1.6 Quantum states and qubits

Quantum states and qubits form the foundation of quantum computing. Unlike classical bits, qubits can exist in superposition, allowing them to represent multiple states simultaneously. This unique property enables quantum computers to perform complex calculations faster than traditional computers.

Quantum states are described using mathematical tools like state vectors and the Bloch sphere. Understanding these concepts is crucial for grasping how quantum computers operate and why they have the potential to revolutionize various fields, from cryptography to drug discovery.

Quantum states

• Quantum states are the fundamental building blocks of quantum systems and are essential for understanding how quantum computers operate
• In quantum mechanics, a system can exist in a superposition of multiple states simultaneously, unlike classical systems where an object can only be in one state at a time
• Quantum states are mathematically represented using complex vectors and matrices, which capture the probabilistic nature of quantum phenomena

Superposition

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• Superposition is a key principle of quantum mechanics where a quantum system can exist in multiple states simultaneously until it is measured
• A qubit, the basic unit of quantum information, can be in a superposition of the $|0\rangle$ and $|1\rangle$ states, represented as $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha$ and $\beta$ are complex numbers
• The coefficients $\alpha$ and $\beta$ determine the probabilities of measuring the qubit in the $|0\rangle$ or $|1\rangle$ state, with $|\alpha|^2 + |\beta|^2 = 1$
• Superposition allows quantum computers to perform many calculations simultaneously, enabling them to solve certain problems much faster than classical computers (quantum parallelism)

Quantum state vectors

• Quantum state vectors are used to mathematically describe the state of a quantum system
• A qubit's state is represented by a 2D complex vector, $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $|0\rangle$ and $|1\rangle$ are the computational basis states
• The state vector captures the probability amplitudes of the qubit being in each basis state
• Quantum state vectors are normalized, meaning the sum of the squares of the absolute values of the amplitudes is equal to 1

Bloch sphere representation

• The Bloch sphere is a geometric representation of a qubit's state, providing a visual tool for understanding single-qubit operations
• In the Bloch sphere, the state of a qubit is represented by a point on the surface of a unit sphere
• The north and south poles of the sphere correspond to the $|0\rangle$ and $|1\rangle$ states, respectively, while any other point on the surface represents a superposition of these states
• The Bloch sphere representation is useful for visualizing single-qubit gates as rotations of the state vector around various axes

Qubits

• Qubits are the fundamental building blocks of quantum computers, analogous to bits in classical computers
• Unlike classical bits, qubits can exist in a superposition of states and exhibit quantum properties such as entanglement
• The state of a qubit is described by a quantum state vector, which captures the probability amplitudes of the qubit being in different basis states

Qubit vs classical bit

• Classical bits can only be in one of two states, 0 or 1, at any given time
• Qubits can exist in a superposition of the $|0\rangle$ and $|1\rangle$ states, allowing them to represent a continuum of values between 0 and 1
• While classical bits are deterministic, qubits are probabilistic, meaning that their state is determined only when measured
• Qubits can be entangled with each other, a property not exhibited by classical bits

Qubit states

• The two basis states of a qubit are denoted as $|0\rangle$ and $|1\rangle$, which correspond to the classical bit values of 0 and 1
• A qubit can also be in a superposition of these basis states, represented as $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha$ and $\beta$ are complex numbers satisfying $|\alpha|^2 + |\beta|^2 = 1$
• The probability of measuring the qubit in the $|0\rangle$ state is $|\alpha|^2$, while the probability of measuring it in the $|1\rangle$ state is $|\beta|^2$

Measuring qubits

• Measuring a qubit collapses its state from a superposition to one of the basis states, either $|0\rangle$ or $|1\rangle$
• The outcome of a measurement is probabilistic and depends on the probability amplitudes of the qubit's state before measurement
• After measurement, the qubit's state is irreversibly altered, and any information about the previous superposition is lost
• Measurement is a crucial step in quantum algorithms, as it allows for extracting classical information from quantum states

Multiple qubits

• Quantum systems can consist of multiple qubits, allowing for more complex quantum states and operations
• Multiple qubits can be combined using the tensor product operation, which creates a larger Hilbert space to describe the composite system
• Qubits can exhibit entanglement, a quantum phenomenon where the state of one qubit cannot be described independently of the others

Tensor product

• The tensor product is a mathematical operation used to combine the state spaces of individual qubits into a larger state space for the composite system
• For two qubits, the tensor product of their state vectors results in a 4D vector representing the state of the two-qubit system
• The tensor product is denoted using the $\otimes$ symbol, for example, $|0\rangle \otimes |1\rangle = |01\rangle$
• The tensor product is essential for describing multi-qubit states and operations in quantum computing

Entanglement

• Entanglement is a quantum phenomenon where the states of multiple qubits are correlated in such a way that the state of one qubit cannot be described independently of the others
• Entangled qubits exhibit non-classical correlations, meaning that measuring one qubit instantly affects the state of the other, regardless of the distance between them
• Entanglement is a key resource in quantum computing, enabling certain quantum algorithms and protocols (quantum teleportation, superdense coding)
• The Bell states are examples of maximally entangled two-qubit states

Bell states

• Bell states, also known as EPR pairs, are four specific maximally entangled two-qubit states
• The four Bell states are:
  • $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$
  • $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$
  • $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$
  • $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$
• Bell states are important in quantum information processing tasks such as quantum teleportation and superdense coding
• They are also used in quantum error correction schemes and quantum cryptography protocols

Quantum gates

• Quantum gates are the basic building blocks of quantum circuits, analogous to classical logic gates
• Quantum gates operate on qubits, manipulating their states to perform quantum computations
• Quantum gates are represented by unitary matrices, which preserve the normalization of quantum states and ensure reversibility

Single-qubit gates

• Single-qubit gates operate on a single qubit and are represented by 2x2 unitary matrices
• Examples of single-qubit gates include:
  • Pauli gates (X, Y, Z): perform rotations around the x, y, and z axes of the Bloch sphere
  • Hadamard gate (H): creates an equal superposition of the $|0\rangle$ and $|1\rangle$ states
  • Phase shift gates (S, T): introduce a phase difference between the $|0\rangle$ and $|1\rangle$ states
• Single-qubit gates are used to manipulate the state of individual qubits and are essential for implementing quantum algorithms

Multi-qubit gates

• Multi-qubit gates operate on two or more qubits simultaneously, allowing for the creation of entanglement and the implementation of complex quantum operations
• The most common two-qubit gate is the controlled-NOT (CNOT) gate, which flips the state of the target qubit if the control qubit is in the $|1\rangle$ state
• Other examples of multi-qubit gates include the controlled-Z (CZ) gate and the SWAP gate
• Multi-qubit gates are essential for creating and manipulating entangled states, which are crucial for many quantum algorithms

Gate sequences

• Quantum algorithms are implemented by applying a sequence of quantum gates to a set of qubits
• The order and combination of gates determine the overall operation performed on the qubits
• Gate sequences can be optimized to minimize the number of gates required or to reduce the depth of the quantum circuit
• Quantum compilers translate high-level quantum algorithms into optimized gate sequences that can be executed on quantum hardware

Quantum circuits

• Quantum circuits are a graphical representation of quantum computations, showing the sequence of quantum gates applied to a set of qubits
• Quantum circuits are the quantum analogue of classical logic circuits and are used to design and analyze quantum algorithms
• A quantum circuit consists of a set of input qubits, a sequence of quantum gates, and a set of output qubits

Circuit diagrams

• Circuit diagrams are visual representations of quantum circuits, using standardized symbols for qubits, quantum gates, and measurements
• In a circuit diagram, qubits are represented by horizontal lines, and quantum gates are represented by boxes or symbols on these lines
• The flow of the circuit is from left to right, with the input qubits on the left and the output qubits on the right
• Measurements are represented by a meter symbol at the end of a qubit line

Quantum algorithms

• Quantum algorithms are procedures that utilize quantum mechanical properties to solve computational problems more efficiently than classical algorithms
• Examples of quantum algorithms include:
  • Shor's algorithm for integer factorization
  • Grover's algorithm for unstructured search
  • Quantum Fourier transform (QFT) for period finding and phase estimation
• Quantum algorithms leverage superposition, entanglement, and interference to achieve speedups over classical algorithms
• Designing efficient quantum algorithms is an active area of research in quantum computing

Quantum computing applications

• Quantum computing has the potential to revolutionize various fields by solving problems that are intractable for classical computers
• Some promising applications of quantum computing include:
  • Cryptography: Shor's algorithm can break certain public-key cryptographic schemes, necessitating the development of quantum-resistant cryptography
  • Drug discovery: Quantum computers can simulate complex molecular systems, aiding in the design of new drugs and materials
  • Optimization: Quantum algorithms can be used to solve optimization problems in logistics, finance, and machine learning
  • Quantum chemistry: Quantum computers can efficiently simulate quantum systems, enabling more accurate predictions of chemical properties and reactions
• As quantum hardware and algorithms continue to improve, the range of practical applications for quantum computing is expected to grow