Glossary

Basis

Bra-ket notation:

Classical bit: A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)

Closed system

Commutator: The commutator of A and B, signified by [A,B], is AB-BA. Its value may be found by implementing the operators of A and B on a test function. If something has a commutator of zero, it is said to commute.

Complex conjugate:

Complex number: A complex number has a real and imaginary part. A complex number can be represented in the form a+bi or Ce^(i\theta).

Controlled not (CNOT gate)

Controlled operation: An operation on a state or set of states that is conditioned on another state or set of states.

Dirac delta function

DiVincenzo's requirements for quantum computing

Dot product: The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed.

Entangled state

Gate (see Quantum gate)

Grover's algorithm

Hadamard gate

Hermitian: An operator whose transpose equals its complex conjugate.

Hilbert-Schmidt inner product (2.4)

i: square root of negative one

Identity matrix:

Isolated system (seen Closed system)

Ket: See bra-ket notation

Matrix transformation

Normalization: A process of scaling some set of numbers or functions in order that an operation including them returns a desired value. For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.

Open system

Ordered basis

Orthogonal

P gate:

Pauli matrices: The X,Y,Z gates.

Phase gate: See Z gate or P gate

Projection postulate

Quantum bit: See Qubit

Quantum gate: A unitary transformation applied to one or more qubits.

Quantum NOT gate: see X gate

Qubit: A Qubit is represented by two states of a quantum mechanical system. (1.3)

RSA encryption

set

Shor's algorithm

Superposition: A qubit state in superposition, \phi may be written as |\phi>=\alpha|0>+\beta|1> where \alpha and \beta are complex numbers.

Tensor product

Trace

Transpose

Unitary transformation: A transformation which leaves the magnitude of any object it transforms the same.

Universal quantum computing

Universal set of gates (universality) (2.6)

X gate (2.3.2)

Vector space

Y gate

Z gate, or phase-flip gate (2.3.2)