Unitary MatrixA square matrix whose conjugate transpose equals its inverse, ensuring reversibility and probability conservation in quantum operations.
Bloch SphereA unit sphere used to geometrically represent the state of a single qubit, where gates correspond to rotations.
Bra-Ket NotationDirac notation for quantum states, where |psi> (ket) represents a column vector and <psi| (bra) represents a row vector (conjugate transpose).
Pauli-X GateThe quantum NOT gate that flips |0> to |1> and vice versa, equivalent to a 180-degree rotation around the X-axis of the Bloch sphere.
Pauli-Y GateA single-qubit gate that combines a bit flip and phase flip, equivalent to a 180-degree rotation around the Y-axis of the Bloch sphere.
Pauli-Z GateA phase flip gate that leaves |0> unchanged and maps |1> to -|1>, equivalent to a 180-degree rotation around the Z-axis.
Hadamard GateCreates an equal superposition from a computational basis state, mapping |0> to (|0>+|1>)/sqrt(2) and |1> to (|0>-|1>)/sqrt(2).
CNOT GateControlled-NOT gate that flips the target qubit if and only if the control qubit is |1>, essential for creating entanglement.
T GateA pi/8 gate that applies a phase of e^(i*pi/4) to |1>, crucial for achieving universal quantum computation with the Clifford+T gate set.
S GateA phase gate that applies a phase of i to |1>, equivalent to the square root of the Z gate.
EigenvalueA scalar associated with a matrix and its eigenvector, representing the factor by which the eigenvector is scaled when the matrix is applied.
EigenvectorA non-zero vector that, when a matrix is applied to it, results in a scaled version of itself (only the magnitude changes, not the direction).
Tensor ProductA mathematical operation that combines two quantum systems into a larger joint system, used to describe multi-qubit states and gates.
Hilbert SpaceA complete complex vector space with an inner product, serving as the mathematical framework for quantum mechanics and quantum computing.
Conjugate TransposeThe matrix obtained by taking the transpose and then the complex conjugate of each entry, also known as the Hermitian adjoint or dagger operation.
Gate FidelityA measure of how closely a physically implemented gate matches the ideal mathematical gate operation, with 1.0 being perfect.
SWAP GateA two-qubit gate that exchanges the states of two qubits, equivalent to three consecutive CNOT operations.
Toffoli GateA three-qubit controlled-controlled-NOT gate that is universal for classical reversible computation and useful in quantum algorithms.
Rotation GateA parameterized single-qubit gate that rotates the qubit state by a specified angle around a given axis of the Bloch sphere.
Clifford GroupThe set of quantum gates that map Pauli operators to Pauli operators under conjugation, efficiently simulatable on classical computers by the Gottesman-Knill theorem.
Inner ProductA generalization of the dot product to complex vector spaces, used to compute transition amplitudes and measurement probabilities in quantum mechanics.
Measurement BasisThe set of orthogonal states used to perform a quantum measurement, with the computational basis (|0>, |1>) being the most common choice.
Phase GateA parameterized single-qubit gate that adds a relative phase between the |0> and |1> components without changing measurement probabilities.
Operator NormA measure of the maximum factor by which a matrix can stretch a vector, used to quantify gate errors and approximation quality.
Computational BasisThe standard measurement basis consisting of the states |0> and |1> for a single qubit, or tensor products thereof for multi-qubit systems.