Symbols & glossary

State notation

Symbol	Meaning
$\lvert\psi\rangle$	A state vector, pronounced “ket psi”
$\langle\psi\rvert$	Its conjugate-transposed row vector, a bra
$\alpha,\beta$	Generally complex probability amplitudes
$\lvert\alpha\rvert^2$	Squared magnitude; a measurement probability in the stated basis
$\otimes$	Tensor product, combining systems
$U^\dagger$	Conjugate transpose of U
$I$	Identity operation
$\theta,\phi$	Angles; check whether a lab uses degrees, radians or turns

Core terms

Amplitude: a complex number whose squared magnitude contributes a measurement probability. Amplitudes also carry relative phase.

Basis: an independent set of vectors used to describe a state. A measurement basis determines which alternatives a projective measurement distinguishes.

Bloch sphere: a geometric picture of a single-qubit state. Pure states lie on its surface; mixed states lie inside. The current explorer models only pure states.

Circuit: an ordered sequence of quantum operations and measurements. Site diagrams run left to right.

Entanglement: a joint state that cannot be described as a product of the individual subsystem states. Classical correlation alone is insufficient to establish it.

Gate: an ideal unitary operation on a quantum state. Hardware implementations have constraints and errors.

Global phase: a common unit-magnitude factor on a whole state; it does not change measurement probabilities.

Interference: the addition of amplitudes, including reinforcement and cancellation due to relative phase.

Measurement: a physical interaction that produces a classical result. It generally changes the state; measuring an eigenstate in its eigenbasis can leave it unchanged.

Oracle: a problem-specific operation supplied to an algorithm. An oracle query count does not automatically include its construction cost.

Qubit: a two-level quantum information system. A pure qubit is represented by two normalised complex amplitudes, up to global phase.

Shot: one preparation and execution followed by measurement. Many shots estimate an outcome distribution; they do not reveal every amplitude directly.

Statevector: the pure-state mathematical model used by the ideal simulator. Its memory grows exponentially with the number of qubits.

Tools for the algorithm chapters

Eigenvector and eigenvalue: a nonzero vector whose direction is preserved by a matrix. If $A|v\rangle=\lambda|v\rangle$, the factor $\lambda$ is its eigenvalue. For a unitary, that factor has magnitude one and can be read as a phase. See phase estimation.

Hermitian operator: a matrix equal to its conjugate transpose. Its eigenvalues are real. Quantum energy operators, called Hamiltonians, are Hermitian. See Hamiltonian simulation.

Expectation value: the average signed or numerical measurement result predicted by a model. For Z, it is $P(0)-P(1)$. One shot returns an outcome; many shots estimate the expectation.

Projector: an operator selecting a subspace. The squared norm of the selected component gives its measurement probability. The conditional state must then be normalised.

Singular value: a nonnegative measure of how much a matrix stretches a particular direction. Left and right singular vectors can differ, so singular-value transformation is not generally ordinary matrix exponentiation. See QSVT.

Condition number: the ratio of largest to smallest relevant singular value for an invertible matrix. A large ratio can make inversion sensitive and expensive. See linear systems.

Block encoding: a unitary on a larger system containing a scaled target matrix in a specified ancilla block. Selecting that block can have an input-dependent success probability. See block encoding.

Ancilla: an additional qubit or register used as workspace, a control or a flag. Its preparation, cleanup and measurement are part of the algorithm.

Query complexity: the number of calls to a specified input-access operation. Gate count, data preparation, memory and physical runtime are additional resources.

Additive error: an absolute difference, such as a probability estimate within 0.01. Relative error measures the difference as a fraction of the target value. These are very different requirements for rare events.

Gibbs state: a thermal mixture whose energy weights are proportional to $e^{-\beta E}$. It is a mixed-state ensemble rather than a coherent superposition with the same basis probabilities. See thermal preparation.

Hardware and software terms

Decoherence: unwanted interaction with the environment that degrades quantum information.

Logical qubit: an encoded information unit protected by a quantum error-correcting scheme. Its physical overhead depends on code, error rate and target reliability.

Physical qubit: an actual hardware system used to encode quantum information.

QAOA / VQE: families of hybrid variational methods using parameterised quantum circuits with classical optimisation. Useful practical advantage is not guaranteed by the method’s name.

Transpilation: rewriting a circuit for a chosen gate set, connectivity and execution model.

Security

QKD: quantum key distribution, using quantum communication and authenticated classical communication under stated security assumptions.

PQC: post-quantum cryptography, classical cryptographic methods designed to resist known quantum attacks. PQC is distinct from QKD.

QBER: quantum bit error rate. In the toy BB84 lab it is the fraction of sifted positions at which Alice and Bob disagree. QBER alone is not a complete security proof.

The original notes include the author’s longer glossary.