Nielsen and Chuang Chap 2 1 Linear Algebra

Nielsen and Chuang Chap 2. 1 Linear Algebra: p 60 -75 Dr. Charles Tappert The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang

2. 1 Linear Algebra n Linear algebra is the study of vector spaces and linear operations on those vector spaces n n n Understanding of quantum mechanics is based on linear algebra Finite vector spaces are closed under addition and scalar multiplication We will use the standard notation of quantum mechanics for linear algebra concepts

2. 1 Linear Algebra Standard quantum mechanical notation Dirac notation A = operator

2. 1 Linear Algebra 2. 1. 1 Bases and linear independence n n n A spanning set of a vector space is a set of vectors such that any vector in the space can be written as a linear combination of the vectors in the set A set of vectors is linearly dependent if there exists coefficients such that Otherwise the set is linearly independent and if it spans the vector space, it is a basis set

2. 1 Linear Algebra 2. 1. 2 Linear operators and matrices n n n A vector space linear operator maps a vector in the space to another vector in the space The most convenient way to understand linear operators is in terms of operator matrices Ex 2. 2: suppose V is a vector space with basis vectors , and A is a linear operator on the space such that then the matrix for A is

2. 1 Linear Algebra 2. 1. 3 The Pauli matrices n Useful for quantum computation

2. 1 Linear Algebra 2. 1. 4 Inner products n The inner product of two vectors is a scalar We write the inner product of n An inner product space has inner products s. t. n

2. 1 Linear Algebra 2. 1. 4 Inner products n n n In finite dimensions, an inner product space is a Hilbert space, the preferred quantum term Two vectors are orthogonal if their inner product is zero Unit vector where Orthogonal unit vectors are orthonormal In quantum computing we deal with an orthonormal basis set of vectors that spans the state space

2. 1 Linear Algebra 2. 1. 4 Inner products n Linear operator outer product representation

2. 1 Linear Algebra 2. 1. 5 Eigenvectors and eigenvalues n An eigenvector of a linear operator A on a vector space is a non-zero vector such that where is the eigenvalue of A corresponding to n n Here we use for both eigenvalue and eigenvector Find eigenvalues by solving the characteristic equation

2. 1 Linear Algebra 2. 1. 5 Eigenvectors and eigenvalues n n n = diagonal representation of A where vectors form orthonormal set of eigenvectors with corresponding eigenvalues A is diagonalizable if it has a diagonal form For example, Pauli Z matrix

2. 1 Linear Algebra 2. 1. 6 Adjoints and Hermitian operators n n n Operator = adjoint conjugate If is know as Hermitian or self-adjoint Projectors are an important class of Hermitian operators n Hermitian matrices have real eigenvalues

2. 1 Linear Algebra 2. 1. 6 Adjoints and Hermitian operators n A operator or matrix U is unitary if The outer product representation of unitary U n Then For example, Pauli Z matrix n

2. 1 Linear Algebra 2. 1. 7 Tensor products n n The tensor product allows putting vector spaces together to form larger vector spaces For example, if V is a 2 D vector space with basis vectors