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## Quantum Computing

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**Quantum Computing**Lecture on Linear Algebra Sources:Angela Antoniu, Bulitko, Rezania, Chuang, Nielsen**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**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**Review: Complex numbers**• A complex numberisof the form where and i2=-1 • Polar representation • With the modulus or magnitude • And the phase • Complex conjugate**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**Review: Complex Exponentiation**• Powers of i are complex units: • Note: ei/2 = i ei= 1 e3 i/2 = i e2 i= e0 = 1 ei +i 1 +1 i Z1=2 e i Z12 = (2 e i)2 = 2 2 (ei)2= 4 (e i )2 = 4 e 2i 2 4**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**Properties of Qubits**• Qubits are computational basis states - orthonormal basis - we cannot examine a qubit to determine its quantum state - A measurement yields**(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**Vectors**• Characteristics: • Modulus (or magnitude) • Orientation • Matrix representation of a vector Operations on vectors This is adjoint, transpose and next conjugate**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**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**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: xy = (yx)* (* = complex conjugate) xx 0 xx= 0 if and only if x = 0 xy 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 xy/|x| “bracket”**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:**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*, sos| can be defined as |s†, and xy = x†y. “Bracket”**Vector Spaces**Complex number field**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**Bases and Linear Independence**Linearly independent vectors in the space Red and blue vectors add to 0, are not linearly independent Always exists!**Linear Operators**So far we talked only about vectors and operations on them. Now we introduce matrices**Linear Operators**New space**Pauli Matrices**X is like inverter • Properties: Unitary and Hermitian This is adjoint**Matrices**Pay attention to this notation**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**Slightly other formalism for Inner Products**Be familiar with various formalisms**Outer Products of vectors**This is Kronecker operation**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**Eigenvectors of linear operators and their Eigenvalues**Eigenvalues of matrices are used in analysis and synthesis**Eigenvalues and Eigenvectors versus diagonalizable matrices**Eigenvector of Operator A**Diagonal Representations of matrices**Diagonal matrix**Adjoint Operators**This is very important, we have used it many times already**Normal and Hermitian Operators**But not necessarily equal identity**Exam Problems**• Diagonalization of unitary matrices**Unitary and Positive Operators: some properties**Other notation for adjoint (Dagger is also used Positive operator Positive definite operator**Hermitian Operators: some properties in different notation**These are important and useful properties of our matrices of circuits**Tensor Products of VectorSpaces**Notation for vectors in space V Note various notations**Tensor Products of vectors and Tensor Products of Operators**Properties of tensor products for vectors Tensor product for operators**Properties of Tensor Products of vectors and operators**These can be vectors of any size We repeat them in different notation here**Functions of Operators**I is the identity matrix Matrix of Pauli rotation X Spectral decomposition eigenvalues**Polar Decomposition**Left polar decomposition Right polar decomposition**Eigenvalues and Eigenvectors**More on Inner Products Hilbert Space: Orthogonality: Norm: Orthonormal basis: