# Quantum Computing Lecture 2. Review of Linear Algebra

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1 Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces and linear operators are studied in linear algebra This lecture reviews definitions from linear algebra that we will need in the rest of the course! Unless stated otherwise, all vector spaces are over C Recall: Any z C is of the form z = a + ib for some a, b R

2 Matrices An n m matrix is an array of numbers a ij arranged as follows: a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm For any A, we write A ij to denote the entry in the i-th row and the j-th column of A In the matrix above we have A ij = a ij We denote the set of all n m complex matrices by M n,m (C) Basic matrix operations Matrices can be added (if they are of the same shape) and multiplied by a constant (scalar) Both operations are performed entry-wise If A, B M n,m (C) and c C then (A + B) ij = A ij + B ij (ca) ij = ca ij Example: ( ) ( ) ( ) a11 a 12 a 13 b11 b + 12 b 13 a11 + b = 11 a 12 + b 12 a 13 + b 13 a 21 a 22 a 23 b 21 b 22 b 23 a 21 + b 21 a 22 + b 22 a 23 + b 23 ( ) ( ) a11 a c 12 a 13 ca11 ca = 12 ca 13 a 21 a 22 a 23 ca 21 ca 22 ca 23

3 Conjugate transpose For A M n,m (C) we define complex conjugate: A and ( Ā ) (A ) ij = (A ij ) a11 a 12 a 13 = a 21 a 22 a 23 ( ) a 11 a 12 a 13 a 21 a 22 a 23 transpose: A T ( ) T a (A T a11 a ) ij = A 12 a 11 a ji = a a 21 a 22 a 12 a a 13 a 23 conjugate transpose (adjoint): A = (A ) T ( ) a (A ) ij = A a11 a 12 a 11 a ji = a a 21 a 22 a 12 a a 13 a 23 Warning: mathematicians often use A to denote A Vectors and inner product Vector is a matrix that has only one row or one column: row vector ( a1 a 2 a n ) column vector b 1 b 2 Just like matrices, vectors of the same shape can be added and multiplied by scalars They form a vector space denoted C n A row vector and a column vector can be multiplied together if they have the same number of entries The result is a number: b 1 ( ) b 2 a1 a 2 a n = a 1b 1 + a 2 b a n b n = b n This is a special case of matrix product b n n a i b i i=1

4 Matrix multiplication If A is n m and B is m l matrix then C = A B is the n l matrix with entries given by m C ik = A ij B jk j=1 for all i = 1,, n and k = 1,, l Note that C ik is just the product of the i-th row of A and the k-th column of B: l Note: It is possible to multiply A and B only if the number of columns of A agrees with the number of rows of B: [n m] [m l] = [n l] n m A m i B C k Properties of matrix multiplication Associativity: (A B) C = A (B C) = ABC Distributivity: A(B + C) = AB + AC and (A + B)C = AB + BC Scalar multiplication: λ(ab) = (λa)b = A(λB) = (AB)λ Complex conjugate: (AB) = A B Transpose: (AB) T = B T A T (reversed order!) Conjugate transpose: (AB) = B A (reversed order!) Warning: Matrix multiplication is not commutative: AB BA In fact, BA might not even make sense even if AB is well-defined (eg, when A is 3 2 and B is 2 1)

5 Dirac bra-ket notation Column vector (ket) a 1 a 2 ψ = a n Row vector (bra) ψ = ( ) a 1 a 2 a n Note: Ket and bra are conjugate transposes of each other: ψ ψ ψ ψ For ψ, ϕ C n and A M n,n (C) the following are valid products: inner product: ψ ϕ ψ ϕ C outer product: ψ ϕ ψ ϕ M n n (C) matrix-vector product: A ψ A ψ and ψ A ψ A Quiz Let ψ, ϕ C n and A M n,n (C) Which of these make sense? (a) ψ + ϕ (h) ψ ϕ A (b) ψ ϕ (i) ψ A ϕ (c) A ψ (j) ψ A ϕ (d) ψ A (k) ψ A ϕ + ψ ϕ (e) ψ A (l) ψ ϕ ψ (f) ψ A + ϕ (m) ψ ϕ A (g) ψ ϕ (n) ψ ψ ϕ = ψ ϕ ψ

6 Inner product The complex number u v is called the inner product of u, v C n If u = a 1 a n and v = Simple observations: b 1 b n u v = ( ) a 1 a n u v = ( v u ) = v u then their inner product is b 1 b n = n a i b i i=1 u u = n i=1 a i 2 is a real number u u and u u = iff u = (the all-zeroes vector) Norm and orthogonality The norm of a vector u is the non-negative real number u = n u u = i=1 a i 2 A unit vector is a vector with norm 1, ie, n i=1 a i 2 = 1 Recall: qubit states α + β 1 are unit vectors since α 2 + β 2 = 1 Cauchy Schwarz inequality: u v u v for any u, v C n If u and v are unit vectors then u v [, 1], so u v = cos θ for some angle θ [, π/2] Vectors u and v are orthogonal if u v = (ie, θ = π/2)

7 Basis A basis of C n is a minimal collection of vectors v 1,, v n such that every vector v C n can be expressed as a linear combination of these: v = α 1 v α n v n for some coefficients α i C In particular, v 1,, v n are linearly independent, meaning that no v i can be expressed as a linear combination of the rest The number n (the size of the basis) is uniquely determined by the vector space and is called its dimension The dimension of C n is n Given a basis, every vector v can be uniquely represented as an n-tuple of scalars α 1,, α n, called the coordinates of v in this basis An orthonormal basis is a basis made up of unit vectors that are pairwise orthogonal: { 1 if i = j v i v j = δ ij if i j Standard basis for C n The following are all bases for C 2 (the last two are orthonormal): ( ( ) ( ( ( ) ( ) ,,, 2) i ) 1) The standard or computational basis for C n is n 1 (Sometimes we will use, 1,, n 1 instead, eg, when n = 2) Any vector can be expanded in the standard basis: a 1 n = a i i a i=1 n

8 Outer product and projectors With every pair of vectors u C n, v C m we can associate a matrix u v M n,m (C) known as the outer product of u and v For any w C m, it satisfies the property ( u v ) w = v w u If u is a unit vector then u u is the projector on the one-dimensional subspace generated by u This can be easily seen if u and v are real and we set cos θ = u v Then u u v = u v u v θ u v u u Expanding a matrix in the standard basis 1 1 ( ) 1 2 = 1 = Similarly, i j has 1 at coordinates (i, j) and zeroes everywhere else Any n m matrix can be expressed as a linear combination of outer products: a 11 a 12 a 1m a 21 a 22 a 2m = n m a ij i j i=1 j=1 a n1 a n2 a nm We can recover the (i, j) entry of A as follows: i A j = a ij

9 Eigenvalues and eigenvectors An eigenvector of A M n,n (C) is a non-zero vector v C n such that A v = λ v for some complex number λ (the eigenvalue corresponding to v ) The eigenvalues of A are the roots of the characteristic polynomial: det(a λi) = where det denotes the determinant and I is the n n identity matrix: I = n i i i=1 Each n n matrix has at least one eigenvalue Diagonal representation Matrix A M n,n (C) is diagonalisable if A = n λ i v i v i i=1 where v 1,, v n is an orthonormal set of eigenvectors of A with corresponding eigenvalues λ 1,, λ n This is called spectral decomposition of A Equivalently, A can be written as a diagonal matrix λ 1 λ n in the basis v 1,, v n of its eigenvectors

10 Normal and Hermitian matrices Matrix A M n,n (C) is normal if AA = A A Theorem: A matrix is diagonalisable if and only if it is normal A is Hermitian if A = A A normal matrix is Hermitian if and only if it has real eigenvalues Unitary matrices Matrix A M n,n (C) is unitary if AA = A A = I where I is the n n identity matrix Unitary operators are normal and therefore diagonalisable Unitary operators preserve inner products: if U is unitary and u = U u and v = U v then u v = (U u ) (U v ) = u U U v = u v All eigenvalues of a unitary operator have absolute value 1

11 Tensor product Let A and B be matrices of arbitrary shape Their tensor product is the following block matrix: A 11 B A 12 B A 1m B A 21 B A 22 B A 2m B A B = A n1 B A n2 B A nm B where the block at coordinates (i, j) is equal to A ij B If A is n m and B is n m then A B is nn mm Tensor product applies to vectors too, eg, ψ ϕ and ψ ϕ Note: ψ ϕ is a shorthand for ψ ϕ Summary Matrix multiplication: AB = C where j A ijb jk = C ik, A, B, C have dimensions [n m] [m l] = [n l] Conjugate transpose: A = ĀT, (AB) = B A Dirac notation: ψ ψ and ψ ψ Inner product: ψ ϕ, it is for orthogonal vectors Norm: ψ = ψ ψ, it is 1 for unit vectors Orthonormal basis: v i v j = δ ij Outer product: ψ ϕ, projector: ψ ψ where ψ = 1 Eigenvalues and eingevectors: A v = λ v Spectral decomposition: A = i λ i v i v i iff A is normal Normal: AA = A A Hermitian: A = A Unitary: AA = A A = I Tensor product: A B is a block matrix with (i, j)-th block A ij B

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