➿Quantum Computing Unit 9 – Quantum Systems: Mathematical Formalism

Key Concepts and Terminology

• Quantum systems mathematical formalism provides a rigorous framework for describing and analyzing quantum phenomena
• Hilbert space is a complex vector space with an inner product, serving as the mathematical foundation for quantum mechanics
• Quantum states are represented by vectors in Hilbert space, known as ket vectors, denoted as $|\psi\rangle$
  • The dual of a ket vector is a bra vector, denoted as $\langle\psi|$
  • The inner product of a bra and a ket vector is written as $\langle\psi|\phi\rangle$
• Operators are mathematical objects that act on quantum states, such as position, momentum, and energy operators
• Observables are physical quantities that can be measured in a quantum system, represented by Hermitian operators
• Eigenvalues and eigenvectors are crucial concepts in quantum mechanics
  • An eigenvector of an operator is a vector that, when acted upon by the operator, yields a scalar multiple of itself
  • The scalar multiple is called the eigenvalue corresponding to the eigenvector
• The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum state

Mathematical Foundations

• Linear algebra is the primary mathematical tool used in quantum mechanics
  • Vectors, matrices, and linear operators are essential components of the quantum formalism
• Hilbert space is a complete, complex inner product space that generalizes the notion of Euclidean space
  • Completeness ensures that limits of sequences of vectors are well-defined within the space
• The inner product of two vectors in Hilbert space is a complex number that satisfies certain properties
  • Conjugate symmetry: $\langle\psi|\phi\rangle = \overline{\langle\phi|\psi\rangle}$
  • Linearity: $\langle\psi|a\phi_1 + b\phi_2\rangle = a\langle\psi|\phi_1\rangle + b\langle\psi|\phi_2\rangle$
  • Positive definiteness: $\langle\psi|\psi\rangle \geq 0$, with equality if and only if $|\psi\rangle = 0$
• The norm of a vector $|\psi\rangle$ is defined as $\sqrt{\langle\psi|\psi\rangle}$, which represents the length of the vector
• Orthonormality is a crucial concept in quantum mechanics
  • Two vectors $|\psi\rangle$ and $|\phi\rangle$ are orthogonal if their inner product is zero: $\langle\psi|\phi\rangle = 0$
  • A set of vectors is orthonormal if they are pairwise orthogonal and normalized (i.e., have unit norm)

Quantum States and Vectors

• Quantum states are represented by vectors in Hilbert space, typically denoted using Dirac notation (bra-ket notation)
  • A ket vector $|\psi\rangle$ represents a quantum state
  • The corresponding bra vector $\langle\psi|$ is the dual of the ket vector, obtained by taking the complex conjugate and transpose
• The superposition principle is a fundamental concept in quantum mechanics
  • A quantum state can be a linear combination of other quantum states: $|\psi\rangle = a|\phi_1\rangle + b|\phi_2\rangle$
  • The coefficients $a$ and $b$ are complex numbers called probability amplitudes
• The Born rule relates the probability amplitudes to the probabilities of measuring a particular outcome
  • The probability of measuring a quantum state $|\psi\rangle$ in the state $|\phi\rangle$ is given by $|\langle\phi|\psi\rangle|^2$
• The Bloch sphere is a geometric representation of a single-qubit quantum state
  • A qubit state is represented by a point on the surface of the Bloch sphere
  • The north and south poles correspond to the computational basis states $|0\rangle$ and $|1\rangle$, respectively

Operators and Observables

• Operators are mathematical objects that act on quantum states and can represent physical quantities or transformations
  • Linear operators satisfy the properties of linearity: $\hat{A}(a|\psi\rangle + b|\phi\rangle) = a\hat{A}|\psi\rangle + b\hat{A}|\phi\rangle$
• Observables are physical quantities that can be measured in a quantum system, such as position, momentum, and energy
  • Observables are represented by Hermitian operators, which satisfy $\hat{A}^\dagger = \hat{A}$
• The eigenvalue equation relates an operator $\hat{A}$, its eigenvectors $|\psi_i\rangle$, and the corresponding eigenvalues $\lambda_i$: $\hat{A}|\psi_i\rangle = \lambda_i|\psi_i\rangle$
  • Eigenvectors of an observable represent the possible measurement outcomes
  • Eigenvalues are the values associated with each measurement outcome
• The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$
  • If the commutator is zero, the operators are said to commute, and they can be simultaneously diagonalized
  • Non-commuting observables, such as position and momentum, are subject to the Heisenberg uncertainty principle

Schrödinger Equation

• The Schrödinger equation is a linear partial differential equation that describes the time evolution of a quantum state $|\psi(t)\rangle$
  • Time-dependent Schrödinger equation: $i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle$
  • Time-independent Schrödinger equation: $\hat{H}|\psi\rangle = E|\psi\rangle$
• The Hamiltonian operator $\hat{H}$ represents the total energy of the quantum system
  • It is the sum of the kinetic and potential energy operators: $\hat{H} = \hat{T} + \hat{V}$
• The time-independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator
  • The eigenstates of the Hamiltonian are called energy eigenstates or stationary states
  • The eigenvalues of the Hamiltonian represent the possible energy levels of the quantum system
• The time evolution of a quantum state is determined by the unitary time-evolution operator $\hat{U}(t)$
  • The time-evolved state is given by $|\psi(t)\rangle = \hat{U}(t)|\psi(0)\rangle$
  • The time-evolution operator is related to the Hamiltonian by $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$

Measurement and Probability

• Measurement in quantum mechanics is a probabilistic process that collapses the quantum state onto an eigenstate of the measured observable
  • The act of measurement changes the quantum state, a phenomenon known as the collapse of the wave function
• The probability of measuring a particular eigenvalue $\lambda_i$ of an observable $\hat{A}$ in a quantum state $|\psi\rangle$ is given by $P(\lambda_i) = |\langle\psi_i|\psi\rangle|^2$
  • $|\psi_i\rangle$ is the eigenvector corresponding to the eigenvalue $\lambda_i$
• The expectation value of an observable $\hat{A}$ in a quantum state $|\psi\rangle$ is the average value of the measurement outcomes
  • Expectation value: $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$
• The uncertainty principle states that certain pairs of observables, such as position and momentum, cannot be simultaneously measured with arbitrary precision
  • Heisenberg uncertainty principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\Delta x$ and $\Delta p$ are the standard deviations of position and momentum, respectively
• The density matrix formalism is used to describe mixed states, which are statistical ensembles of pure quantum states
  • The density matrix $\rho$ is a positive semidefinite, Hermitian operator with unit trace
  • Pure states have density matrices that satisfy $\rho^2 = \rho$, while mixed states have $\rho^2 \neq \rho$

Quantum Entanglement

• Quantum entanglement is a phenomenon in which two or more quantum systems exhibit correlations that cannot be explained by classical physics
  • Entangled states cannot be described as a product of individual quantum states
• The Bell states are maximally entangled two-qubit states, which form a basis for the two-qubit Hilbert space
  • The four Bell states are: $|\Phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)$ and $|\Psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)$
• Entanglement is a crucial resource in quantum information processing, enabling tasks such as quantum teleportation and superdense coding
• The Schmidt decomposition is a tool for quantifying entanglement in bipartite quantum systems
  • Any bipartite pure state can be written as $|\psi\rangle = \sum_i \sqrt{\lambda_i} |i_A\rangle \otimes |i_B\rangle$, where $\lambda_i$ are the Schmidt coefficients and $|i_A\rangle$ and $|i_B\rangle$ are orthonormal bases for the two subsystems
• Entanglement measures, such as entanglement entropy and concurrence, quantify the amount of entanglement in a quantum state
  • Entanglement entropy: $S = -\sum_i \lambda_i \log_2 \lambda_i$, where $\lambda_i$ are the eigenvalues of the reduced density matrix
  • Concurrence: $C = \max\{0, \sqrt{\lambda_1} - \sqrt{\lambda_2} - \sqrt{\lambda_3} - \sqrt{\lambda_4}\}$, where $\lambda_i$ are the eigenvalues of a matrix related to the density matrix

Applications in Quantum Computing

• Quantum computing leverages the principles of quantum mechanics to perform computations that are intractable for classical computers
  • Quantum bits (qubits) are the fundamental building blocks of quantum computers
  • Qubits can be realized using various physical systems, such as superconducting circuits, trapped ions, and photons
• Quantum algorithms exploit quantum phenomena, such as superposition and entanglement, to solve specific problems more efficiently than classical algorithms
  • Shor's algorithm for integer factorization and Grover's algorithm for unstructured search are examples of quantum algorithms with significant speedups over classical counterparts
• Quantum error correction is essential for building reliable quantum computers
  • Quantum errors can be caused by decoherence, which is the uncontrolled interaction between a quantum system and its environment
  • Quantum error correction codes, such as the surface code and the color code, use redundancy to detect and correct errors without disturbing the encoded quantum information
• Quantum simulation is an important application of quantum computers
  • Quantum systems can be efficiently simulated using quantum computers, enabling the study of complex physical and chemical systems
  • Examples include simulating the electronic structure of molecules and modeling strongly correlated materials
• Quantum machine learning aims to develop quantum algorithms for machine learning tasks
  • Quantum algorithms can potentially offer speedups for tasks such as data classification, clustering, and dimensionality reduction
  • Variational quantum algorithms, which combine classical optimization with quantum circuits, are a promising approach for near-term quantum devices