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: Two expressions, consisting of a real number ($\alpha \!$ ) and imaginary number ($\beta i\!$), where the $\beta \!$ component is of the same magnitude, but different sign. For some $\alpha ,\beta \in \mathbb {R} ,\alpha i+\beta$ and $\alpha i-\beta \!$.

Complex number: A complex number has a real and imaginary quantity. A complex number can be represented in the form $\alpha +\beta i\!$ or, $ce^{(i\theta )}\!$, where $\alpha ,\beta ,c,\theta \in \mathbb {R}$

Controlled not (CNOT gate): Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is $\left\vert {1}\right\rangle \,\!$.

Controlled operation: An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an "If-then" or "If-then-else" statement.

Coset of a group: Used in group theory where ${\mathcal {S}}\,\!$ is a subgroup and $G\,\!$ be an element of the group ${\mathcal {G}}\,\!$. The left coset is a subset of the group $G{\mathcal {S}}=\{GS|S\in {\mathcal {S}}\}.\,\!$ One can similarly define the right coset.

CSS codes: Short for Calderbank-Shor-Steane codes. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, ${\mathcal {C}}_{1}\,\!$ and ${\mathcal {C}}_{2}\,\!$, that are of the form $[[n,k,d]]\!$. Protects against both phase- and bit-flips.

Cyclic group:

Dagger: See Hermitian Conjugate, Adjoint (of a matrix)

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 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 $AA^{-1}=mathbb{I}=A^{-1}A\!$, where $mathbb{I}\!$ 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, only one or the other.)

Invertible matrix: A matrix for which an inverse exists.

Isolated system: See Closed system.

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

Isotropy group: See Stabilizer.

Isotropy subgroup: See Stabilizer.

Jacobi identity:

Ket: See Bra-Ket Notation.

Kraus representation: 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 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 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 matrix: Matrix whose eigenvalues are all greater than zero.

Semi-definite matrix: Matrix whose eigenvalues are nonnegative.

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: One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.

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: Operation for which every state on which the operator can act there exists an operation which restores it to its original state.

RSA encryption:

Schmidt decomposition:

Schrodinger's Equation:

Set: Any mathematical construct, often represented by a capital letter.

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, where $\phi$ 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: Phase-flip gate. (2.3.2)