The basic unit of quantum information

A two-level quantum system (e.g., electron spin, photon polarization, superconducting circuit state)

Written as: |ψ⟩ = α|0⟩ + β|1⟩

α and β are complex numbers called probability amplitudes

Must satisfy normalization: |α|² + |β|² = 1

Produces a classical outcome: 0 or 1

Probability of measuring 0 is |α|²; of measuring 1 is |β|²

Measurement typically collapses the state into the observed basis state

A geometric way to visualize a single qubit state

Any pure qubit state corresponds to a point on a sphere surface

Rotations on the Bloch sphere correspond to single-qubit gates

A qubit can be in a linear combination of |0⟩ and |1⟩ simultaneously before measurement

Superposition describes amplitudes, not “being 0 and 1” in a classical sense

Applying quantum gates (e.g., the Hadamard gate)

Example: H|0⟩ = ( |0⟩ + |1⟩ ) / √2

Enables quantum states to encode information across many possibilities at once

With n qubits, the system state is a superposition over 2ⁿ basis states

You do not directly read out all 2ⁿ values; measurement yields one outcome

Quantum advantage comes from manipulating amplitudes so correct answers become more likely

Computation is a sequence of gates acting on qubits

Ends with measurements that convert quantum information into classical bits

Single-qubit gates

Multi-qubit gates

Unitary evolution

Amplitudes can add or cancel based on phase

Algorithms exploit interference to amplify correct outcomes and suppress incorrect ones

A joint state of multiple qubits that cannot be factored into independent single-qubit states

Enables non-classical correlations used for quantum speedups, error correction, and communication tasks

Bell state: ( |00⟩ + |11⟩ ) / √2

Quantum simulation

Factoring and cryptanalysis (Shor’s algorithm)

Search and amplitude amplification (Grover’s algorithm)

Optimization and sampling (context-dependent)

Replacing classical computers for everyday tasks

Many workloads remain faster/cheaper on classical hardware

Qubits are fragile; interactions with the environment destroy quantum information

Gates and measurements have nonzero error probabilities

Uses many physical qubits to build one logical (more reliable) qubit

Requires fault-tolerant thresholds and substantial overhead

Building systems with many high-quality qubits and reliable control is difficult

They evolve a superposition, but measurement yields one result; advantage comes from interference and algorithm design

Superposition is a quantum state with amplitudes; classical interpretation is limited

Speedups apply to specific problem classes and require sufficiently large, error-corrected devices