Takes a definite value: 0 or 1

Operations are deterministic logical gates (AND, OR, NOT, etc.)

Can exist in a combination of basis states (|0⟩ and |1⟩)

Operations are quantum gates (unitary transformations) that change probability amplitudes

Classical computers explore possibilities largely sequentially or via parallel hardware

Quantum computers manipulate a quantum state that can encode many possibilities at once, then extract information via measurement

The fundamental unit of quantum information

A two-level quantum system with basis states |0⟩ and |1⟩

Examples of physical implementations

A bit stores a definite state (0/1)

A qubit stores a quantum state described by amplitudes

|ψ⟩ = α|0⟩ + β|1⟩

α and β are complex amplitudes

Normalization: |α|² + |β|² = 1

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

This is not “half 0 and half 1” but a state with amplitudes that determine measurement probabilities

Measuring a qubit yields a classical outcome: 0 or 1

Probabilities are given by amplitudes

After measurement, the qubit collapses to the observed basis state

If |ψ⟩ = (1/√2)|0⟩ + (1/√2)|1⟩

Before measurement, the system can exhibit interference effects when combined with other operations

Enables quantum algorithms to create and transform combinations of many computational paths

Allows interference to amplify correct answers and suppress incorrect ones

Any pure qubit state corresponds to a point on a sphere

North pole: |0⟩, South pole: |1⟩

Other points: superpositions with different phases

Latitude relates to probability of measuring 0 vs 1

Longitude encodes relative phase between |0⟩ and |1⟩

Single-qubit operations can be viewed as rotations around axes (X, Y, Z)

Computation expressed as a sequence of gates applied to qubits, then measurement

X gate (bit-flip)

Z gate (phase-flip)

H gate (Hadamard)

S/T gates (phase gates)

CNOT (controlled-NOT)

Controlled phase gates (CZ, etc.)

Quantum evolution (excluding measurement) is unitary and reversible

n qubits represent a state over 2^n basis states

Example

Simulating large quantum states becomes difficult for classical computers as n grows

Quantum computers manipulate this large state directly (subject to noise and control constraints)

A correlation between qubits where the joint state cannot be factored into independent single-qubit states

(|00⟩ + |11⟩)/√2

Measuring one qubit instantaneously determines correlated outcomes with the other

Enables non-classical correlations used in algorithms, error correction, and communication protocols

Amplifies amplitudes of desired outcomes

Cancels amplitudes of undesired outcomes

Superposition alone does not guarantee speedup; carefully designed interference patterns do

Measurement returns only one outcome per run

Advantage comes from shaping amplitudes so the right answers are more likely

Prepare superposition

Apply transformations that encode problem structure

Use interference to concentrate probability on correct results

Measure and repeat as needed for confidence

Efficient factoring of large integers (theoretical exponential speedup vs best known classical)

Impacts RSA-style public-key cryptography

Quadratic speedup for unstructured search

Finds a marked item in ~O(√N) queries vs O(N)

Simulating molecules/materials and quantum systems more naturally than classical approaches

VQE (Variational Quantum Eigensolver) for chemistry/optimization

QAOA (Quantum Approximate Optimization Algorithm) for combinatorial optimization

Superconducting qubits

Trapped ions

Photonic approaches

Spin-based qubits

Qubit count

Gate fidelity (error rates)

Coherence time (how long quantum information lasts)

Connectivity (which qubits can interact directly)

Measurement fidelity and speed

Interaction with environment causes loss of coherence (decoherence)

Control imperfections introduce gate errors

Limits circuit depth (how many operations can be done reliably)

Requires repeated runs and statistical analysis

Encode one logical qubit into many physical qubits to detect/correct errors

Use redundancy via entanglement without directly copying unknown states (no-cloning theorem)

Surface codes (widely studied for scalability)

Stabilizer codes (framework for many codes)

Fault-tolerant quantum computing needs many more physical qubits than logical qubits

Noisy Intermediate-Scale Quantum devices: tens to thousands of noisy qubits without full error correction

Research into chemistry simulation, optimization heuristics, sampling tasks

Benchmarking and exploring quantum advantage in specific tasks

Noise restricts algorithm depth and reliability

Hard to demonstrate broad, consistent advantage over classical methods

Threat to classical public-key systems (RSA, ECC) via Shor’s algorithm

Motivation for post-quantum cryptography

Better modeling of molecular energies, reactions, catalysts, batteries

Potential improvements for certain structured problems; active research

Quantum kernels, sampling-based methods, potential speedups in niche settings

Better understanding of quantum many-body systems and high-energy physics models

Only certain problem classes show proven or expected speedups

Measurement returns limited information; algorithms rely on interference

Correlations do not transmit usable information instantaneously