a wave function: \[\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle\] where $\alpha$ and $\beta$ are complex numbers called probability amplitudes, satisfying the normalization condition $|\alpha|^2 + |\beta|^2 = 1$.

Information Storage and Access

While the coefficients $\alpha$ and $\beta$ can, in principle, encode infinite information, measurement of a qubit yields only classical outcomes, either 0 or 1. The probabilities of these outcomes are given by $|\alpha|^2$ for state $\left|{0}\right\rangle$ and $|\beta|^2$ for state $\left|{1}\right\rangle$.

Advantage of Superposition

Superposition enables quantum computers to perform operations in parallel on multiple states simultaneously. This inherent parallelism is a key source of quantum computational power, allowing for potentially exponential speedups for certain algorithms compared to classical approaches. However, accessing this information is constrained by the measurement process, which always results in a classical bit.

Limitations and Applications

Due to the nature of measurement and information access in quantum systems, directly translating classical algorithms to quantum algorithms is not always straightforward or advantageous. Quantum computing excels in specific types of problems, particularly those where superposition and entanglement can be effectively utilized, such as problems with yes/no answers or those requiring parallel exploration of multiple possibilities.

Physical Realizations of Cubits

Cubits can be physically implemented using various quantum systems, including:

Photon Polarization: Using horizontal and vertical polarizations of photons to represent $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$.
Electron Spin: Utilizing the spin-up and spin-down states of electrons.
Nuclear Spin: Employing the spin states of atomic nuclei, such as hydrogen nuclei.
Atomic Energy Levels: Using two distinct energy levels of an atom to define qubit states.

These physical systems allow for the manipulation and measurement of qubit states, forming the hardware for quantum computation.

Cubits: Quantum Bits

Classical Bits vs. Cubits

Classical bits are deterministic, existing in either state 0 or 1. In contrast, qubits, the quantum analogue, can exist in a superposition of states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$.

Superposition and Cubit State

Superposition Principle

The superposition principle is described as:

A qubit can be in a linear combination of basis states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$, described by the superposition principle. This allows a qubit to exist in both states simultaneously until measured.

Wave Function

The state of a qubit is represented by a wave function or quantum state $\left|{\psi}\right\rangle$: \[\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle\] Here, $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$ are the computational basis states, and $\alpha, \beta \in \mathbb{C}$ are complex probability amplitudes.

Normalization Condition

The coefficients $\alpha$ and $\beta$ must satisfy the normalization condition: \[|\alpha|^2 + |\beta|^2 = 1\] This ensures that the total probability of measurement outcomes is unity.

Information and Measurement

Information Capacity vs. Measurement

While qubit states are described by complex numbers $\alpha$ and $\beta$, potentially encoding infinite information, measurement outcomes are limited to classical states $\left|{0}\right\rangle$ or $\left|{1}\right\rangle$. Measurement does not directly reveal $\alpha$ and $\beta$.

Probabilistic Measurement

Probabilistic measurement is described as:

Measurement of a qubit in state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ is probabilistic. The probabilities are:

Probability of measuring $\left|{0}\right\rangle$: $|\alpha|^2$
Probability of measuring $\left|{1}\right\rangle$: $|\beta|^2$

The sum of probabilities is $|\alpha|^2 + |\beta|^2 = 1$.

State Representation and Bloch Sphere

Phase Factor Representation

Phase factor representation is given by:

A qubit state can be written incorporating phase factors: \[\left|{\psi}\right\rangle = \cos\left(\frac{\theta}{2}\right)\left|{0}\right\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\left|{1}\right\rangle\] where $\theta$ and $\phi$ are real numbers.

Remark. Remark 3. A global phase factor $e^{i\gamma}$ can be factored out without affecting measurement probabilities: \[\left|{\psi}\right\rangle = e^{i\gamma}\left( \cos\left(\frac{\theta}{2}\right)\left|{0}\right\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\left|{1}\right\rangle \right)\]

Bloch Sphere

The Bloch sphere provides a visual representation of qubit states. A qubit state is mapped to a point on the sphere, defined by angles $\theta$ (polar angle) and $\phi$ (azimuthal angle).

$\theta$ determines the probabilities of measuring $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$.
$\phi$ represents the relative phase between $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$.

$\left|{0}\right\rangle$ and $\left|{1}\right\rangle$ are conventionally placed at the poles of the sphere.

Physical Cubits

Physical systems that can represent qubits include:

Photon Polarization: Polarization states of photons.
Electron Spin: Spin states of electrons.
Nuclear Spin: Spin states of atomic nuclei (e.g., Hydrogen).
Atomic Energy Levels: Two energy levels within an atom.

These systems can be manipulated and measured to perform quantum computations. State preparation typically involves initializing the system into a known quantum state.

Multiple Cubits and Entanglement

Multi-Qubit Systems and Tensor Products

Quantum computers utilize multiple qubits to achieve computational power beyond single qubits. The state space of a multi-qubit system is constructed from the tensor product of individual qubit state spaces. For a two-qubit system composed of qubit 1 and qubit 2, if qubit 1 is in state $\left|{\psi_1}\right\rangle$ and qubit 2 is in state $\left|{\psi_2}\right\rangle$, the combined state is given by the tensor product $\left|{\psi_1}\right\rangle \otimes \left|{\psi_2}\right\rangle$, often written as $\left|{\psi_1, \psi_2}\right\rangle$ or $\left|{\psi_1}\right\rangle\left|{\psi_2}\right\rangle$.

Two-Qubit State Representation

Basis States for Two Cubits

The computational basis for a two-qubit system is formed by the tensor product of the single-qubit basis states $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$:

$\left|{00}\right\rangle = \left|{0}\right\rangle \otimes \left|{0}\right\rangle$ (Both qubits in state $\left|{0}\right\rangle$)
$\left|{01}\right\rangle = \left|{0}\right\rangle \otimes \left|{1}\right\rangle$ (Qubit 1 in $\left|{0}\right\rangle$, Qubit 2 in $\left|{1}\right\rangle$)
$\left|{10}\right\rangle = \left|{1}\right\rangle \otimes \left|{0}\right\rangle$ (Qubit 1 in $\left|{1}\right\rangle$, Qubit 2 in $\left|{0}\right\rangle$)
$\left|{11}\right\rangle = \left|{1}\right\rangle \otimes \left|{1}\right\rangle$ (Both qubits in state $\left|{1}\right\rangle$)

A general two-qubit state $\left|{\Psi}\right\rangle$ is a superposition of these basis states: \[\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle\] where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ are complex coefficients.

Normalization

For $\left|{\Psi}\right\rangle$ to represent a valid quantum state, the coefficients must satisfy the normalization condition: \[|\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1\]

Measurement Probabilities

Joint Measurement Probabilities

Joint measurement probabilities are given by:

For a state $\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle$, the probability of measuring each basis state is given by the squared modulus of the corresponding coefficient:

$P(00) = |\alpha|^2$
$P(01) = |\beta|^2$
$P(10) = |\gamma|^2$
$P(11) = |\delta|^2$

Marginal Probabilities

Marginal probabilities are described as:

Marginal probabilities refer to the probability of measuring a specific state for one qubit, irrespective of the other qubit’s state. For example, the probability of measuring the first qubit in state $\left|{0}\right\rangle$ is: \[P(\text{Qubit 1 = 0}) = P(00) + P(01) = |\alpha|^2 + |\beta|^2\] Similarly, for the second qubit being in state $\left|{0}\right\rangle$: \[P(\text{Qubit 2 = 0}) = P(00) + P(10) = |\alpha|^2 + |\gamma|^2\]

Wave Packet Reduction and Conditional Probability

Measurement on a qubit alters the state of the system, a process known as wave packet reduction or state collapse. If a measurement of the first qubit yields $\left|{0}\right\rangle$ for the state $\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle$, the state collapses to: \[\left|{\Psi'}\right\rangle = \frac{\alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle}{\sqrt{|\alpha|^2 + |\beta|^2}}\] The probabilities for subsequent measurements on the second qubit are now conditional on the first measurement’s outcome. For example, the conditional probability of measuring the second qubit in state $\left|{0}\right\rangle$, given the first qubit was measured as $\left|{0}\right\rangle$, is $|\alpha|^2 / (|\alpha|^2 + |\beta|^2)$.

Quantum Entanglement

Non-Separability

Entanglement occurs when a multi-qubit state cannot be represented as a tensor product of single-qubit states. These entangled states exhibit quantum correlations that are fundamentally non-classical.

Non-Classical Correlations

Entangled qubits are correlated in a way that measuring one qubit instantaneously influences the state of the other, regardless of physical separation. This correlation is stronger than any classical correlation and is a crucial resource in quantum computing and communication.

Bell States

Bell states are maximally entangled two-qubit states, essential for quantum communication protocols. An example is the $\left|{\Phi^+}\right\rangle$ Bell state: \[\left|{\Phi^+}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)\] In $\left|{\Phi^+}\right\rangle$, measuring the first qubit in state $\left|{0}\right\rangle$ guarantees that the second qubit will also be measured in state $\left|{0}\right\rangle$, and similarly for $\left|{1}\right\rangle$.

Bell Inequalities and Violation

Bell inequalities define the limits of classical correlations. Quantum mechanics predicts, and experiments confirm, that entangled states violate Bell inequalities. This violation demonstrates that quantum correlations are fundamentally different from classical correlations and cannot be explained by classical local realism.

Quantum Gates: Operations on Cubits

Quantum Gates as Transformations

Quantum gates are fundamental operations in quantum circuits, analogous to classical logic gates. Physically, they represent transformations applied to qubits, achieved by manipulating the quantum system (e.g., using electromagnetic pulses on electron spins). These operations evolve the quantum state of qubits over time.

Properties of Quantum Gates

Reversibility

All quantum gates are inherently reversible. This is a key difference from some classical gates (like AND or NOT in certain contexts) which can be irreversible. Quantum gates are described by unitary transformations, which are always invertible, ensuring no information loss. This reversibility stems from the fundamental principles of quantum mechanics, where time evolution is unitary.

Parallel Processing

Quantum gates naturally perform parallel processing. When a gate is applied to a qubit in superposition, the transformation acts simultaneously on all components of the superposition. For a qubit in state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ and a gate $U$, the operation is $U\left|{\psi}\right\rangle = \alpha U\left|{0}\right\rangle + \beta U\left|{1}\right\rangle$. This parallelism extends to multi-qubit systems, where a gate can operate on up to $2^n$ states in parallel for $n$ qubits, offering significant computational advantages.

Single-Qubit Gates

Single-qubit gates operate on individual qubits. We introduce some essential single-qubit gates:

Pauli-X Gate (NOT Gate, X Gate)

The Pauli-X gate, or X gate (also known as NOT gate), is a fundamental single-qubit gate.

Action

The X gate flips the computational basis states: \[\begin{aligned} X\left|{0}\right\rangle &= \left|{1}\right\rangle \\ X\left|{1}\right\rangle &= \left|{0}\right\rangle \end{aligned}\] For a superposition $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$, it acts as: \[X\left|{\psi}\right\rangle = \alpha\left|{1}\right\rangle + \beta\left|{0}\right\rangle\] Physically, this can be visualized as a rotation of the qubit state.

Matrix Representation

The matrix representation of the X gate in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$ is: \[X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\] Applying X to a state vector $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ is matrix multiplication: \[X \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} \beta \\ \alpha \end{pmatrix}\]

Hadamard Gate (H Gate)

The Hadamard gate (H gate) is crucial for creating superposition states and is widely used in quantum algorithms.

Action

The Hadamard gate transforms basis states as: \[\begin{aligned} H\left|{0}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{0}\right\rangle + \left|{1}\right\rangle) \\ H\left|{1}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{0}\right\rangle - \left|{1}\right\rangle) \end{aligned}\] It creates an equal superposition from basis states, with a phase flip for $\left|{1}\right\rangle$.

Matrix Representation

The matrix representation of the Hadamard gate is: \[H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\] Applying H to state vectors: \[\begin{aligned} H\left|{0}\right\rangle &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}(\left|{0}\right\rangle + \left|{1}\right\rangle) \\ H\left|{1}\right\rangle &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \frac{1}{\sqrt{2}}(\left|{0}\right\rangle - \left|{1}\right\rangle) \end{aligned}\]

$H^2 = I$

Applying the Hadamard gate twice yields the identity gate $I$: \[\begin{aligned} H^2 &= H \cdot H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \end{aligned}\]

Pauli-Z Gate (Z Gate)

The Pauli-Z gate, or Z gate, introduces a phase shift to the $\left|{1}\right\rangle$ state.

Action

The Z gate acts on basis states as: \[\begin{aligned} Z\left|{0}\right\rangle &= \left|{0}\right\rangle \\ Z\left|{1}\right\rangle &= -\left|{1}\right\rangle \end{aligned}\] It leaves $\left|{0}\right\rangle$ unchanged and applies a phase of $-1$ to $\left|{1}\right\rangle$. For a superposition $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$: \[Z\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle - \beta\left|{1}\right\rangle\]

Matrix Representation

The matrix representation of the Z gate is: \[Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\] Applying Z to a state vector: \[Z \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} \alpha \\ -\beta \end{pmatrix}\]

Example: Normalization and Phase Factor of a Cubit State

Normalization of Quantum States

Quantum states must be normalized to ensure that the probabilities of all possible measurement outcomes sum to unity. For a qubit state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$, normalization requires dividing by its norm $\left\|{\left|{\psi}\right\rangle}\right\| = \sqrt{|\alpha|^2 + |\beta|^2}$. The normalized state is given by: \[\left|{\psi}\right\rangle_{\text{norm}} = \frac{\left|{\psi}\right\rangle}{\left\|{\left|{\psi}\right\rangle}\right\|} = \frac{\alpha}{\left\|{\left|{\psi}\right\rangle}\right\|}\left|{0}\right\rangle + \frac{\beta}{\left\|{\left|{\psi}\right\rangle}\right\|}\left|{1}\right\rangle\]

Polar Form of Complex Coefficients

Expressing complex coefficients in polar form $z = re^{i\phi}$, where $r = |z|$ is the modulus and $\phi = \arg(z)$ is the phase, provides insights into their magnitude and phase components. For a complex number $z = a + ib$:

Modulus: $r = |z| = \sqrt{a^2 + b^2}$
Phase: $\phi = \arg(z) = \arctan\left(\frac{b}{a}\right)$, adjusted for the correct quadrant.

Example: Normalizing $\left|{\psi}\right\rangle = (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle$

Consider the non-normalized qubit state $\left|{\psi}\right\rangle = (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle$.

Normalization Calculation

First, calculate the squared norm: \[\begin{aligned} \left\|{\left|{\psi}\right\rangle}\right\|^2 &= |1+i|^2 + |3+2i|^2 \\ &= (1^2 + 1^2) + (3^2 + 2^2) \\ &= 2 + (9 + 4) \\ &= 15 \end{aligned}\] Thus, the norm is $\left\|{\left|{\psi}\right\rangle}\right\| = \sqrt{15}$. The normalized state is: \[\left|{\psi}\right\rangle_{\text{norm}} = \frac{1}{\sqrt{15}} \left( (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle \right) = \frac{1+i}{\sqrt{15}}\left|{0}\right\rangle + \frac{3+2i}{\sqrt{15}}\left|{1}\right\rangle\]

Polar Form and Phase Factor Extraction

Express the coefficient of $\left|{0}\right\rangle$, $\alpha = \frac{1+i}{\sqrt{15}}$, in polar form:

Modulus: $|\alpha| = \frac{|1+i|}{\sqrt{15}} = \frac{\sqrt{2}}{\sqrt{15}} = \sqrt{\frac{2}{15}}$
Phase: $\phi_\alpha = \arg(1+i) = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}$

So, $\alpha = \sqrt{\frac{2}{15}} e^{i\pi/4}$.

For the coefficient of $\left|{1}\right\rangle$, $\beta = \frac{3+2i}{\sqrt{15}}$:

Modulus: $|\beta| = \frac{|3+2i|}{\sqrt{15}} = \frac{\sqrt{13}}{\sqrt{15}} = \sqrt{\frac{13}{15}}$
Phase: $\phi_\beta = \arg(3+2i) = \arctan\left(\frac{2}{3}\right)$

So, $\beta = \sqrt{\frac{13}{15}} e^{i\arctan(2/3)}$.

The normalized state in polar form is: \[\left|{\psi}\right\rangle_{\text{norm}} = \sqrt{\frac{2}{15}} e^{i\pi/4}\left|{0}\right\rangle + \sqrt{\frac{13}{15}} e^{i\arctan(2/3)}\left|{1}\right\rangle\] Factoring out the global phase $e^{i\pi/4}$: \[\left|{\psi}\right\rangle_{\text{norm}} = e^{i\pi/4} \left( \sqrt{\frac{2}{15}} \left|{0}\right\rangle + \sqrt{\frac{13}{15}} e^{i(\arctan(2/3) - \pi/4)}\left|{1}\right\rangle \right)\] Here, $\sqrt{\frac{2}{15}} = \cos(\theta/2)$ and $\sqrt{\frac{13}{15}} = \sin(\theta/2)$ for some angle $\theta$, and $\phi = \arctan(2/3) - \frac{\pi}{4}$ is the relative phase between the coefficients.

Conclusion

This lecture introduced foundational concepts essential for quantum computing and quantum communication. We began with complex numbers, the mathematical basis for quantum mechanics, and transitioned to qubits, contrasting them with classical bits and exploring the principle of superposition. We examined the wave function representation, the probabilistic nature of quantum measurements, and visualized qubit states using the Bloch sphere. Physical realizations of qubits were discussed, alongside the crucial concepts of multi-qubit systems, tensor products, and quantum entanglement, emphasizing entanglement’s unique non-classical correlations. Finally, we introduced quantum gates, specifically single-qubit gates (X, Hadamard, and Z), and illustrated state normalization and phase factor representation with a detailed example.

Key Takeaways:

Superposition: Qubits can exist in superpositions of states, unlike classical bits, enabling richer information encoding.
Probabilistic Measurement: Quantum measurement is probabilistic, collapsing superpositions into definite states.
Entanglement: Entanglement is a uniquely quantum correlation with no classical analogue, crucial for quantum technologies.
Reversible Quantum Gates: Quantum gates are reversible and enable parallel operations on quantum states.
Complex Numbers in Quantum Mechanics: Complex numbers are indispensable for describing quantum states and operations.

In the next lecture, we will expand upon these foundations by exploring multi-qubit gates and constructing quantum circuits to perform computations.

Prepare for the next lecture by considering:

Multi-Qubit Gates: How do gates operate on multiple qubits, and what are examples like CNOT?
Quantum Circuits: How are quantum gates combined to form circuits for quantum algorithms?
Quantum Algorithm Advantages: What specific problems can quantum algorithms solve more efficiently than classical algorithms?

--- title: "Introduction to Quantum Computing" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture serves as an introductory course to quantum mechanics, specifically tailored for quantum computing and quantum communication. As these fields are under very active development, we will focus on the fundamental principles. This course aims to provide a comprehensive overview, establishing a foundation for more advanced topics. We will begin with a general introduction to the subject, covering essential formalism and key concepts. This will be followed by a formal introduction to linear algebra and the principles of quantum mechanics. The course will then progress through quantum statistical mechanics, density matrices, quantum circuits, quantum complexity, quantum algorithms, and quantum communication, concluding with an overview of quantum computer implementations. ## Course Textbook The primary textbook for this course is: - *Quantum Computation and Quantum Information* by Michael A. Nielsen and Isaac L. Chuang, Cambridge University Press. This is a standard reference in the field, offering a comprehensive resource for further study. ## Course Program The program will cover the following topics: - **General Introduction**: An overview of quantum computing to provide context and motivation. - **Linear Algebra**: A formal introduction to the necessary mathematical tools. - **Quantum Mechanics Principles**: Fundamental principles of quantum mechanics, focusing on systems relevant to quantum computing, particularly qubits. - **Quantum Statistical Mechanics and Density Matrix**: Introduction to quantum statistical mechanics and the density matrix formalism for systems with incomplete information. - **Quantum Circuits**: Representation and operation of quantum circuits, and their differences from classical circuits. - **Quantum Complexity**: An analysis of quantum computational complexity compared to classical complexity. - **Quantum Algorithms**: Exploration of key quantum algorithms. - **Quantum Communication**: Introduction to the principles of quantum communication. - **Quantum Computer Realizations**: A brief overview of the physical implementations of quantum computers. # Mathematical Preliminaries: Complex Numbers ## Motivation for Complex Numbers Complex numbers extend the real number system to provide solutions for equations like $x^2 = -1$. This extension is analogous to how number systems evolved from natural numbers to integers, rationals, and reals to overcome limitations in mathematical operations. Complex numbers are thus introduced to solve equations that have no solutions within the real numbers. ## Definition and Components ::: tcolorbox A complex number $z$ is expressed as $z = a + ib$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined by $i^2 = -1$. ::: ### Real and Imaginary Parts ::: tcolorbox For $z = a + ib$, the **real part** of $z$, denoted as $\operatorname{Re}(z)$, is $a$. We write $\operatorname{Re}(z) = a$. ::: :::: tcolorbox For $z = a + ib$, the **imaginary part** of $z$, denoted as $\operatorname{Im}(z)$, is $b$. We write $\operatorname{Im}(z) = b$. ::: remark **Remark 1**. *Note: the imaginary part is the real number $b$, not $ib$.* ::: :::: ## Operations with Complex Numbers ### Multiplication ::: tcolorbox For two complex numbers $z_1 = a + ib$ and $z_2 = c + id$, their product is: $$\begin{aligned} z_1 \cdot z_2 &= (a + ib)(c + id) \\ &= ac + iad + ibc + i^2bd \\ &= (ac - bd) + i(ad + bc) \end{aligned}$$ The real part of the product is $ac - bd$, and the imaginary part is $ad + bc$. ::: ## Complex Conjugation ::: tcolorbox The **complex conjugate** of $z = a + ib$, denoted as $z^*$, is obtained by reversing the sign of the imaginary part: ::: ## Modulus of a Complex Number ::: tcolorbox The **modulus** (or absolute value) of $z = a + ib$, denoted as $|z|$, is a non-negative real number given by: $$\begin{aligned} |z|^2 &= z \cdot z^* = (a + ib)(a - ib) = a^2 + b^2 \\ |z| &= \sqrt{a^2 + b^2} = \sqrt{z \cdot z^*} \end{aligned}$$ The modulus represents the distance from the origin to $z$ in the complex plane. ::: ## Geometric Representation: Complex Plane The **complex plane** represents complex numbers geometrically: - The horizontal axis is the real axis. - The vertical axis is the imaginary axis. A complex number $z = a + ib$ corresponds to a point $(a, b)$ or a vector from the origin to $(a,b)$. The conjugate $z^* = a - ib$ is the reflection of $z$ across the real axis, and $|z|$ is the vector's length. In polar coordinates $(r, \phi)$, where $r = |z|$ and $\phi$ is the argument, we have: $$\begin{aligned} a &= r \cos(\phi) \\ b &= r \sin(\phi) \end{aligned}$$ Thus, $z = r(\cos(\phi) + i\sin(\phi))$. ## Exponential Form: Euler's Formula :::: tcolorbox **Euler's formula** states that for any real $\phi$: This is derived from Taylor series expansions. Using Euler's formula, the polar form becomes the exponential form: $$z = re^{i\phi}$$ ::: remark **Remark 2**. *Note that $|e^{i\phi}| = \sqrt{\cos^2(\phi) + \sin^2(\phi)} = 1$.* ::: :::: For multiplication of $z_1 = r_1e^{i\phi_1}$ and $z_2 = r_2e^{i\phi_2}$: The argument $\phi$ can be found using: $$\phi = \arctan\left(\frac{b}{a}\right) = \arctan\left(\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right)$$ # Introduction to Quantum Computing and Cubits ## Quantum Mechanics for Quantum Computing Quantum computing utilizes principles of quantum mechanics to perform computations, offering advantages over classical computing for specific problems. Unlike classical bits, which are either 0 or 1, quantum computers use **qubits**. ## Cubits: Superposition and Quantum Information ### Superposition ::: tcolorbox Qubits can exist in a **superposition** of states, representing 0 and 1 simultaneously. This is a fundamental departure from classical bits and allows qubits to hold more information and perform computations in fundamentally different ways. A qubit state is described as a linear combination of the basis states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$. ::: ### Wave Function Representation ::: tcolorbox The state of a qubit is represented by a wave function: $$\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$$ where $\alpha$ and $\beta$ are complex numbers called probability amplitudes, satisfying the normalization condition $|\alpha|^2 + |\beta|^2 = 1$. ::: ### Information Storage and Access While the coefficients $\alpha$ and $\beta$ can, in principle, encode infinite information, measurement of a qubit yields only classical outcomes, either 0 or 1. The probabilities of these outcomes are given by $|\alpha|^2$ for state $\left|{0}\right\rangle$ and $|\beta|^2$ for state $\left|{1}\right\rangle$. ### Advantage of Superposition Superposition enables quantum computers to perform operations in parallel on multiple states simultaneously. This inherent parallelism is a key source of quantum computational power, allowing for potentially exponential speedups for certain algorithms compared to classical approaches. However, accessing this information is constrained by the measurement process, which always results in a classical bit. ### Limitations and Applications Due to the nature of measurement and information access in quantum systems, directly translating classical algorithms to quantum algorithms is not always straightforward or advantageous. Quantum computing excels in specific types of problems, particularly those where superposition and entanglement can be effectively utilized, such as problems with yes/no answers or those requiring parallel exploration of multiple possibilities. ## Physical Realizations of Cubits Cubits can be physically implemented using various quantum systems, including: - **Photon Polarization**: Using horizontal and vertical polarizations of photons to represent $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$. - **Electron Spin**: Utilizing the spin-up and spin-down states of electrons. - **Nuclear Spin**: Employing the spin states of atomic nuclei, such as hydrogen nuclei. - **Atomic Energy Levels**: Using two distinct energy levels of an atom to define qubit states. These physical systems allow for the manipulation and measurement of qubit states, forming the hardware for quantum computation. # Cubits: Quantum Bits ## Classical Bits vs. Cubits Classical bits are deterministic, existing in either state 0 or 1. In contrast, qubits, the quantum analogue, can exist in a superposition of states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$. ## Superposition and Cubit State ### Superposition Principle The superposition principle is described as: ::: tcolorbox A qubit can be in a linear combination of basis states $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$, described by the superposition principle. This allows a qubit to exist in both states simultaneously until measured. ::: ### Wave Function ::: tcolorbox The state of a qubit is represented by a wave function or quantum state $\left|{\psi}\right\rangle$: $$\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$$ Here, $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$ are the computational basis states, and $\alpha, \beta \in \mathbb{C}$ are complex probability amplitudes. ::: ### Normalization Condition ::: tcolorbox The coefficients $\alpha$ and $\beta$ must satisfy the normalization condition: $$|\alpha|^2 + |\beta|^2 = 1$$ This ensures that the total probability of measurement outcomes is unity. ::: ## Information and Measurement ### Information Capacity vs. Measurement While qubit states are described by complex numbers $\alpha$ and $\beta$, potentially encoding infinite information, measurement outcomes are limited to classical states $\left|{0}\right\rangle$ or $\left|{1}\right\rangle$. Measurement does not directly reveal $\alpha$ and $\beta$. ### Probabilistic Measurement Probabilistic measurement is described as: ::: tcolorbox Measurement of a qubit in state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ is probabilistic. The probabilities are: - Probability of measuring $\left|{0}\right\rangle$: $|\alpha|^2$ - Probability of measuring $\left|{1}\right\rangle$: $|\beta|^2$ The sum of probabilities is $|\alpha|^2 + |\beta|^2 = 1$. ::: ## State Representation and Bloch Sphere ### Phase Factor Representation Phase factor representation is given by: :::: tcolorbox A qubit state can be written incorporating phase factors: $$\left|{\psi}\right\rangle = \cos\left(\frac{\theta}{2}\right)\left|{0}\right\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\left|{1}\right\rangle$$ where $\theta$ and $\phi$ are real numbers. ::: remark **Remark 3**. *A global phase factor $e^{i\gamma}$ can be factored out without affecting measurement probabilities: $$\left|{\psi}\right\rangle = e^{i\gamma}\left( \cos\left(\frac{\theta}{2}\right)\left|{0}\right\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\left|{1}\right\rangle \right)$$* ::: :::: ### Bloch Sphere ::: tcolorbox The **Bloch sphere** provides a visual representation of qubit states. A qubit state is mapped to a point on the sphere, defined by angles $\theta$ (polar angle) and $\phi$ (azimuthal angle). - $\theta$ determines the probabilities of measuring $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$. - $\phi$ represents the relative phase between $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$. $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$ are conventionally placed at the poles of the sphere. ::: ## Physical Cubits Physical systems that can represent qubits include: - **Photon Polarization**: Polarization states of photons. - **Electron Spin**: Spin states of electrons. - **Nuclear Spin**: Spin states of atomic nuclei (e.g., Hydrogen). - **Atomic Energy Levels**: Two energy levels within an atom. These systems can be manipulated and measured to perform quantum computations. State preparation typically involves initializing the system into a known quantum state. # Multiple Cubits and Entanglement ## Multi-Qubit Systems and Tensor Products ::: tcolorbox Quantum computers utilize multiple qubits to achieve computational power beyond single qubits. The state space of a multi-qubit system is constructed from the tensor product of individual qubit state spaces. For a two-qubit system composed of qubit 1 and qubit 2, if qubit 1 is in state $\left|{\psi_1}\right\rangle$ and qubit 2 is in state $\left|{\psi_2}\right\rangle$, the combined state is given by the tensor product $\left|{\psi_1}\right\rangle \otimes \left|{\psi_2}\right\rangle$, often written as $\left|{\psi_1, \psi_2}\right\rangle$ or $\left|{\psi_1}\right\rangle\left|{\psi_2}\right\rangle$. ::: ## Two-Qubit State Representation ### Basis States for Two Cubits The computational basis for a two-qubit system is formed by the tensor product of the single-qubit basis states $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$: - $\left|{00}\right\rangle = \left|{0}\right\rangle \otimes \left|{0}\right\rangle$ (Both qubits in state $\left|{0}\right\rangle$) - $\left|{01}\right\rangle = \left|{0}\right\rangle \otimes \left|{1}\right\rangle$ (Qubit 1 in $\left|{0}\right\rangle$, Qubit 2 in $\left|{1}\right\rangle$) - $\left|{10}\right\rangle = \left|{1}\right\rangle \otimes \left|{0}\right\rangle$ (Qubit 1 in $\left|{1}\right\rangle$, Qubit 2 in $\left|{0}\right\rangle$) - $\left|{11}\right\rangle = \left|{1}\right\rangle \otimes \left|{1}\right\rangle$ (Both qubits in state $\left|{1}\right\rangle$) A general two-qubit state $\left|{\Psi}\right\rangle$ is a superposition of these basis states: $$\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle$$ where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ are complex coefficients. ### Normalization ::: tcolorbox For $\left|{\Psi}\right\rangle$ to represent a valid quantum state, the coefficients must satisfy the normalization condition: $$|\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1$$ ::: ## Measurement Probabilities ### Joint Measurement Probabilities Joint measurement probabilities are given by: ::: tcolorbox For a state $\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle$, the probability of measuring each basis state is given by the squared modulus of the corresponding coefficient: - $P(00) = |\alpha|^2$ - $P(01) = |\beta|^2$ - $P(10) = |\gamma|^2$ - $P(11) = |\delta|^2$ ::: ### Marginal Probabilities Marginal probabilities are described as: ::: tcolorbox Marginal probabilities refer to the probability of measuring a specific state for one qubit, irrespective of the other qubit's state. For example, the probability of measuring the first qubit in state $\left|{0}\right\rangle$ is: $$P(\text{Qubit 1 = 0}) = P(00) + P(01) = |\alpha|^2 + |\beta|^2$$ Similarly, for the second qubit being in state $\left|{0}\right\rangle$: $$P(\text{Qubit 2 = 0}) = P(00) + P(10) = |\alpha|^2 + |\gamma|^2$$ ::: ## Wave Packet Reduction and Conditional Probability Measurement on a qubit alters the state of the system, a process known as wave packet reduction or state collapse. If a measurement of the first qubit yields $\left|{0}\right\rangle$ for the state $\left|{\Psi}\right\rangle = \alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle + \gamma\left|{10}\right\rangle + \delta\left|{11}\right\rangle$, the state collapses to: $$\left|{\Psi'}\right\rangle = \frac{\alpha\left|{00}\right\rangle + \beta\left|{01}\right\rangle}{\sqrt{|\alpha|^2 + |\beta|^2}}$$ The probabilities for subsequent measurements on the second qubit are now conditional on the first measurement's outcome. For example, the conditional probability of measuring the second qubit in state $\left|{0}\right\rangle$, given the first qubit was measured as $\left|{0}\right\rangle$, is $|\alpha|^2 / (|\alpha|^2 + |\beta|^2)$. ## Quantum Entanglement ### Non-Separability ::: tcolorbox **Entanglement** occurs when a multi-qubit state cannot be represented as a tensor product of single-qubit states. These entangled states exhibit quantum correlations that are fundamentally non-classical. ::: ### Non-Classical Correlations Entangled qubits are correlated in a way that measuring one qubit instantaneously influences the state of the other, regardless of physical separation. This correlation is stronger than any classical correlation and is a crucial resource in quantum computing and communication. ### Bell States Bell states are maximally entangled two-qubit states, essential for quantum communication protocols. An example is the $\left|{\Phi^+}\right\rangle$ Bell state: $$\left|{\Phi^+}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$$ In $\left|{\Phi^+}\right\rangle$, measuring the first qubit in state $\left|{0}\right\rangle$ guarantees that the second qubit will also be measured in state $\left|{0}\right\rangle$, and similarly for $\left|{1}\right\rangle$. ### Bell Inequalities and Violation Bell inequalities define the limits of classical correlations. Quantum mechanics predicts, and experiments confirm, that entangled states violate Bell inequalities. This violation demonstrates that quantum correlations are fundamentally different from classical correlations and cannot be explained by classical local realism. # Quantum Gates: Operations on Cubits ## Quantum Gates as Transformations ::: tcolorbox **Quantum gates** are fundamental operations in quantum circuits, analogous to classical logic gates. Physically, they represent transformations applied to qubits, achieved by manipulating the quantum system (e.g., using electromagnetic pulses on electron spins). These operations evolve the quantum state of qubits over time. ::: ## Properties of Quantum Gates ### Reversibility ::: tcolorbox All quantum gates are inherently **reversible**. This is a key difference from some classical gates (like AND or NOT in certain contexts) which can be irreversible. Quantum gates are described by unitary transformations, which are always invertible, ensuring no information loss. This reversibility stems from the fundamental principles of quantum mechanics, where time evolution is unitary. ::: ### Parallel Processing ::: tcolorbox Quantum gates naturally perform **parallel processing**. When a gate is applied to a qubit in superposition, the transformation acts simultaneously on all components of the superposition. For a qubit in state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ and a gate $U$, the operation is $U\left|{\psi}\right\rangle = \alpha U\left|{0}\right\rangle + \beta U\left|{1}\right\rangle$. This parallelism extends to multi-qubit systems, where a gate can operate on up to $2^n$ states in parallel for $n$ qubits, offering significant computational advantages. ::: ## Single-Qubit Gates Single-qubit gates operate on individual qubits. We introduce some essential single-qubit gates: ### Pauli-X Gate (NOT Gate, X Gate) ::: tcolorbox The Pauli-X gate, or X gate (also known as NOT gate), is a fundamental single-qubit gate. #### Action The X gate flips the computational basis states: $$\begin{aligned} X\left|{0}\right\rangle &= \left|{1}\right\rangle \\ X\left|{1}\right\rangle &= \left|{0}\right\rangle \end{aligned}$$ For a superposition $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$, it acts as: $$X\left|{\psi}\right\rangle = \alpha\left|{1}\right\rangle + \beta\left|{0}\right\rangle$$ Physically, this can be visualized as a rotation of the qubit state. #### Matrix Representation The matrix representation of the X gate in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$ is: $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$ Applying X to a state vector $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ is matrix multiplication: $$X \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} \beta \\ \alpha \end{pmatrix}$$ ::: ### Hadamard Gate (H Gate) ::: tcolorbox The Hadamard gate (H gate) is crucial for creating superposition states and is widely used in quantum algorithms. #### Action The Hadamard gate transforms basis states as: $$\begin{aligned} H\left|{0}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{0}\right\rangle + \left|{1}\right\rangle) \\ H\left|{1}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{0}\right\rangle - \left|{1}\right\rangle) \end{aligned}$$ It creates an equal superposition from basis states, with a phase flip for $\left|{1}\right\rangle$. #### Matrix Representation The matrix representation of the Hadamard gate is: $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$ Applying H to state vectors: $$\begin{aligned} H\left|{0}\right\rangle &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}(\left|{0}\right\rangle + \left|{1}\right\rangle) \\ H\left|{1}\right\rangle &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \frac{1}{\sqrt{2}}(\left|{0}\right\rangle - \left|{1}\right\rangle) \end{aligned}$$ #### $H^2 = I$ Applying the Hadamard gate twice yields the identity gate $I$: $$\begin{aligned} H^2 &= H \cdot H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \end{aligned}$$ ::: ### Pauli-Z Gate (Z Gate) ::: tcolorbox The Pauli-Z gate, or Z gate, introduces a phase shift to the $\left|{1}\right\rangle$ state. #### Action The Z gate acts on basis states as: $$\begin{aligned} Z\left|{0}\right\rangle &= \left|{0}\right\rangle \\ Z\left|{1}\right\rangle &= -\left|{1}\right\rangle \end{aligned}$$ It leaves $\left|{0}\right\rangle$ unchanged and applies a phase of $-1$ to $\left|{1}\right\rangle$. For a superposition $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$: $$Z\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle - \beta\left|{1}\right\rangle$$ #### Matrix Representation The matrix representation of the Z gate is: $$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ Applying Z to a state vector: $$Z \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} \alpha \\ -\beta \end{pmatrix}$$ ::: # Example: Normalization and Phase Factor of a Cubit State ## Normalization of Quantum States ::: tcolorbox Quantum states must be normalized to ensure that the probabilities of all possible measurement outcomes sum to unity. For a qubit state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$, normalization requires dividing by its norm $\left\|{\left|{\psi}\right\rangle}\right\| = \sqrt{|\alpha|^2 + |\beta|^2}$. The normalized state is given by: $$\left|{\psi}\right\rangle_{\text{norm}} = \frac{\left|{\psi}\right\rangle}{\left\|{\left|{\psi}\right\rangle}\right\|} = \frac{\alpha}{\left\|{\left|{\psi}\right\rangle}\right\|}\left|{0}\right\rangle + \frac{\beta}{\left\|{\left|{\psi}\right\rangle}\right\|}\left|{1}\right\rangle$$ ::: ## Polar Form of Complex Coefficients ::: tcolorbox Expressing complex coefficients in polar form $z = re^{i\phi}$, where $r = |z|$ is the modulus and $\phi = \arg(z)$ is the phase, provides insights into their magnitude and phase components. For a complex number $z = a + ib$: - Modulus: $r = |z| = \sqrt{a^2 + b^2}$ - Phase: $\phi = \arg(z) = \arctan\left(\frac{b}{a}\right)$, adjusted for the correct quadrant. ::: ## Example: Normalizing $\left|{\psi}\right\rangle = (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle$ ::: tcolorbox Consider the non-normalized qubit state $\left|{\psi}\right\rangle = (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle$. ### Normalization Calculation First, calculate the squared norm: $$\begin{aligned} \left\|{\left|{\psi}\right\rangle}\right\|^2 &= |1+i|^2 + |3+2i|^2 \\ &= (1^2 + 1^2) + (3^2 + 2^2) \\ &= 2 + (9 + 4) \\ &= 15 \end{aligned}$$ Thus, the norm is $\left\|{\left|{\psi}\right\rangle}\right\| = \sqrt{15}$. The normalized state is: $$\left|{\psi}\right\rangle_{\text{norm}} = \frac{1}{\sqrt{15}} \left( (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle \right) = \frac{1+i}{\sqrt{15}}\left|{0}\right\rangle + \frac{3+2i}{\sqrt{15}}\left|{1}\right\rangle$$ ### Polar Form and Phase Factor Extraction Express the coefficient of $\left|{0}\right\rangle$, $\alpha = \frac{1+i}{\sqrt{15}}$, in polar form: - Modulus: $|\alpha| = \frac{|1+i|}{\sqrt{15}} = \frac{\sqrt{2}}{\sqrt{15}} = \sqrt{\frac{2}{15}}$ - Phase: $\phi_\alpha = \arg(1+i) = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}$ So, $\alpha = \sqrt{\frac{2}{15}} e^{i\pi/4}$. For the coefficient of $\left|{1}\right\rangle$, $\beta = \frac{3+2i}{\sqrt{15}}$: - Modulus: $|\beta| = \frac{|3+2i|}{\sqrt{15}} = \frac{\sqrt{13}}{\sqrt{15}} = \sqrt{\frac{13}{15}}$ - Phase: $\phi_\beta = \arg(3+2i) = \arctan\left(\frac{2}{3}\right)$ So, $\beta = \sqrt{\frac{13}{15}} e^{i\arctan(2/3)}$. The normalized state in polar form is: $$\left|{\psi}\right\rangle_{\text{norm}} = \sqrt{\frac{2}{15}} e^{i\pi/4}\left|{0}\right\rangle + \sqrt{\frac{13}{15}} e^{i\arctan(2/3)}\left|{1}\right\rangle$$ Factoring out the global phase $e^{i\pi/4}$: $$\left|{\psi}\right\rangle_{\text{norm}} = e^{i\pi/4} \left( \sqrt{\frac{2}{15}} \left|{0}\right\rangle + \sqrt{\frac{13}{15}} e^{i(\arctan(2/3) - \pi/4)}\left|{1}\right\rangle \right)$$ Here, $\sqrt{\frac{2}{15}} = \cos(\theta/2)$ and $\sqrt{\frac{13}{15}} = \sin(\theta/2)$ for some angle $\theta$, and $\phi = \arctan(2/3) - \frac{\pi}{4}$ is the relative phase between the coefficients. ::: # Conclusion This lecture introduced foundational concepts essential for quantum computing and quantum communication. We began with complex numbers, the mathematical basis for quantum mechanics, and transitioned to qubits, contrasting them with classical bits and exploring the principle of superposition. We examined the wave function representation, the probabilistic nature of quantum measurements, and visualized qubit states using the Bloch sphere. Physical realizations of qubits were discussed, alongside the crucial concepts of multi-qubit systems, tensor products, and quantum entanglement, emphasizing entanglement's unique non-classical correlations. Finally, we introduced quantum gates, specifically single-qubit gates (X, Hadamard, and Z), and illustrated state normalization and phase factor representation with a detailed example. **Key Takeaways:** - **Superposition**: Qubits can exist in superpositions of states, unlike classical bits, enabling richer information encoding. - **Probabilistic Measurement**: Quantum measurement is probabilistic, collapsing superpositions into definite states. - **Entanglement**: Entanglement is a uniquely quantum correlation with no classical analogue, crucial for quantum technologies. - **Reversible Quantum Gates**: Quantum gates are reversible and enable parallel operations on quantum states. - **Complex Numbers in Quantum Mechanics**: Complex numbers are indispensable for describing quantum states and operations. In the next lecture, we will expand upon these foundations by exploring multi-qubit gates and constructing quantum circuits to perform computations. **Prepare for the next lecture by considering:** - Multi-Qubit Gates: How do gates operate on multiple qubits, and what are examples like CNOT? - Quantum Circuits: How are quantum gates combined to form circuits for quantum algorithms? - Quantum Algorithm Advantages: What specific problems can quantum algorithms solve more efficiently than classical algorithms?

Introduction

Course Textbook

Course Program

Mathematical Preliminaries: Complex Numbers

Motivation for Complex Numbers

Definition and Components

Real and Imaginary Parts

Operations with Complex Numbers

Multiplication

Complex Conjugation

Modulus of a Complex Number

Geometric Representation: Complex Plane

Exponential Form: Euler’s Formula

Introduction to Quantum Computing and Cubits

Quantum Mechanics for Quantum Computing

Cubits: Superposition and Quantum Information

Superposition

Wave Function Representation

Information Storage and Access

Advantage of Superposition

Limitations and Applications

Physical Realizations of Cubits

Cubits: Quantum Bits

Classical Bits vs. Cubits

Superposition and Cubit State

Superposition Principle

Wave Function

Normalization Condition

Information and Measurement

Information Capacity vs. Measurement

Probabilistic Measurement

State Representation and Bloch Sphere

Phase Factor Representation

Bloch Sphere

Physical Cubits

Multiple Cubits and Entanglement

Multi-Qubit Systems and Tensor Products

Two-Qubit State Representation

Basis States for Two Cubits

Normalization

Measurement Probabilities

Joint Measurement Probabilities

Marginal Probabilities

Wave Packet Reduction and Conditional Probability

Quantum Entanglement

Non-Separability

Non-Classical Correlations

Bell States

Bell Inequalities and Violation

Quantum Gates: Operations on Cubits

Quantum Gates as Transformations

Properties of Quantum Gates

Reversibility

Parallel Processing

Single-Qubit Gates

Pauli-X Gate (NOT Gate, X Gate)

Action

Matrix Representation

Hadamard Gate (H Gate)

Action

Matrix Representation

\(H^2 = I\)

Pauli-Z Gate (Z Gate)

Action

Matrix Representation

Example: Normalization and Phase Factor of a Cubit State

Normalization of Quantum States

Polar Form of Complex Coefficients

Example: Normalizing \(\left|{\psi}\right\rangle = (1+i)\left|{0}\right\rangle + (3+2i)\left|{1}\right\rangle\)

Normalization Calculation

Polar Form and Phase Factor Extraction

Conclusion