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Glossary

Every load-bearing term across the 24 lessons, defined crisply in the lessons' own words and searchable.

Written by Claude Fable 5

Every load-bearing term across the 24 lessons, defined crisply. Filter to find one fast, or skim it to see how the pieces connect.

AES-256: A widely used symmetric cipher with a 256-bit key. Grover only halves its effective strength to 128 bits, which stays far beyond any feasible attack.
Amplitude: A complex number attached to each possible outcome. Unlike a probability it can be negative or complex, which is what lets outcomes cancel. Its squared magnitude is the probability.
Amplitude amplification: Grover's technique: two reflections per step, an oracle sign-flip of the marked amplitude then diffusion, rotate the state toward the answer. It is the source of the quadratic speedup.
Argument: Written arg z, the angle a complex number makes with the positive real axis. Also called its phase.
Bell state: A maximally entangled two-qubit state, such as (|00> + |11>)/root2, whose qubits are perfectly correlated on measurement but individually random.
Bijection: A one-to-one and onto map: every input has a distinct output and every output is used once. On bitstrings, a gate is reversible if and only if it is a bijection.
Bit flip: The X gate, the quantum NOT. It swaps the amplitudes of |0> and |1>, a half-turn about the Bloch x-axis.
Bloch sphere: The ball whose surface holds every pure one-qubit state. |0> is the north pole, |1> the south pole, and the equator is a ring of equal superpositions differing only in phase.
Born rule: The bridge from amplitudes to probabilities: measuring alpha|0> + beta|1> returns 0 with probability |alpha|^2 and 1 with probability |beta|^2.
BPP: Bounded-error probabilistic polynomial time: what a classical computer solves efficiently when it may flip coins and be wrong with small probability. Believed to equal P.
BQP: Bounded-error quantum polynomial time: the problems a polynomial-size quantum circuit decides correctly with probability at least two-thirds. The formal set of what quantum makes easy.
CNOT: Controlled-NOT: a two-qubit gate that applies X to the target if and only if the control qubit is 1. On a superposed control it produces entanglement.
Collapse: On measurement the state jumps to the basis state matching the outcome. Measuring again then returns the same result with certainty.
Decoherence: A qubit accidentally entangling with an untracked environment, so that from your point of view its superposition has been measured and lost.
Diffusion operator: The second Grover reflection: it replaces every amplitude with 2*mean minus itself, inverting each about the average. Geometrically a reflection across the equal-superposition direction.
Entangled: A joint state that cannot be written as a tensor product of a state for each qubit. Its parts have no independent state of their own.
Euler's formula: The identity e^(i theta) = cos theta + i sin theta. Read geometrically, e^(i theta) is the point on the unit circle at angle theta.
Gate: A fixed rule that transforms a quantum state. Classically AND, OR and NOT act on bits; a quantum gate is the reversible, linear, unitary generalization.
Global phase: An overall factor e^(i gamma) multiplying the entire state. It has magnitude 1, cancels in every measurement probability, and is physically undetectable.
Hadamard: The H gate. It maps |0> to |+> and |1> to |->, turning a definite bit into an equal superposition and back. It is its own inverse, so HH = I.
Harvest now, decrypt later: An attack that records encrypted traffic today and stores it, to be decrypted once a quantum computer capable of running Shor exists.
Histogram: A bar chart of how often each outcome bitstring appeared across all shots. It approximates the circuit's true output probabilities.
Interference: Amplitudes for the same outcome combining before they are squared: equal signs reinforce and opposite signs cancel. The engine every quantum speedup runs on.
Ket: Dirac notation for a state vector: |v> is a column vector standing for a quantum state, read 'ket v'.
L1 norm: The plain sum of the absolute values of a vector's entries. Classical probability normalizes a distribution so its L1 norm equals 1.
L2 norm: The square root of the sum of squared magnitudes: ordinary Euclidean length. Quantum states are unit vectors in the L2 norm, which is what lets amplitudes carry a sign.
Lattice-based cryptography: The leading family of post-quantum schemes, whose security rests on hard problems about points in high-dimensional lattices, with no known efficient quantum attack.
Logical qubit: One protected qubit of information encoded across many physical qubits, so that errors on a few of them can be detected and corrected.
Magnitude: Written |z|, the distance from the origin to the complex number z. For z = x + iy it is the hypotenuse, sqrt(x^2 + y^2).
Measurement: The read-out step. It samples one basis state by the Born rule and yields one classical bitstring per run. It is not a gate and cannot be reversed.
NISQ era: Noisy Intermediate-Scale Quantum: today's devices with tens to hundreds of qubits and no full error correction, so results carry noise from decoherence and imperfect gates.
No-cloning theorem: There is no unitary that copies an arbitrary unknown quantum state. Cloning a general superposition would violate the linearity of quantum gates.
No-signaling: A local operation on one half of an entangled pair leaves the other half's own measurement statistics unchanged, so entanglement carries no faster-than-light message.
Normalization: The constraint |alpha|^2 + |beta|^2 = 1: total probability across all outcomes must equal 1. Geometrically it means the state is a unit vector.
NP-complete: The hardest problems in NP, such as SAT and scheduling. BQP is not believed to contain them, so quantum computers are not expected to solve them efficiently.
Oracle: A reversible gate U_f that computes f without destroying its input: it maps |x>|y> to |x>|y XOR f(x)>, so running it twice returns the original state.
Orthogonal: Perfectly distinguishable: a single measurement in the right basis tells the two states apart with certainty. |0> and |1> are orthogonal, sitting at opposite poles.
P: Polynomial time: the problems an ordinary computer solves in a number of steps that grows only polynomially with input size. The practical meaning of efficient.
Period: The order r of a modulo N: the smallest positive integer with a^r = 1 mod N. The sequence a^x mod N repeats every r steps.
Period finding: Finding the order r of a modulo N, the smallest r with a^r = 1 mod N. It is the single quantum step of Shor's algorithm, done with the QFT.
Phase flip: The Z gate. It leaves |0> alone and multiplies |1> by -1, changing no measurement probability yet flipping the relative phase.
Phase kickback: With the oracle's output qubit in |->, the query multiplies the input |x> by the phase (-1)^f(x). The function's value is written as a sign on the input, not a bit to read out.
Phi: The azimuthal angle around the Bloch equator. It sets the relative phase e^(i phi) between the |0> and |1> parts and never changes standard-basis probabilities.
Physical qubit: A single real hardware qubit, noisy and error-prone. Many of them together encode one logical qubit.
Post-quantum cryptography: Public-key schemes built on problems believed hard for classical and quantum computers alike, chiefly structured-lattice problems. They run on ordinary hardware today.
PSPACE: The problems solvable using a polynomial amount of memory, however much time it takes. BQP sits inside it: P is contained in BPP in BQP in PSPACE.
Quadratic speedup: A speedup that replaces a cost of N with a cost of sqrt(N). Real and useful, but it only divides the exponent by two, so an exponential cost stays exponential.
Quantum Fourier transform: A change of basis that sends |j> to an even spread of all basis states, each carrying a phase that winds at rate j. It is the discrete Fourier transform realized as a circuit.
Quantum parallelism: A single oracle query on a superposed input produces a superposition holding every input paired with its output. The evaluations coexist, but only one pair survives measurement.
Qubit: The quantum unit of information: a unit vector alpha|0> + beta|1>, one specific state pinned down by two amplitudes, not a bit that is secretly 0 or 1.
Relative phase: A phase between the components of a superposition, such as e^(i phi) on |1> only. It is observable through interference and is the resource every quantum algorithm uses.
Reversible: A gate is reversible when its input can always be recovered from its output, equivalently when no two inputs share an output. Every quantum gate is reversible.
Shots: Repeated identical runs of the same circuit. Each shot yields one bitstring; many shots build up the output distribution.
Standard basis: The pair |0> and |1> (also called the computational basis) in which measurement outcomes are read. A qubit's amplitudes are its coordinates in this basis.
State vector: The list of amplitudes that fully describes a quantum state, written as a column vector. For one qubit it is alpha|0> + beta|1>.
Superposition: A qubit that is a genuine blend of |0> and |1> at once, with amplitudes on both. Measuring it yields a single 0 or 1 with odds set by those amplitudes.
Surface code: A 2D grid of physical qubits with local parity checks encoding one logical qubit. Below threshold, adding more qubits drops the logical error rate exponentially.
Syndrome: The pattern of parity-check results in error correction. It pinpoints which qubit is out of line without revealing the logical value the qubits encode.
T1: The relaxation time: how long a qubit keeps any excitation before it decays back toward |0>.
T2: The dephasing time: how long the relative phase, the equatorial part of the Bloch vector, survives before it decays away.
Tensor product: The operation that combines the state spaces of two systems. Two qubits combine into a 4-dimensional space; n qubits into 2^n dimensions.
Theta: The polar angle on the Bloch sphere, measured down from |0>. It sets the measurement odds through P(0) = cos^2(theta/2).
Threshold theorem: If the physical error rate per gate is below a fixed threshold (roughly 0.1 to 1 percent), adding physical qubits drives the logical error rate down exponentially. Above threshold, more qubits make things worse.
Toffoli: The three-bit reversible gate (CCNOT) that flips the third bit only when the first two are both 1. It is universal for reversible computing.
Unitary: A matrix U whose conjugate-transpose is its inverse, U-dagger U = I. It preserves vector length and is therefore reversible. Every quantum gate is unitary.