FOUNDATIONS
Amplitude
A complex coefficient of a basis state. Its squared magnitude gives the probability of that outcome when measuring in the corresponding basis.
THE REFERENCE DESK
Quantum, defined.
Concise definitions for the mathematics, circuits, algorithms, and experimental methods used throughout the course.
46OF 46 CONCEPTS
ALGORITHMS
Amplitude amplification
A family of algorithms that increases the probability of marked outcomes by iteratively changing relative phases and reflecting amplitudes.
CIRCUITS
Ancilla
An auxiliary qubit used as workspace in a circuit. It may need to be returned to a known state so it does not remain entangled with the answer.
MATHEMATICS
Basis
A set of vectors used to represent states in a vector space. Measurement probabilities depend on the measurement basis.
ENTANGLEMENT
Bell inequality
A bound obeyed by correlations in a class of local hidden-variable models under stated assumptions. Quantum predictions can violate suitable Bell inequalities.
ENTANGLEMENT
Bell state
One of four maximally entangled two-qubit states. For example, |Φ⁺⟩ = (|00⟩ + |11⟩)/√2.
GEOMETRY
Bloch sphere
A geometric representation of pure single-qubit states up to global phase. The surface is pure states; mixed qubit states lie inside.
FOUNDATIONS
Born rule
The rule that assigns a measurement probability equal to the squared magnitude of the associated amplitude, or Tr(ρΠ) for a density operator ρ and projector Π.
CIRCUITS
CNOT
A two-qubit controlled-NOT gate: it applies X to the target if the control is |1⟩ and leaves the target unchanged if the control is |0⟩.
ERROR CORRECTION
Code distance
The minimum weight of a physical operator that acts nontrivially on encoded logical information in an error-correcting code.
NOISE
Coherence
The phase relationships between components of a quantum state that allow interference. Noise or entanglement with an environment can reduce observable coherence.
FOUNDATIONS
Computational basis
The standard basis {|0⟩, |1⟩} for a qubit, extended to bit strings such as |00⟩ and |11⟩ for multiple qubits.
CIRCUITS
Controlled gate
A gate whose operation on a target depends on the state of one or more control qubits.
MATHEMATICS
Density matrix
An operator ρ describing a pure state or statistical mixture. It is Hermitian, positive semidefinite, and has trace one.
ALGORITHMS
Deutsch–Jozsa algorithm
A quantum algorithm that distinguishes constant from balanced Boolean functions with one oracle query, given the problem's promise.
NOISE
Decoherence
The loss of locally observable phase coherence through interaction with degrees of freedom that are not tracked, often the environment.
ENTANGLEMENT
Entanglement
A property of a joint state that prevents it from being written as a product of states for its subsystems.
ERROR CORRECTION
Error syndrome
The outcomes of check measurements in a quantum error-correcting code. They provide information about errors without directly revealing the encoded logical state.
ERROR CORRECTION
Fault tolerance
A method of carrying out computation while correcting errors as they occur, so logical errors remain controlled as a circuit grows.
CIRCUITS
Gate
A quantum operation represented by a unitary matrix for ideal closed-system evolution. Measurement and noise channels are different kinds of operations.
FOUNDATIONS
Global phase
A common complex phase multiplying every component of a state vector. It has no observable effect on that isolated state.
CIRCUITS
Hadamard gate
A single-qubit unitary that maps |0⟩ to |+⟩ and |1⟩ to |−⟩. It changes between the Z and X bases.
MATHEMATICS
Inner product
A complex-valued overlap between vectors. For normalized states, the squared magnitude of an overlap gives a projection probability.
ERROR CORRECTION
Logical qubit
An encoded qubit represented across multiple physical qubits by a quantum error-correcting code.
FOUNDATIONS
Measurement
An interaction that produces a classical outcome according to a specified measurement and the system's state. The measurement basis is part of the question asked.
MATHEMATICS
Mixed state
A state described by a density matrix that represents statistical uncertainty or a subsystem of a larger entangled state.
NOISE
Noise channel
A mathematical map describing how a quantum state changes because of interactions, imperfect control, or an unobserved environment.
ALGORITHMS
Oracle
A defined black-box operation used in an algorithm model to answer a specific query. Its construction and implementation costs still matter.
CIRCUITS
Pauli gates
The X, Y, and Z matrices. They are useful as single-qubit operations, rotation axes, and a basis for describing common error models.
MATHEMATICS
Partial trace
The operation TrB(ρAB) that obtains the reduced state of subsystem A when subsystem B is not observed.
FOUNDATIONS
Phase
The argument of a complex amplitude. Relative phase can affect interference; a phase shared by every amplitude is global and unobservable.
ALGORITHMS
Phase kickback
A technique in which a function value applied to a target prepared in a phase-sensitive state appears as a phase on the input register.
ALGORITHMS
Phase estimation
An algorithmic procedure that estimates an eigenphase of a unitary using controlled powers and a Fourier-transform step.
HARDWARE
Physical qubit
A hardware degree of freedom engineered to behave approximately as a two-level quantum system.
FOUNDATIONS
Probability amplitude
A complex value associated with a basis outcome. The Born rule converts its squared magnitude into a probability.
ALGORITHMS
Quantum Fourier transform
A unitary discrete Fourier transform over a register. It is used in phase estimation and in period-finding algorithms.
ENTANGLEMENT
Quantum teleportation
A protocol that transfers an unknown quantum state using shared entanglement, a joint measurement, and classical communication. It does not enable faster-than-light messaging.
FOUNDATIONS
Relative phase
The phase difference between components of a superposition. Unlike global phase, it can change later measurement statistics through interference.
EXPERIMENTS
Shot
One circuit execution followed by measurement. Repeated shots produce a sample used to estimate an outcome distribution.
ALGORITHMS
Shor's algorithm
A quantum algorithm for factoring integers that reduces the problem to period finding. Its cryptographic implications require a sufficiently large fault-tolerant machine.
ERROR CORRECTION
Stabilizer
A commuting observable whose specified eigenvalue defines part of a quantum error-correcting code space.
MATHEMATICS
State vector
A normalized vector of complex amplitudes representing a pure quantum state in a chosen basis.
FOUNDATIONS
Superposition
A coherent linear combination of basis states. It does not mean measurement returns every component or answer at once.
MATHEMATICS
Tensor product
The operation that combines state spaces. n qubits have a 2ⁿ-dimensional joint state space.
HARDWARE
Transpilation
The compilation of an abstract circuit into gates and connections supported by a particular device architecture.
MATHEMATICS
Unitary
A matrix U satisfying U†U = I. Unitary operations preserve inner products and normalization.
Definitions are kept short here; the curriculum develops each idea with examples and experiments.
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