### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Course Overview

This course is an introduction to quantum computational complexity theory, the study of the fundamental capabilities and limitations of quantum computers. Topics include complexity classes, lower bounds, communication complexity, proofs, advice, and interactive proof systems in the quantum world. The objective is to bring students to the research frontier.

### Requirements

Grades will be determined roughly as follows:

ACTIVITIES | PERCENTAGES |
---|---|

Project | 45% |

Problem sets | 45% |

Class participation | 10% |

### Projects

All students are expected to complete a course project. This will involve submitting a written report, as well as giving a 10-minute presentation toward the end of class. The projects can be either original research or literature surveys on some topic in quantum complexity theory (or related areas of quantum information science). Survey projects should be individual, while research projects can be done either individually or in teams of two. Many possibilities for projects will be discussed as the course progresses. Solving a significant open problem pretty much guarantees an A in the course.

### Problem Sets

There will be 4-5 problem sets. Problem sets will be due about two weeks after being assigned. Students are welcome to collaborate on problem sets; however, if they do so, they must list the names of collaborators.

### Scribe Notes

Notes (taken by the students) from a previous offering of the course are available in the Lecture Notes section. We will not follow the 2008 notes exactly this semester, but will follow them for perhaps 70-80% of the course. Since no book exists that covers much of the material, the notes should be an extremely useful resource. However, please be warned that the notes haven’t been carefully proofread and contain some errors and omissions.

### Textbooks

The course has no official textbook. However, students may find the following books helpful:

Nielsen, Michael, and Issac Chuang. *Quantum Computation and Quantum Information*. Cambridge University Press, 2011. ISBN: 9781107002173.

Mermin, David. *Quantum Computer Science: An Introduction*. Cambridge University Press, 2007. ISBN: 9780521876582.

Arora, Sanjeev, and Boaz Barak. *Computational Complexity: A Modern Approach*. Cambridge University Press, 2009. ISBN: 9780521424264.

Sipser, Michael. *Introduction to the Theory of Computation*. 2nd ed. Course Technology, 2005. ISBN: 9780534950972.

In addition, students might benefit from the following online resources:

Watrous, John. “Quantum Computational Complexity.”

Quantum Computing Since Democritus: Lecture Notes

### Prerequisites

No prior knowledge of quantum mechanics is assumed. Open to students who have taken a previous course on computational complexity theory (such as 6.045 or 6.840), or a previous course on quantum computing and information (such as 18.435).

### Schedule of Topics (extremely approximate)

“Who Ordered Quantum Mechanics?”

Classical Complexity Theory Crash Course

Defining BQP: Bounded-Error Quantum Polynomial-Time

Universal Sets of Quantum Gates

Basic Closure Properties of BQP

How BQP Relates to Classical Complexity Classes: P, BPP, PP, P#P, PSPACE

Coping with Imprecision

Basic Quantum Algorithms: Deutsch-Jozsa, Bernstein-Vazirani, Shor, Grover

The Hidden Subgroup Framework

**II. The Quantum Black-Box Model**

Defining and Motivating the Quantum Black-Box Model

Oracle Separations: The Baker-Gill-Solovay Paradigm

Oracle Separation of BQP from BPP

Oracle Separation of NP from BQP: The BBBV Hybrid Argument

The Polynomial Method for Quantum Lower Bounds

Quantum/Classical Relation for Total Boolean Functions

Ambainis’s Adversary Method, with Application to Game-Tree Search

Quantum Lower Bound for the Collision Problem

**III. The Zoo of Quantum Complexity Classes**

BQPSPACE: Quantum Polynomial Space

QMA: Quantum Merlin-Arthur (and QMA-completeness)

QCMA: Quantum Classical Merlin-Arthur

QIP: Quantum Interactive Proofs

BQP/qpoly: Quantum Computing with Quantum Advice

PostBQP: Quantum Computing with Postselection

**IV. Other Topics**

Quantum Communication Complexity: Separations and Lower Bounds

Dense Quantum Coding and Learnability of Quantum States

The Stabilizer Formalism

Alternative Quantum Computing Paradigms: Adiabatic, Cluster States, …

Quantum Computing with Noninteracting Bosons and Fermions

Hypothetical Models Beyond Quantum Computing: Nonlinear QM, Closed Timelike Curves, …

More depending on student interest

**V. Student Project Presentations**