Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64CSEP 590tv: Quantum ComputingDave BaconJune 29, 2005Today’s MenuAdministriviaComplex NumbersBra’s Ket’s and All ThatQuantum CircuitsAdministrivia Changes: slowing down.Mailing list: sign up on sheet being passed around.In class problems: hardness on the same order of magnitude as the homework problems.Problem Set 1: has been posted. Anyone who didn’t get myemail about the first homework being canceled, please let meknow and we will arrange accordingly.Think: Physics without CalculusQuantum theory with a minimal of linear algebraOffice Hours: Ioannis Giotis, 5:30-6:30 Wednesday in 430 CSELast WeekLast week we saw that there is a big motivation for understandingquantum computers. BIG PICTURE: understanding quantum information processing machines is the goal of this class!We also saw that there were there funny postulates describing quantum systems.This week we will be slowing down and understanding the basicworkings of quantum theory by understanding one qubit and twoqubit systems.Quantum Theory’s Language“Complex linear algebra” is the language of quantum theoryToday we will go through this slowly1. Complex numbers2. Complex vectors3. Bras, Kets, and all that(in class problem)4. Qubits5. Measuring Qubits6. Evolving Qubits(in class problem)7. Two qubits: the tensor product8. Quantum circuits(in class problem)MathMathematics as a series of discoveries of objects whoat first you don’t believe exist, and then after you findout they do exist, you discover that they are actually useful!irrationalnumbersComplex Numbers, DefinitionComplex numbers are numbers of the formrealreal“square root of minus one”Examples:“purely real”“purely imaginary”roots ofComplex Numbers, GeometryComplex numbers are numbers of the formrealreal“square root of minus one”Complex plane:real axisimaginary axisComplex Numbers, MathComplex numbers can be addedExample:and multipliedExample:Complex Numbers, That * ThingWe can take the complex conjugate of a complex numberExample:We can find its modulusExample:Complex Numbers, ModulusModulus is the length of the complex number in the complexplane:real axisimaginary axisModulusComplex Numbers, EulerEuler’s formulaExample:The modulus ofSome important cases:Complex Numbers, PhasesEuler’s formula geometricallyreal axisimaginary axisphase angleMultiplying phases is beautiful:Conjugating phases is also beautiful:Complex Numbers, GeometryAll complex numbers can be expressed as:real axisimaginary axisphase anglemodulus, magnitudeComplex Numbers, GeometryAll complex numbers can be expressed as:Example:real axisComplex Numbers, MultiplyingAll complex numbers can be expressed as:It is easy to multiply complex numbers when they are in this formExample:Complex VectorsN dimensional complex vector is a list of N complex numbers:Example:3 dimensional complex vectors(we start counting at 0 because eventually N will be a a power of 2)is the th component of the vector“column vector”“ket”Complex Vectors, Scalar TimesComplex numbers can be multiplied by a complex numberExample:3 dimensional complex vector multiplied by a complex numberis a complex numberComplex Vectors, AdditionComplex numbers can be addedAddition and multiplication by a scalar:Complex Vectors, AdditionExamples:Vectors, AdditionRemember adding real vectors looks geometrically like:We should have a similar picture in mind for complex vectorsBut the components of our vector are now complex numbersComputational BasisSome special vectors:Example: 2 dimensional complex vectors (also known as: a qubit!)Computational BasisVectors can be “expanded” in the computational basis:Example:Computational Basis MathExample:Computational Basis MathExample:Bras and KetsFor every “ket,” there is a corresponding “bra” & vice versaExamples:Bras, MathMultiplied by complex numberExample:AddedExample:Computational BrasComputational Basis, but now for bras:Example:The Inner ProductGiven a “bra” and a “ket” we can calculate an “inner product”This is a generalization of the dot product for real vectorsThe result of taking an inner product is a complex numberThe Inner ProductExample:Complex conjugate of inner product:The Inner Product in Comp. BasisKronecker deltaInner product of computational basis elements:The Inner Product in Comp. BasisExample:In Class Problem # 1Norm of a VectorNorm of a vector: which is always a positive real numberExample:it is (roughly) the length of the complex vectorQuantum Rule 1Rule 1: The wave function of a N dimensional quantum system is given by an N dimensional complex vector with norm equal to one.Example:a valid wave function for a 3 dimensional quantum systemQubitsTwo dimensional quantum systems are called qubitsA qubit has a wave function which we write asExamples:Valid qubit wave functions:Invalid qubit wave function:Measuring QubitsA bit is a classical system with two possible states, 0 and 1A qubit is a quantum system with two possible states, 0 and 1When we observe a qubit, we get the result 0 or the result 10 1orIf before we observe the qubit the wave function of the qubit isthen the probability that we observe 0 is and the probability that we observe 1 is“measuring in the computational basis”Measuring QubitsWe are given a qubit with wave functionIf we observe the system in the computational basis, then weget outcome 0 with probabilityand we get outcome 1 with probability:Example:Measuring Qubits ContinuedWhen we observe a qubit, we get the result 0 or the result 10 1orIf before we observe the qubit the wave function of the qubit isthen the probability that we observe 0 is and the probability that we observe 1 is“measuring in the computational basis” and the new wave function for the qubit is and the new wave function for the qubit isMeasuring Qubits Continued01probabilityprobabilitynew wave functionnew wave functionThe wave function is a description of our system.When we measure the system we find the system in one stateThis happens
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