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 group
Abelian subgroup (see abelian group and 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:
Centralizer
Checksum (see dot product)
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
Code
Codewords
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.
Coset of a group
CSS codes
Cyclic group
Dagger (see hermitian conjugate)
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.
Diagonalizable: A matrix M is diagonalizable when it can be put into the form D=S^(-1)MS, where S and S^(-1) exist and are inverses.
Differentiable manifold
Dirac delta function
Dirac notation (see bra-ket notation)
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 matrix
Dual of a code
Eigenfunction, eigenvalue, eigenvector: If HY=EY where H is a matrix, E is a scalar and Y is a vector, then Y is the eigenvector and E is the eigenvalue. If Y is a function, it is called an eigenfunction.
Entangled state
Environment system (see Bath system)
EPR paradox
Epsilon tensor
Equivalent representation
Error syndrome
Euler angle parametrization
Euler's law: sine(x)+i*cos(x)=e^(ix)
Expectation value:
Exponentiating a matrix (see matrix exponential)
Faithful representation
Field
Gate (see Quantum gate)
General linear group
Generators of a group
Generator matrix
Gram-Schmidt decomposition (see Schmidt decomposition)
Group
Grover's algorithm
H bar: Planck's constant divided by 2\pi
Hadamard gate
Hamming bound
Hamming code
Hamming distance
Hamming weight
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.
Hermitian conjugate: The transpose complex conjugate of an operator.
Hidden variable theory (see also local hidden variable theory):
Hilbert-Schmidt inner product (2.4)
Hilbert space
Homomorphism
i: square root of negative one
Identity matrix: A matrix of zeros except for the diagonal, where each element is 1. Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.
Inner product (see dot product)
Inverse of a matrix: A matrix's inverse is the matrix which, when they are multiplied together, yield an identity matrix.
Invertible matrix: A matrix for which an inverse exists.
Isolated system (see Closed system)
Isomorphism
Isotropy group or Isotropy subgroup (see stabilizer)
Jacobi identity
Ket: See bra-ket notation
Kraus representation (or Kraus decomposition) (see SMR representation)
Kronecker delta
Levi-Civita symbol (see epsilon tensor)
Lie algebra
Lie group
Linear code
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.
Little group (see stabilizer)
Local actions
Local hidden variable theory (see also hidden variable theory):
Logical bit
Matrix exponential
Matrix properties
Matrix transformation
Measurement
Minimum distance
Modular arithmetic: When a number is divided into another and does not go evenly, there is left a remainder. Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number. For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced "three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer. This usage of modulo has nothing to do with the physics usage of modulus.
Modulus
n,k,d code (see [n,k,d] code)
No cloning theorem: No operator can duplicate an arbitrary quantum state.
Noise
Non-degenerate code
Normalizer:
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.
One-to-one: A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.
Onto: A mapping where each domain element mapped to at most one range element.
Open system
Operator
Operator-sum representation (see SMR representation)
Order of a group
Ordered basis
Orthogonal: Two vectors are orthogonal when their dot product is zero.
Outer product
P gate (not the phase gate):
Parity
Parity check (see inner product)
Parity check matrix
Partial trace
Partition of a group
Pauli group
Pauli matrices: The X,Y,Z gates.
Permutation:
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
Representation space
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: Any mathematical construct.
Shor's algorithm
Shor's nine-bit quantum error correcting code
Similarity transformation
Singular values
Singular value decomposition
SMR representation
Special unitary matrix
Spin
Spooky action at a distance
Stabilizers of a group
Stabilizer code
Standard deviation
Stationary subgroup (see stabilizer)
Stirling's formula
Subgroup
Superposition: A qubit state in superposition, \phi may be written as |\phi>=\alpha|0>+\beta|1> where \alpha and \beta are complex numbers.
Syndrome measurement
Taylor expansion
Teleportation
Tensor product
Trace: The sum of the diagonal elements of a matrix.
Transpose
Trivial representation
Turing machine
Uncertainty principle
Unitary matrix
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)
Variance
Vector: A directed quantity.
Vector space
Weight of a vector (see Hamming weight)
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)
