Glosarium

A complex number whose squared magnitude gives a measurement probability.

The coefficient α in |ψ⟩ = α|0⟩ + β|1⟩ (and its multi-qubit generalization). Amplitudes carry both magnitude and phase; probabilities are |α|², so global phase is unobservable while relative phase drives interference.

Entries of the state vector in a chosen basis of the Hilbert space; more generally, matrix elements ⟨x|ψ⟩. All quantum-algorithmic advantage lives in the structured manipulation of amplitude phases before Born-rule sampling.

An analog quantum approach that evolves a system toward low-energy configurations.

Encode a cost function into an Ising-type Hamiltonian, initialize in an easy ground state, and slowly deform toward the problem Hamiltonian so the system ends near a low-energy (good) solution. Not gate-based.

Adiabatic-inspired analog optimization; distinct from universal adiabatic quantum computation. Practical advantage remains problem-specific and contested against classical heuristics such as simulated annealing and parallel tempering.

A geometric representation of a single pure qubit state.

A pure qubit state cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩ maps to the point (θ, φ) on a unit sphere. It is a map of state space, not a picture of a spinning ball.

The CP¹ ≅ S² representation of single-qubit pure states; mixed states fill the interior ball via the density-matrix Bloch vector. The picture does not generalize simply beyond one qubit.

The number of sequential gate layers in a circuit.

The longest path of dependent operations after compilation to the device's native gates and connectivity. The quantity that must fit inside coherence and error budgets.

Post-transpilation depth (including routing SWAPs) is the operative figure; for fault-tolerant algorithms, T-depth and magic-state supply typically dominate the wall-clock estimate.

Preservation of quantum phase relationships.

The persistence of definite phase relations between amplitudes, characterized by relaxation and dephasing times (T1, T2). Interference (and therefore quantum computation) requires it.

Off-diagonal structure of the density matrix in the relevant basis; decays under environmental coupling per the system's noise channels. Benchmarked via T1/T2, Ramsey and echo experiments; a necessary but not sufficient system-quality metric.

Loss of quantum coherence through interaction with the environment.

Unwanted system-environment entanglement that decays relative phases, turning quantum superpositions into effectively classical mixtures: the reason qubits live in shielded, cold, dark places.

Formally a CPTP channel contracting off-diagonal density-matrix elements; the practical ceiling on circuit depth in NISQ devices and the phenomenon error correction exists to outpace.

A nonclassical correlation among quantum systems.

A joint state that cannot be written as a product of individual states, e.g. the Bell state (|00⟩+|11⟩)/√2. Measuring one side yields correlations no independent local description reproduces. It cannot transmit information faster than light.

Non-separability across a tensor-product decomposition; quantified via Schmidt rank and entanglement entropy. A resource for computation, teleportation and device-independent protocols; certified experimentally by Bell-inequality violations (2022 Nobel Prize).

Encoding logical information so physical errors can be detected and corrected.

Spreading one logical qubit across many physical qubits and repeatedly measuring stabilizer 'syndromes' that reveal errors without reading the protected information. One logical qubit can cost hundreds to thousands of physical qubits.

Stabilizer codes define the codespace as a joint +1 eigenspace of commuting Pauli checks; below threshold, logical error falls exponentially with code distance. Universality then hinges on non-Clifford resources (T gates, magic-state distillation), which dominate overhead.

Statistical techniques that reduce noise bias without full fault tolerance.

Post-processing tricks (readout correction, zero-noise extrapolation, symmetry verification, probabilistic error cancellation) that clean up expectation values from noisy circuits at the price of extra samples. Mitigation is not correction.

Bias-reduction estimators whose sampling overhead generally scales exponentially with circuit size/noise; useful for NISQ experiments, but no route to unbounded depth. Distinguish sharply from active QEC in any diligence exercise.

The ability to perform arbitrarily long computations reliably despite component errors, provided errors are below a threshold.

Architectures where encoded operations, measurement and correction are all performed so that errors don't cascade: enabling logical error rates far below physical ones.

The threshold theorem's regime: with physical error p below threshold p_th and poly-log overhead, arbitrarily long quantum computation is possible. The field's central engineering race is demonstrating this at scale with acceptable constants.

A controlled transformation of a quantum state.

A unitary operation on one or more qubits: X flips, Z shifts phase, H makes superpositions, CNOT entangles. Circuits are time-ordered gate sequences; two-qubit gates are usually the noisiest.

Elements of U(2ⁿ) generated by a universal set (e.g. Clifford + T). Native gate sets differ per hardware; compilation into them (and the resulting depth) often decides practicality.

Reinforcement or cancellation of quantum amplitudes.

Amplitudes add like waves: in phase they reinforce, out of phase they cancel. Quantum algorithms choreograph phases so wrong answers cancel and right answers amplify. This, not 'trying everything at once', is the mechanism of speedup.

Coherent sums over computational paths; the guide's one-sentence definition stands: algorithms reshape probability amplitudes through interference so fewer steps are needed to extract the desired property. Destroyed by which-path information (decoherence).

An error-corrected qubit encoded using multiple physical qubits.

The qubit an algorithm actually sees: encoded across many physical qubits with continuous syndrome measurement and decoding. Logical qubit count and logical error rate are the milestones that matter.

Codespace qubit of a QEC code (typically surface-code patches today); quality measured by logical error per cycle and its scaling with distance. 'Below threshold' demonstrations (2023–24 onward) mark the field's honest scoreboard.

Extraction of a classical outcome from a quantum state.

Projects the state onto a basis and returns one bit string per run with Born-rule probabilities: you cannot read the amplitudes themselves. Hence circuits are repeated over many 'shots'.

Generalized as POVMs; basis choice (Z, X, Y, …) determines the information extracted, which underlies tomography and many algorithms. Output-extraction cost is a recurring killer of naive quantum-advantage claims.

Noisy intermediate-scale quantum: current systems with limited size and significant noise.

Devices with tens to hundreds of physical qubits and no full error correction. Capable of experiments and shallow circuits, not arbitrarily long computation.

Preskill's 2018 coinage. The open question of the era: whether any NISQ workload achieves practical, economically relevant advantage before fault tolerance; dequantization results keep raising the classical bar.

An individual hardware element used to store and manipulate quantum information.

One actual device qubit (a transmon, trapped ion, atom, photon mode or spin) with its own error rates. Many physical qubits make one logical qubit.

The raw resource whose fidelity, connectivity and stability determine QEC overhead; headline physical counts without error data are marketing, not capability.

Classical cryptographic algorithms designed to resist attacks by quantum computers.

New public-key schemes (lattice-, hash-, code-based) standardized by NIST. FIPS 203 ML-KEM, FIPS 204 ML-DSA, FIPS 205 SLH-DSA (Aug 2024), plus HQC (2025) — that run on today's computers. Distinct from quantum cryptography.

Security reductions rest on lattice/code/hash problems believed hard for both classical and quantum adversaries. Migration is an enterprise architecture program: inventory, crypto-agility, hybrid deployment, protocol and certificate updates.

The fundamental two-level unit of quantum information.

A two-level quantum system with state |ψ⟩ = α|0⟩ + β|1⟩, |α|²+|β|² = 1. It carries amplitude and phase. Not merely an unknown classical value, and not literally '0 and 1 at the same time'.

A ray in ℂ²; physically any controllable two-level system with sufficient coherence, initialization, gates and readout (DiVincenzo criteria). The engineering, not the definition, is the hard part.

A demonstrated benefit over the best practical classical method for a defined task.

The bar is end-to-end: against the best classical algorithm on appropriate hardware (including GPUs and specialized solvers), counting data loading, repetitions and readout: not against naive brute force.

A moving target: dequantization and improved classical methods have repeatedly eroded claimed advantages. Distinguish contrived-benchmark advantage from advantage on a problem someone will pay to solve: the field's dominant conflation.

An older term for performing a task infeasible for classical computers, regardless of immediate usefulness.

Demonstrations on contrived sampling tasks (Google 2019 onward). Milestones for hardware control, deliberately not claims of useful advantage.

Claims are benchmark-and-baseline dependent; the 2019 result drew a credible IBM challenge on classical runtime, and later tensor-network methods narrowed gaps further. Treat every such claim as a (claim, baseline, date) triple.

One execution and measurement of a quantum circuit.

Circuits are run thousands of times; the distribution over shots is the actual output. Estimating an observable to precision ε costs O(1/ε²) shots: a real budget line in any workflow.

Sampling cost frequently dominates variational and estimation workloads; amplitude estimation trades shots for coherent depth (O(1/ε)), which is exactly why fault tolerance matters for it.

A coherent linear combination of basis states.

n qubits carry 2ⁿ amplitudes at once. Measurement returns a single n-bit outcome per shot. Superposition is a resource for interference, not an automatic answer generator.

Linearity of the state space; computationally valuable only in combination with entanglement and engineered interference. 'Reads all answers in parallel' is the misconception this site exists to correct.

A leading quantum error-correcting code based on a two-dimensional arrangement of qubits.

Local stabilizer checks on a 2-D lattice; code distance sets how many errors are correctable, and (below threshold) bigger patches mean exponentially fewer logical errors.

High threshold (~1% circuit-level, hardware-dependent) and planar locality made it the default; costs include large qubit overhead and non-transversal non-Clifford gates, hence lattice surgery and magic-state factories. qLDPC alternatives are the active frontier.

A hybrid algorithm in which a classical optimizer tunes a parameterized quantum circuit.

The loop: choose an ansatz, run it, measure an objective, update parameters classically, repeat. VQE targets energies; QAOA targets combinatorial optimization. Shallow circuits, many noisy evaluations.

Trainability is the catch: barren plateaus, rugged landscapes and measurement cost can erase the point. A 'variational' label establishes nothing about advantage: ansatz structure and classical competition decide.