About this course

Welcome to 8.371x Quantum Information Science II.

This is an advanced, graduate-level course on quantum computation and quantum information, based on the eponymous course at MIT, which has been offered since 2001. The course assumes you already have a firm grasp of quantum mechanics, including linear algebra and operator mechanics, e.g. via the full sequence of MIT 8.05x courses on edX. You should also have taken a first semester or introductory course on quantum computation, e.g. Berkeley's CS191 course. The Caltech-TU Delft course on quantum cryptography may also be helpful.

This course comprises seven units, each with video lectures and in-depth, automatically graded problems:

Unit 1: Quantum circuits and quantum operations – after a very brief review of quantum gates and circuits, we dive in depth into the operator sum representation, which formally describes the dynamics of open quantum systems, and provides an essential foundation for quantum error correction. We explicitly cover density matrices and measurement, including the preferred ensemble fallacy, and the infinity of von Neumann unravelings of both states and operations.

Unit 2: Stabilizer quantum codes – the largest and most well studied family of quantum error correction codes is known as stabilizer codes; these are the quantum analogues of classical additive codes, and we study them, beginning with the stabilizer formalism, and moving to a definition of the quantum error correction criteria in terms of stabilizer codes. Graph states, an important subgroup of stabilizers which have an intuitive graphical representation, are also introduced and used to define graph codes. We touch on decoherence free subspaces and topological quantum codes, in the problems.

Unit 3: Fault-Tolerant Quantum Computation – one of the most fascinating aspects of computation is that arbitrarily reliable, large-scale computations can be constructed from noisy components, as long as the gate failure probability is below a certain threshold. This remarkable threshold theorem holds for both classical and quantum computation, as we see, by investigating fault-tolerant NAND gate constructions, and by utilizing Calderbank-Shor-Steane (CSS) codes for quantum fault tolerant procedure constructions. This is made possible with the concept of normalizer operations, which transform encoded states to encoded states, typically with a nontrivial mapping of logical states. In the problems, we study such normalizer operations on stabilizer states, formalize the set of such possible gates as Clifford operations, and investigate the Gottesman-Knill theorem, showing that Clifford gate operations alone can be efficiently simulated, on stabilizer states and for stabilizer measurements.

Unit 4: Models of Computation – ordinarily, computation is thought of as a dynamical process, typically realized using a sequence of logic gates. Quantum computation, however, admits a richer variety of models. For example, as we study, gates can be teleported onto states. We also see how computation can also be performed solely by (single qubit!) measurements alone, acting on a generic initial (“cluster") state, with suitable changes of measurement bases. These alternative models of computation allow a concept of “quantum software" to be introduced - software which is encoded into a quantum state, and is consumed when executed.

Unit 5: QFT Algorithms – quantum algorithms are an appealing alternative to classical algorithms, because they may execute exponentially faster, but only for certain classes of problems. The quantum Fourier transform plays a distinct role in essentially every such exponentially fast quantum algorithm, as we see in this unit. We revisit Shor's quantum factoring algorithm from the more general perspective of phase estimation, and show how the central construct of Shor's algorithm, which employs the QFT, is also an essential ingredient in fast quantum simulation of Hamiltonians. As an excursion, the problem set explores quantum factoring as a feedback process, showing how the QFT emerges from an almost Kalman-filter like optimization. We also briefly touch on the quantum search algorithm (which has a polynomial speedup over its classical analogue), for comparison.

Unit 6: Quantum Communication – quantum information science also exhibits remarkable differences from classical scenarios in communication theory, as we see by reviewing Shannon's noisy and noiseless coding theorems, and considering their quantum analogues. We explore Schumacher compression and Shannon entropy (and quantum typical sequences), and show how classical information theory constructs extend to apply to manipulation of quantum entanglement; because of how it can be distilled and diluted (converted to and from a “gold standard", the “e-bit") entanglement is a physical resource.

Unit 7: Quantum Protocols – quantum information also provides new scenarios for classical cryptography and multi-party protocols; we touch only briefly on this, covering the proof of the impossibility of secure quantum bit commitment and quantum games with entanglement. Problems are also provided, which explore concepts in quantum communication complexity, and distributed quantum algorithms.

Assessments and Deadlines

There are seven problem sets - one for each unit. In total, there are about 120 problems (most of which are quite short).

All the course content, including problems, videos, and text, are available now. The problem sets are all due by the end of the course, Monday, November 28, at 21:30 UTC.

Grading and Certificates

The course grade is determined entirely by the problem set grades, with the following cutoffs:

90% : A

80% : B

70% : C

The minimum passing grade for a verified-ID certificate is 70%.

Honor Code

As described in the edX Honor code, you are expected to:

Complete all tests and assignments on my own, unless collaboration on an assignment is explicitly permitted.

Maintain only one user account and not let anyone else use my username and/or password.

Not engage in any activity that would dishonestly improve my results, or improve or hurt the results of others.

Not post answers to problems that are being used to assess student performance.

Textbook and References

No textbook is required for this course, but you may find it helpful to do optional readings from the textbook Quantum Computation and Quantum Information, by Nielsen and Chuang. There are also excellent, freely available lecture notes by John Preskill, and superb video lectures by Daniel Gottesman.

This is an advanced, graduate-level course, and you are expected to largely learn the material on your own. The discussion forums may be a good avenue for help from peers; there will be occasional (but not full-time) help from course staff, on the forums.

Entrance Survey

We would greatly appreciate if you could please fill in the entrance survey as you begin the course. This will help us improve understand who is using this material, and how it may be improved for future users like you.