Notation

In this section, we detail some of the notation and conventions used throughout the book. In writing our work, we have attempted to use the same notation as the original cited publications to maintain a high fidelity to the original literature. However, in some cases, we have modified the notation to make equations easier to read.

STANDARD NOTATION

• denotes the set of nonnegative integers.

• denotes the set of real numbers.

• denotes the space of complex numbers.

• denotes the N-dimensional space of complex numbered vectors.

COMPUTER SCIENCE

We make extensive use of asymptotic notation. For two functions f, g : → , we write:

• f(x) = O(g(x)) if and only if there exists a positive constant C and an integer x₀ such that |f(x)| ≤ C|g(x)| for all x ≥ x₀.

• f(x) = Ω(g(x)) if and only if g(x) = O(f(x)).

• f(x) = Θ(g(x)) if and only if f(x) = O(g(x)) and f(x) = Ω(g(x)).

• a ⊕ b refers to the XOR operation between two binary bits, a and b.

CALCULUS

• generally refers to the first derivative of the differentiable function f(x).

• generally refers to the second derivative of the differentiable function f(x).

LINEAR ALGEBRA

• I generally refers to the identity matrix of appropriate size (unless otherwise stated).

• det(A) refers to the determinant of the matrix A.

• A^† refers to the conjugate transpose of A.

• F⁺ denotes the Moore-Penrose pseudoinverse of F.

QUANTUM MECHANICS

• ℏ refers to Planck’s constant.

• |ψ refers to a ket, which is generally a state vector for a quantum state.

• ψ| refers to a bra, the conjugate transpose of the vector |ψ.

• ρ generally refers to the density matrix of a quantum system (unless otherwise stated).

• U generally refers to a unitary matrix where U^†U = UU^† = I (unless otherwise stated).

• ϕ|ψ refers to the inner product between the vectors |ϕ and |ψ).

• ϕ|A|ψ refers to the inner product between ϕ and Aψ.

• |ϕ ⊗ |ψ refers to a tensor product between |ϕ and |ψ.

• |ϕ |ψ also refers to the tensor product between |ϕ and |ψ.

• |ψ^⊗N refers to the quantum state (in superposition) of the composite system with N interacting quantum systems, each having quantum state |ψ.

• σ_x often refers to the Pauli-X matrix .

• σ_y often refers to the Pauli-Y matrix where i = .

• σ_z often refers to the Pauli-Z matrix .

• [A, B] often refers to a commutation operator [A, B] = AB – BA.

• A set of matrices {K_i} is a set of Kraus matrices if it satisfies