Representation of Qubits

Detailed Description

A qubit, or quantum bit, is the fundamental unit of quantum information, analogous to a classical bit but capable of existing in superpositions. In Hilbert space, a qubit is represented as a vector in a two-dimensional complex vector space, allowing for states that are linear combinations of basis states.

History

The concept of qubits was formalized by David Deutsch in the 1980s, building on quantum mechanics by Dirac and von Neumann in the 1930s.

Derivation of Formula

From the postulates of quantum mechanics, states are unit vectors in Hilbert space. For two levels, the general state is a superposition: derive by normalizing α|0⟩ + β|1⟩ such that |α|^2 + |β|^2 = 1.

Formula: \( |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \)

Where: |\psi\rangle = qubit state vector, \alpha, \beta = complex amplitudes (probabilities |α|^2 + |β|^2 = 1), |0\rangle = ground state (column vector [1,0]^T), |1\rangle = excited state ([0,1]^T).

Real-Life Example

In quantum computers like IBM's Qiskit, qubits are implemented using superconducting circuits, where superposition enables parallel computation for algorithms like Shor's factoring.

Bloch Sphere

Detailed Description

The Bloch sphere is a geometrical representation of a qubit's state space, mapping the two-dimensional Hilbert space onto a unit sphere, where pure states lie on the surface.

History

Introduced by Felix Bloch in 1946 for nuclear magnetic resonance, adapted to qubits in quantum information in the 1990s.

Derivation of Formula

Parametrize |\psi\rangle = cos(θ/2) |0\rangle + e^{iφ} sin(θ/2) |1\rangle, mapping to spherical coordinates (θ, φ) on the unit sphere.

Formula: \( |\psi\rangle = \cos\left(\frac{\theta}{2}\right) |0\rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right) |1\rangle \)

Where: θ = polar angle (0 to π), φ = azimuthal angle (0 to 2π).

Real-Life Example

Used in visualizing qubit manipulations in quantum simulators like QuTiP, aiding in designing quantum error correction codes.

Multi-Qubit Systems

Detailed Description

Multi-qubit systems are described by the tensor product of individual Hilbert spaces, enabling entanglement and exponentially larger state spaces.

History

Tensor products from quantum mechanics in 1920s; applied to quantum computing by Feynman in 1982.

Derivation of Formula

For two qubits, the joint space is span{|00⟩, |01⟩, |10⟩, |11⟩}, derived as |ψ⟩ ⊗ |φ⟩.

Formula: \( |\psi_{AB}\rangle = |\psi_A\rangle \otimes |\psi_B\rangle \)

Where: ⊗ = tensor product, expanding dimensions from 2 to 4.

Real-Life Example

In Google's Sycamore processor, multi-qubit states enable quantum supremacy demonstrations.

Single Qubit Gates (X, Y, Z, H, S, T, Rφ)

Detailed Description

Single-qubit gates are unitary operations on a qubit, represented as 2x2 matrices, rotating the state on the Bloch sphere.

History

Pauli matrices from 1927; Hadamard from 1893 math, adapted to quantum in 1990s.

Derivation of Formula

X gate flips |0> to |1>, derived as σ_x Pauli matrix.

Formula: \( X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), similarly for others.

Where: Matrix elements define action on basis states.

Real-Life Example

Hadamard gate creates superposition in quantum random number generators.

Multi-Qubit Gates (CNOT, Toffoli, Fredkin, Swap)

Detailed Description

Multi-qubit gates operate on multiple qubits, enabling entanglement and conditional operations.

History

CNOT proposed by Feynman in 1985; Toffoli in 1980 for reversible computing.

Derivation of Formula

CNOT: Control on first, target flip if control 1, matrix extension of X.

Formula: \( CNOT = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \)

Where: Acts on |control target⟩.

Real-Life Example

CNOT used in quantum teleportation protocols for secure communication.

Universal Gate Sets

Detailed Description

Universal gate sets can approximate any quantum operation, like {H, S, CNOT, T} for fault-tolerant computing.

History

Proven by Barenco et al. in 1995.

Derivation of Formula

From Solovay-Kitaev theorem, dense in SU(2^n).

Real-Life Example

Used in compiling quantum algorithms for NISQ devices.

Projective Measurements

Detailed Description

Projective measurements collapse the state to an eigenvector, with probability given by Born rule.

History

From von Neumann's 1932 formalism.

Derivation of Formula

Projector P_m = |m⟩⟨m|, outcome m with p(m) = ⟨ψ|P_m|ψ⟩.

Formula: \( p(m) = \langle \psi | P_m | \psi \rangle \)

Where: P_m = projector, m = outcome.

Real-Life Example

Measuring spin in Stern-Gerlach experiment, basis for quantum sensors.

POVMs (Positive Operator Valued Measures)

Detailed Description

POVMs generalize measurements with non-orthogonal outcomes, using positive operators summing to identity.

History

Introduced by Helstrom in 1976 for optimal detection.

Derivation of Formula

E_m ≥ 0, ∑ E_m = I, p(m) = Tr(ρ E_m).

Formula: \( p(m) = \mathrm{Tr}(\rho E_m) \)

Where: ρ = density matrix, E_m = POVM element.

Real-Life Example

Used in quantum tomography for state reconstruction in experiments.

EPR Pairs

Detailed Description

EPR pairs are maximally entangled two-qubit states, highlighting non-locality in quantum mechanics.

History

Proposed by Einstein, Podolsky, Rosen in 1935 to question completeness of QM.

Derivation of Formula

From singlet state, invariant under rotation.

Formula: \( |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle) \)

Where: One of the Bell states.

Real-Life Example

Used in quantum key distribution protocols like BB84 for secure encryption.

Bell States

Detailed Description

Bell states are four maximally entangled two-qubit states, forming a basis for entanglement.

History

Named after John Bell, who in 1964 showed inequalities violated by QM.

Derivation of Formula

Orthonormal basis from tensor product, with ± combinations.

Formula: \( |\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|00\rangle \pm |11\rangle), |\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|01\rangle \pm |10\rangle) \)

Where: ± denote symmetric/asymmetric phases.

Real-Life Example

Entanglement swapping in quantum repeaters for long-distance networks.

GHZ States

Detailed Description

GHZ states are multi-partite entangled states, like three-qubit |000⟩ + |111⟩, used for testing non-locality.

History

Proposed by Greenberger, Horne, Zeilinger in 1989.

Derivation of Formula

Generalize Bell: equal superposition of all-zero and all-one.

Formula: \( |GHZ\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle) \)

Where: For three qubits.

Real-Life Example

Implemented in ion traps for quantum metrology, enhancing precision measurements.