Mathematic terminologies in quantum computing
Introduction to vectors:
A vector is a two-dimensional entity with magnitude (length or size) and direction. It can be represented as an arrow in geometry or as an ordered list of numbers in algebra.
Vector Components: A vector is commonly represented by its components along a coordinate system’s coordinate axes. A vector can be expressed in two dimensions as (x, y), where x is the component in the horizontal direction (x-axis) and y is the component in the vertical direction (y-axis). A vector in three dimensions is represented as (x, y, z), where x, y, and z are the components along the x, y, and z axes, respectively.
Vectors in quantum computing are used to represent the quantum states of qubits and quantum systems. They provide a mathematical framework for describing the quantum state, applying quantum gates, performing computations, and calculating measurement probabilities. By manipulating and transforming these vectors, quantum computers can perform quantum computations and leverage the unique properties of quantum mechanics for solving specific problems more efficiently than classical computers.
A linear combination refers to the sum of scalar multiples of vectors. It involves multiplying each vector in a set by a scalar coefficient and then summing the results. The scalars are typically real numbers, but they can also be complex numbers in certain contexts. Linear combinations are widely used in linear algebra, as they help describe vector spaces and solve systems of linear equations.
Formally, let {v₁, v₂, …, vₙ} be a set of vectors, and let {c₁, c₂, …, cₙ} be a set of scalars. The linear combination of these vectors is expressed as:
c₁v₁ + c₂v₂ + … + cₙvₙ
- Linear Dependence: A set of vectors is said to be linearly dependent if one or more of the vectors in the set can be expressed as a linear combination of the others. In other words, at least one vector in the set can be written as a linear combination of the remaining vectors. This implies that there is redundancy or duplication in the set.
Mathematically, a set of vectors {v₁, v₂, …, vₙ} is linearly dependent if there exist scalars {c₁, c₂, …, cₙ}, not all zero, such that the equation
c₁v₁ + c₂v₂ + … + cₙvₙ = 0
has a non-trivial solution, where the scalars c₁, c₂, …, cₙ are not all zero. The non-trivial solution means that at least one of the coefficients is nonzero.
2. Linear Independence: A set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the other vectors. In other words, the vectors are distinct and provide unique information.
Mathematically, a set of vectors {v₁, v₂, …, vₙ} is linearly independent if the equation
c₁v₁ + c₂v₂ + … + cₙvₙ = 0
has only the trivial solution, where the scalars c₁, c₂, …, cₙ are all zero. The trivial solution means that all coefficients are zero.
- Span: The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it represents the entire space that can be reached by linearly combining the given vectors.
Formally, let {v₁, v₂, …, vₙ} be a set of vectors in a vector space. The span of these vectors, denoted as span{v₁, v₂, …, vₙ}, is the set of all possible vectors that can be expressed as linear combinations of the given vectors:
span{v₁, v₂, …, vₙ} = {c₁v₁ + c₂v₂ + … + cₙvₙ | c₁, c₂, …, cₙ are scalars}
The span is always a subspace of the original vector space and contains all possible linear combinations of the given vectors.
2. Basis: A basis for a vector space is a set of linearly independent vectors that span the entire vector space. In other words, a basis is a set of vectors that can be used to uniquely represent any vector in the vector space.
Formally, let V be a vector space. A set of vectors {v₁, v₂, …, vₙ} is a basis for V if it satisfies the following conditions:
- The vectors are linearly independent, meaning no vector in the set can be expressed as a linear combination of the other vectors.
- The vectors span the entire vector space, meaning any vector in V can be expressed as a linear combination of the basis vectors.
A key property of a basis is that any vector in the vector space can be written uniquely as a linear combination of the basis vectors.
3. Dimension: The dimension of a vector space is the number of vectors in any basis for that vector space. It represents the minimum number of vectors required to span the entire vector space without redundancy.
Mathematically, if {v₁, v₂, …, vₙ} is a basis for a vector space V, then the dimension of V, denoted as dim(V), is equal to the number of vectors in the basis.
The dimension of a vector space provides important information about its size and complexity. It is a fundamental property used to classify and compare different vector spaces.
A matrix is composed of entries, which are the individual numbers or symbols within the array. Each entry is identified by its position in the matrix using row and column indices. For example, in a matrix A, the entry in the ith row and jth column is denoted as Aᵢⱼ.
In the context of vectors, projection refers to the process of finding the component of one vector onto another vector. The projection of a vector onto another vector represents the part of the first vector that lies in the direction of the second vector. It is often used to analyze vector relationships and decompose vectors into their components.
To understand vector projection, let’s consider two vectors: a target vector (v) and a reference vector (u). The projection of v onto u is denoted as projᵤ(v) and is calculated using the following formula:
projᵤ(v) = ((v · u) / ||u||²) * u
where:
v · u represents the dot product of vectors v and u.
||u||² represents the squared magnitude (length) of vector u.
((v · u) / ||u||²) is a scalar that scales the reference vector u.
The resulting vector ((v · u) / ||u||²) * u is the projection of v onto u.
Geometrically, the projection of v onto u is the vector that starts at the origin and extends in the same direction as u, reaching the point on the line spanned by u that is closest to v.
