Glossary

[ [n,k,d] ] code: A code using n qubits to encode k logical qubits to correct (d-1)/2 errors (d will be odd).

Abelian subgroup

Adjoint: The transpose complex conjugate of an operator.

Ancilla

Angular momentum

Anti-commutation: Two operators anti-commute when AB+BA=0.

Basis

Bath system:

Bell's theorem

Bit flip error

Bloch sphere

Block diagonal matrix:

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.

CSS codes

Definite matrix (see matrix properties)

Degenerate: Having more than one of the same eigenvalue.

Density matrix

Density operator

Depolarizing error

Determinant: When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms. Determinants only exist for square matrices.

Dirac delta function

Disjointness condition

Distance of a quantum error correcting code

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.

Dual of a code

Eigenfunction

Eigenvalue

Eigenvector

Entangled state

Environment system (see Bath system)

EPR paradox

Error syndrome

Euler's law

Expectation value:

Exponentiating a matrix (see matrix exponential)

Field

Gate (see Quantum gate)

Generator

Generator matrix

Gram-Schmidt decomposition (see Schmidt decomposition)

Group

Grover's algorithm

H bar: Planck's constant divided by 2\pi

Hadamard gate

Hamiltonian: The operator for all conservative (able to be transformed) energy in the system. In quantum mechanics most energy is conservative.

Heisenberg exchange interaction (8.5.2:

Heisenberg uncertainty principle (see uncertainty principle)

Hermitian: An operator whose transpose equals its complex conjugate.

Hidden variable theory (see also local hidden variable theory):

Hilbert-Schmidt inner product (2.4)

Hilbert space

i: square root of negative one

Identity matrix:

Isolated system (see Closed system)

Ket: See bra-ket notation

Kraus representation (or Kraus decomposition) (see SMR representation)

Linear combination: A set of vectors each multiplied by a scalar and summed to equal a desired vector. A complete basis has a linear combination for all vectors of that dimension.

Linear map: A transformation from one vector to another using one operator once.

Local actions

Local hidden variable theory (see also hidden variable theory):

Logical bit

Matrix exponential

Matrix properties

Matrix transformation

Measurement

n,k,d code (see [n,k,d] code)

No cloning theorem: No operator can duplicate an arbitrary quantum state.

Noise

Non-degenerate code

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

Operator-sum representation (see SMR representation)

Ordered basis

Orthogonal

Outer product

P gate (not the phase gate):

Parity

Parity check matrix

Partial trace

Pauli group

Pauli matrices: The X,Y,Z gates.

Phase flip error

Phase gate: See Z gate

Planck's constant:

Polarization

Positive definite and semidefinite matrix (see matrix properties)

Probability for existing in a state:

Projector: A transformation such that P^2=P.

Projection postulate

Pure state

QKD: See quantum key distribution

Quantum bit: See Qubit

Quantum cryptography

Quantum dense coding

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

Quantum hamming bound

Quantum key distribution:

Quantum NOT gate: see X gate

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

Rank

Rate of a code

Reduced density operator

Reversibility of a quantum operation: For every operation on a qubit there exists an operation which restores the state to its original function.

RSA encryption

Schmidt decomposition

Schrodinger's Equation

set

Shor's algorithm

Shor's nine-bit quantum error correcting code

SMR representation

Spin

Spooky action at a distance

Stabilizer code

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

Taylor expansion

Teleportation

Tensor product

Trace

Transpose

Turing machine

Uncertainty principle

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)

Vector space

Weight of an operator: The number of non-identity elements in the tensor product.

Wigner-Clebsch-Gordon Coefficients

X gate (2.3.2)

Y gate

Z gate, or phase-flip gate (2.3.2)