Quantum Computing

1. Quantum Computing Lecture on Linear Algebra Sources:Angela Antoniu, Bulitko, Rezania, Chuang, Nielsen

2. Introduction to Quantum Mechanics • This can be found in Marinescu and in Chuang and Nielsen • Objective • To introduce all of the fundamental principles of Quantum mechanics • Quantum mechanics • The most realistic known description of the world • The basis for quantum computing and quantum information • Why Linear Algebra? • LA is the prerequisite for understanding Quantum Mechanics • What is Linear Algebra? • … is the study of vector spaces… and of • linear operations on those vector spaces

3. Linear algebra -Lecture objectives • Review basic concepts from Linear Algebra: • Complex numbers • Vector Spaces and Vector Subspaces • Linear Independence and Bases Vectors • Linear Operators • Pauli matrices • Inner (dot) product, outer product, tensor product • Eigenvalues, eigenvectors, Singular Value Decomposition (SVD) • Describe the standard notations (the Dirac notations) adopted for these concepts in the study of Quantum mechanics • … which, in the next lecture, will allow us to study the main topic of the Chapter: the postulates of quantum mechanics

4. Review: Complex numbers • A complex numberisof the form where and i2=-1 • Polar representation • With the modulus or magnitude • And the phase • Complex conjugate

5. Review: The Complex Number System • Another definitions: It is the extension of the real number system via closure under exponentiation. • (Complex) conjugate: c* = (a + bi)*  (a bi) • Magnitude or absolute value: |c|2= c*c =a2+b2 The “imaginary”unit +i c b  + a “Real” axis “Imaginary”axis i

6. Review: Complex Exponentiation • Powers of i are complex units: • Note: ei/2 = i ei= 1 e3 i/2 =  i e2 i= e0 = 1 ei +i  1 +1 i Z1=2 e i Z12 = (2 e i)2 = 2 2 (ei)2= 4 (e i )2 = 4 e 2i 2 4

7. Recall: What is a qubit? • A qubit has two possible states • Unlike bits, a qubit can be in a state other than • We can form linear combinations of states • A qubit state is a unit vector in a two-dimensionalcomplex vector space

8. Properties of Qubits • Qubits are computational basis states - orthonormal basis - we cannot examine a qubit to determine its quantum state - A measurement yields

9. (Abstract) Vector Spaces • A concept from linear algebra. • A vector space, in the abstract, is any set of objects that can be combined like vectors, i.e.: • you can add them • addition is associative & commutative • identity law holds for addition to zero vector 0 • you can multiply them by scalars (incl. 1) • associative, commutative, and distributive laws hold • Note: There is no inherent basis (set of axes) • the vectors themselves are the fundamental objects • rather than being just lists of coordinates

10. Vectors • Characteristics: • Modulus (or magnitude) • Orientation • Matrix representation of a vector Operations on vectors This is adjoint, transpose and next conjugate

11. Vector Space, definition: • A vector space (of dimension n) is a set of n vectors satisfying the following axioms (rules): • Addition: add any two vectors and pertaining to a vector space, say Cn,obtain a vector, the sum, with the properties : • Commutative: • Associative: • Any has a zero vector (called the origin): • To every in Cncorresponds a unique vector - v such as • Scalar multiplication:  next slide Operations on vectors

12. Vector Space (cont) • Scalar multiplication: for any scalar • Multiplication by scalars is Associative: distributive with respect to vector addition: • Multiplication by vectors is distributive with respect to scalar addition: • A Vector subspace in an n-dimensional vector space is a non-empty subset of vectors satisfying the same axioms in such way that Operations on vectors

13. Hilbert spaces • A Hilbert space is a vector space in which the scalars are complex numbers, with an inner product (dot product) operation  : H×H  C • Definition of inner product: xy = (yx)* (* = complex conjugate) xx  0 xx= 0 if and only if x = 0 xy is linear, under scalar multiplication and vector addition within both x and y Black dot is an inner product “Component”picture: y Another notation often used: x xy/|x| “bracket”

14. Vector Representation of States • Let S={s0, s1, …} be a maximal set of distinguishable states, indexed by i. • The basis vector vi identified with the ith such state can be represented as a list of numbers: s0s1s2si-1si si+1 vi = (0, 0, 0, …, 0, 1, 0, … ) • Arbitrary vectors v in the Hilbert space can then be defined by linear combinations of the vi: • And the inner product is given by:

15. Dirac’s Ket Notation You havr to be familiar with these three notations • Note: The inner productdefinition is the same as thematrix product of x, as aconjugated row vector, timesy, as a normal column vector. • This leads to the definition, for state s, of: • The “bra” s| means the row matrix [c0* c1* …] • The “ket” |s means the column matrix  • The adjoint operator† takes any matrix Mto its conjugate transpose M†MT*, sos| can be defined as |s†, and xy = x†y. “Bracket”

16. Linear Algebra

17. Vector Spaces Complex number field

18. Cn

19. Spanning Set and Basis vectors • Or SPANNING SET for Cn: any set of n vectors such that any vector in the vector space Cn can be written using the n base vectors • Example for C2 (n=2): Spanning set is a set of all such vectors for any alpha and beta which is a linear combination of the 2-dimensionalbasis vectors and

20. Bases and Linear Independence Linearly independent vectors in the space Red and blue vectors add to 0, are not linearly independent Always exists!

21. Basis

22. Bases for Cn

23. Linear Operators So far we talked only about vectors and operations on them. Now we introduce matrices

24. Linear Operators New space

25. Pauli Matrices X is like inverter • Properties: Unitary and Hermitian This is adjoint

26. Matrices Pay attention to this notation

27. Examples of operators

28. This is new, we did not use inner products yet Inner Products of vectors We already talked about this when we defined Hilbert space Complex numbers Be able to prove these properties from definitions

29. Slightly other formalism for Inner Products Be familiar with various formalisms

30. Example: Inner Product on Cn

31. Norms

32. Outer Products of vectors This is Kronecker operation

33. Outer Products of vectors |u> <v| is an outer product of |u> and |v> |u> is from U, |v> is from V. |u><v| is a map V U We will illustrate how this can be used formally to create unitary and other matrices

34. Eigenvectors of linear operators and their Eigenvalues Eigenvalues of matrices are used in analysis and synthesis

35. Eigenvalues and Eigenvectors versus diagonalizable matrices Eigenvector of Operator A

36. Diagonal Representations of matrices Diagonal matrix

37. Adjoint Operators This is very important, we have used it many times already

38. Normal and Hermitian Operators But not necessarily equal identity

39. Unitary Operators

40. Exam Problems • Diagonalization of unitary matrices

41. Unitary and Positive Operators: some properties Other notation for adjoint (Dagger is also used Positive operator Positive definite operator

42. Hermitian Operators: some properties in different notation These are important and useful properties of our matrices of circuits

43. Tensor Products of VectorSpaces Notation for vectors in space V Note various notations

44. Tensor Products of vectors and Tensor Products of Operators Properties of tensor products for vectors Tensor product for operators

45. Tensor Product of two Matrices

46. Properties of Tensor Products of vectors and operators These can be vectors of any size We repeat them in different notation here

47. Functions of Operators I is the identity matrix Matrix of Pauli rotation X Spectral decomposition eigenvalues

48. Trace and Commutator

49. Polar Decomposition Left polar decomposition Right polar decomposition

50. Eigenvalues and Eigenvectors More on Inner Products Hilbert Space: Orthogonality: Norm: Orthonormal basis: