  
Student Learning Outcomes 
 A successful student will be able define the problem classes P and NP, and compare classical vs. quantum mechanical algorithms in this context.
 A successful student will be able define the stabilizer code, and test whether given error correction codes satisfy certain analyticallydefined bounds.

Description  
 This course presents classical and quantum information theory as applied to the encoding and error correction of quantum data. The third in a sequence, it begins by presenting quantum entanglement and nonorthogonal measurement. Key results of classical information theory are stated in terms of the complexity classes P and NP, and Shannon entropy is defined and extended to include quantum information. Students receive instruction in different distance measures and bounds for comparing fidelity. Several error correction and stabilizer codes are presented and analyzed.


Course Objectives  
 The student will be able to:
 Use the mathematical tools of linear algebra, tensor products and probability to calculate and analyze a number of systems that model quantum behavior.
 Describe the difference between the density matrix and statevector formulation of quantum mechanics and demonstrate the strengths of each.
 State what it means for a problem to be NPcomplete, and explain the primary conjecture of information theory regarding the complexity classes P and NP.
 Define information entropy and compute the minimum entropy gained by the environment for each bit of information lost (erased) by a computer.
 Define the distance measures “fidelity”, “entanglement fidelity” and “trace” and, given a description of an encoding algorithm, compute the value of each.
 Write an algorithm for some basic error correction schemes utilizing “flip” and Shore codes.
 Explain the difference between classical linear errorcorrecting codes and quantum errorcorrecting codes.
 Test whether a given errorcorrection code satisfies certain bounds defined by an associated inequality.
 Write the formal description of a stabilizer code and give three examples of a stabilizer code, stating the strengths and limitations of each.

Special Facilities and/or Equipment  
  access to a networked computer laboratory which can access quantum computer simulators and tools either via the Web or loaded onto lab computers.
 a website or course management system with an assignment posting component, a forum component. This applies to all sections, including oncampus (i.e., facetoface) offerings.
 When taught via Foothill Global Access on the Internet, course management system through which the instructor and students can interact.
 When taught via Foothill Global Access on the Internet, students must have currently existing email accounts and ongoing access to computers with internet capabilities.

Course Content (Body of knowledge)  
  Mathematical Tools
 2D Hilbert space
 Hermitian operators
 Eigenvectors and eigenvalues
 Tensor product spaces
 Fundamental results from probability and statistics
 Quantum Mechanical Tools
 State vector and density matrix formulations of quantum axioms
 Orthogonal and nonorthogonal measurements
 Positive operator valued measures (POVMs)
 Spin1/2 Hilbert Space and Bloch sphere representation
 Purifications survey and Schmidt number
 GHJW theorem
 Superoperators and their operator sum representations
 Complexity Classes P and NP
 Decision Problems
 Classes P and NP
 NPcomplete problems
 The “P notequal NP” problem
 Examples in number factoring and graph theory
 Classical Information Theory
 Random variables and probabilities
 Shannon entropy
 Mutual information
 Data compression and code words
 Data transmission over noiseless channels
 Data transmission over noisy channels
 Distance
 Trace
 Fidelity
 Entanglement fidelity
 Basic Error Correction Examples
 3qubit bit flip code
 3 qubit phase flip code
 The Shor code
 Fundamental Quantum Error Codes
 Classical linear codes
 CSS Codes
 7Qubit code
 Constraints on Code Parameters
 Quantum Hamming bound
 Nocloning bound
 Quantum Singleton bound
 Stabilizer Codes
 Formalism
 Unitary gates in the stabilizer formalism
 Measurement in the stabilizer formalism
 5qubit code
 3 qubit bitflip code revisited
 9qubit Shor code

Methods of Evaluation  
  Tests and quizzes
 Written assignments which include algorithms, mathematical derivations, logical circuits, and essay questions.
 Final examination

Representative Text(s)  
 Nielsen, M. and Chuang, I.: Quantum Computation and Quantum Information, Cambridge University Press, 2010. Kaye, P., Laflamme, R. and Mosca, M.: An Introduction to Quantum Computing, Oxford University Press, 2007.

Disciplines  
 Computer Science


Method of Instruction  
  Lectures which include mathematical foundations, theoretical motivation and coding implementation of quantum mechanical algorithms.
 Detailed review of assignments which includes model solutions and specific comments on the student submissions.
 In person or online discussion which engages students and instructor in an ongoing dialog pertaining to all aspects of designing, implementing and analyzing programs.
 When course is taught fully online:
 Instructorauthored lecture materials, handouts, syllabus, assignments, tests, and other relevant course material will be delivered through a college hosted course management system or other departmentapproved Internet environment.
 Additional instructional guidelines for this course are listed in the attached addendum of CS department online practices.


Lab Content  
 Not applicable.


Types and/or Examples of Required Reading, Writing and Outside of Class Assignments  
  Reading
 Textbook assigned reading averaging 30 pages per week.
 Reading the supplied handouts and modules averaging 10 pages per week.
 Reading online resources as directed by instructor though links pertinent to programming.
 Reading library and reference material directed by instructor through course handouts.
 Writing
 Writing technical prose documentation that supports and describes the programs that are submitted for grades.
