Glossary

$[[n,k,d]]$ code: A code that uses n physical qubits to encode k logical qubits and will correct ${\frac {(d-1)}{2}}$ errors.

Abelian group: A group for which all elements commute.

Abelian subgroup: See Abelian Group, Subgroup

Adjoint (of a matrix): The transpose complex conjugate of an operator. (Also referred to as the conjugate or Hermitian conjugate.)

Ancilla: An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.

Angular momentum: A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.

Anti-commutation: Two operators anti-commute when $AB+BA=0\!$

Basis: Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.

Bath system: Describes a system that has had an unwanted interaction with an open quantum system. Environment.

Bell's theorem: "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics." See Full Article

Bit flip error: An error which takes $\left|0\right\rangle$ to $\left|1\right\rangle$ and $\left|1\right\rangle$ to $\left|0\right\rangle$. The operator which does this is the Pauli operator $\sigma _{x}$.

Bloch sphere: Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. Image

Block diagonal matrix: A matrix which has non-zero elements only in blocks along the diagonal.

Bra-ket notation: Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, bra, and the right symbol, ket. Bra: $\left\langle \phi \right\vert$; Ket: $\left\vert \psi \right\rangle$

Centralizer: Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some $v\in S$ and all $c\in S$, $vc=cv\!$, for some group $S\!$.

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 quantum system: A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.

Code: Short for Quantum Code. Used in correcting errors in quantum systems. Where $[n,k]\!$ is the way we describe a code $C\!$ with $n\!$ bits that are used to encode $k\!$ bits.

Codewords: Used to describe the set of all elements in a code $C\!$. There are $2^{n}\,\!$ $n\,\!$-bit words in the space.

Commutator: The commutator of $A\!$ and $B\!$ is denoted $[A,B]\!$, which means $AB-BA\!$. Its value may be found by implementing the operators of $A\!$ and $B\!$ on a test function. If the commutator of $A\!$ and $B\!$ is zero, they are said to commute.

Complex conjugate: For some $\alpha ,\beta \in \mathbb {R} ,\alpha i+\beta$ and $\alpha i-\beta \!$ are complex conjugates.

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 or adjoint)

Definite matrix (see matrix properties)

Degenerate: Having two or more eigenvalues that are equal.

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: See Section

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 (see Bath system)

EPR paradox

Epsilon tensor

Equivalent representation

Error syndrome

Euler angle parametrization

Euler's law: $\sin(x)+i\cos(x)=e^{ix}$

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 ($\hbar$): Planck's constant divided by $2\pi$, $h$ is Planck's constant

Hadamard gate

Hamming bound

Hamming code

Hamming distance

Hamming weight

Hamiltonian: The Hamiltonian operator gives the total energy of the system.

Heisenberg exchange interaction (8.5.2):

Heisenberg uncertainty principle (see uncertainty principle)

Hermitian: An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.

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

$i$: Denotes the 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: The inverse of a square matrix $A$ is the matrix, denoted $A^{-1}$, such that $AA^{-1}=I=A^{-1}A$, where $I$ is identity matrix. (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, but only one or the other.)

Invertible matrix: A matrix for which an inverse exists.

Isolated system (see Closed system): A system which does not interact with any other system.

Isomorphism: A one-to-one and onto mapping.

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.

Linear map:

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: There is no universal copying machine. See Section 5.2.

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: A positive definite matrix is one whose eigenvalues are all greater than zero. A positive semidefinite matrix has no negative eigenvalues.

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, but be careful, the NOT gate is only defined for qubits.

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

Reversible quantum operation: An operation is reversible if for every state on which the operator can act, there exists an operation which restores the state to its original.

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)