2.1 Quantum bits (qubits) and quantum gates

Quantum bits, or qubits, are the building blocks 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 more efficiently than classical computers.

Quantum gates are the tools used to manipulate qubits and create quantum circuits. These gates, such as the Hadamard and CNOT gates, can create superpositions, entangle qubits, and perform quantum operations. Understanding qubits and quantum gates is crucial for grasping quantum computation's potential.

Qubit Properties vs Classical Bits

Fundamental Differences

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• Qubits are the fundamental unit of quantum information analogous to classical bits in classical computing
• Classical bits can only exist in one of two states (0 or 1) while qubits can exist in a superposition of states (a linear combination of 0 and 1)
• Qubits are typically represented using the Dirac notation with |0⟩ and |1⟩ denoting the computational basis states
• The state of a qubit can be visualized using the Bloch sphere where the north and south poles represent the basis states |0⟩ and |1⟩ respectively

Measurement and Entanglement

• Measuring a qubit collapses its state to either |0⟩ or |1⟩ with probabilities determined by the amplitudes of the superposition
  • For example, if a qubit is in the state |ψ⟩ = $\frac{1}{\sqrt{2}}$(|0⟩ + |1⟩), measuring it will yield |0⟩ or |1⟩ with equal probability
• Unlike classical bits, qubits can exhibit quantum entanglement where the state of one qubit is correlated with the state of another qubit
  • Entanglement allows for the creation of states like the Bell states (e.g., $\frac{1}{\sqrt{2}}$(|00⟩ + |11⟩)) which cannot be described by individual qubit states

Superposition in Quantum Computation

Superposition States

• Superposition is a fundamental principle of quantum mechanics that allows a qubit to exist in a linear combination of its basis states (|0⟩ and |1⟩)
• The general state of a qubit can be written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex amplitudes satisfying |α|^2 + |β|^2 = 1
  • For example, a qubit in the state |ψ⟩ = $\frac{1}{\sqrt{2}}$(|0⟩ + |1⟩) is in an equal superposition of |0⟩ and |1⟩
• The amplitudes α and β determine the probabilities of measuring the qubit in the |0⟩ or |1⟩ state respectively
  • The probability of measuring |0⟩ is |α|^2, and the probability of measuring |1⟩ is |β|^2

Quantum Parallelism and Algorithms

• Superposition allows quantum computers to perform many calculations simultaneously enabling quantum parallelism
  • A single quantum operation on a superposition state effectively performs the operation on all basis states simultaneously
• Quantum algorithms such as Shor's algorithm (for factoring integers) and Grover's algorithm (for searching unstructured databases) leverage superposition to achieve speedups over classical algorithms
• Superposition also enables the creation of entangled states which are crucial for many quantum algorithms and protocols (quantum teleportation, superdense coding)

Quantum Gates for Qubit Manipulation

Single-Qubit Gates

• Quantum gates are the building blocks of quantum circuits analogous to logic gates in classical computing
• Single-qubit gates operate on a single qubit and include:
  • Pauli gates (X, Y, Z): Perform rotations around the x, y, and z axes of the Bloch sphere respectively
    • The X gate (NOT gate) maps |0⟩ to |1⟩ and |1⟩ to |0⟩
  • Hadamard gate (H): Creates an equal superposition of basis states
    • Applying an H gate to |0⟩ results in $\frac{1}{\sqrt{2}}$(|0⟩ + |1⟩)
  • Phase gate (S) and its inverse (S†): Apply a phase shift to the |1⟩ state
  • T gate and its inverse (T†): Apply a phase shift of π/4 and -π/4 to the |1⟩ state respectively

Multi-Qubit Gates

• Multi-qubit gates operate on two or more qubits and include:
  • Controlled gates (e.g., CNOT, CZ): Apply a single-qubit gate to a target qubit based on the state of a control qubit
    • The CNOT gate applies an X gate to the target qubit if the control qubit is |1⟩
  • Swap gate: Exchanges the states of two qubits
    • Applying a Swap gate to |01⟩ results in |10⟩
  • Toffoli gate (CCNOT): A three-qubit gate that applies an X gate to the target qubit if both control qubits are in the |1⟩ state
• Multi-qubit gates are essential for creating entanglement between qubits and implementing complex quantum algorithms

Matrix Representations of Quantum Gates

State Vectors and Unitary Matrices

• Quantum gates can be represented using unitary matrices that act on the state vector of a qubit or a multi-qubit system
• The state vector of a single qubit can be represented as a 2x1 column vector, while the state vector of an n-qubit system is a 2^n x 1 column vector
  • For example, the state vector of a single qubit in the state |ψ⟩ = α|0⟩ + β|1⟩ is $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$
• Single-qubit gates are represented by 2x2 unitary matrices, while multi-qubit gates are represented by 2^n x 2^n unitary matrices, where n is the number of qubits involved
  • The Hadamard gate is represented by the matrix $\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

Gate Application and State Evolution

• To apply a quantum gate to a qubit state, the matrix representation of the gate is multiplied with the state vector
  • Applying the X gate to the state |0⟩ $=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ results in $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ which is |1⟩
• The resulting state vector represents the new state of the qubit(s) after the gate application
• Matrix multiplication of quantum gates allows for the analysis of the effects of multiple gates applied in sequence
• The final state vector can be used to calculate the probabilities of measuring the qubits in different basis states