Test start at 11:00 AM EST on March 18th, 2021. Quantum Computing course with python.
1. a. What will be special about if it is Hermitian? (Hint: what is its relationship to b. What is special about the eigenvalues of a Hermitian operator? 2. a. What is the relationship between and ? b. What is special about the relationship between and ? c. Suppose acts on and . Specifically, let and . What is the relationship between and ? 3. We have learned that complex numbers may be written in the form where and a real numbers. We have also learned that a complex number may be written in polar form as . In the following, consider the two complex numbers and a. What are the magnitudes of and ? b. Please write in polar form c. Multiply using the "" form. Report the result in that form. d. Compute using the polar form. Report the result in polar form. 4. In a 3 dimensional vector space let an orthonormal set of basis vectors be . Consider an operator given in the Dirac notation as shown below. a. (3 points) Write the matrix, , that represents . 1/h = 1/x b. (2 points) Show that is Hermitian. M = 1/3 c. (3 points) Show that is unitary. M = 2/3 d. (1 point) Consider the vector Write down the column vector, , that represents CS 298 Section01 __________________ March 18, 2021 Name Please read the entire examination before starting to solve the problems. As you do so, please note the point values for each problem and plan your time accordingly. If there is a something that you feel is not clear enough, please ask for clarification. Good Luck! Problem 1: (15 points, 5 points each part) Consider the vector . Please answer the following questions. a. Please write a vector parallel to , but normalized to have a length of 1. That is, find the number such that and that the length of is 1. b. Find another vector that is orthogonal to . (Hint: write then solve for by using what you have learned about orthogonal vectors.) c. Write a vector that is parallel to but again of length 1. Problem 2: (15 points, 5 points each part) We have studied a number of operators, including and . When these are represented by matrices, they are given by the following in the computational basis a. Compute the matrix representing the operator b. Show that is unitary c. Compute the result of applying to the state . Please show the result in Dirac notation, but you may compute it however you wish. d. Compute the result of applying to the state . Please show the result in Dirac notation. Problem 3: (6 points, 3 points each part) Consider the following vectors i. ii. iii. iv. Please answer the following questions. a. Which of these are "legal" to describe a qubit's state? If one or more are not legal, please explain why. b. Of the four, is any equivalent to another? If so, please state which pairs of vectors are equivalent, and why they are equivalent. Problem 4: (30 points, 5 points each part) Consider the two vectors, and shown in the table below. The first row shows their expansions in the computational basis. The second row shows the vectors represented on the Bloch sphere. The third row shows a different representation of the Bloch sphere with much of the detail removed to make it perhaps easier to "read". It may be hard to see, but lies in the plane formed by and the axis. These two vectors represent the two possible outcomes of a quantum algorithm. Please answer the following questions. a. Please show that the visualizations are correct. (This would perhaps be done most easily by "plugging" the angles into the formula for a vector and showing that the result is correct.) b. Show that and are orthogonal. c. Suppose you measure the vectors with our " meter", that is, suppose you measure them as we learned in class. What would the outcomes be, with what probabilities? d. On the basis of one or more repetitions, could you distinguish whether the outcome of the algorithm was or ? (You might have to run the algorithm more than once, but that is fine and to be expected.) e. When we studied the Deutsch algorithm, we saw that employing an operator before we measured the state greatly improved our ability to distinguish the results. Does it help here, too? Please apply the operator to and , and then measure them. What are the outcomes now? f. Please compare the results of part c with those of part e. In particular, explain if adding the operator helped, or hindered your ability to discriminate the two results, or if it made no difference. For you convenience, the following is the operator in both Dirac and matrix form. You may use whichever you find easier. x|0ۧπ/3 π|1ۧy x |0⟩ π/3 π |1⟩ y x|0ۧ2π/3|1ۧy x |0⟩ 2π/3 |1⟩ y CS 298 Quantum ComputingUseful formulae Complex numbers Any complex number, may be represented in polar form as as shown below The magnitude or length, , is . The polar angle is measured in a counter-clockwise direction from the horizontal axis. Numerically, it has the value but care must be taken that the angle falls in the correct quadrant. Angles are usually reported in the range , but angles in quadrants III and IV may also be reported as negative numbers; the negative and positive values are related through the equation Some values of sine, cosine and tangent Some values of arctangent The values given here are in quadrant I for positive values and quadrant IV for negative values. If the desired angle is in quadrant II, add to the value in quadrant IV. If the desired angle is in quadrant III, add to the value in quadrant I. The Bloch Sphere Any vector representing the state of a qubit may be written in the form This form may be represented as a point on the surface of a sphere of radius 1. The parameters and may be chosen to be the familiar latitude and longitude. The polar angle, , is measured from the north pole, and is in the range . The azimuthal angle, , is measured from the positive axis towards the positive axis and is in the range . Useful formulae for vector spaces Let be an -dimensional vector space with basis vectors . In this basis, the matrix elements of an operator are The column vector representing a vector has components Measurement Suppose you wish to measure an operator for a state . Let the eigenvectors of M be with associated eigenvalues , , , , . The probability that a measurement of will yield eigenvalue is This is has an equivalent formulation in terms of the projection operator After the measurement, assuming the value obtained by the measurement was , the state will be simply . ab ϕ a b ϕ Quadrant IQuadrant IIQuadrant IIIQuadrant IV 0