# PHYC 571   Quantum Computation   Spring 2018

## General information

Monday and Wednesday, 17:30-18:45, P&A 184

The course is cross-listed with CS 591 and NSMS 595.

Lectures:

Professor Akimasa Miyake

office: P&A 25,   email: amiyake_at_unm.edu,   office hours: Monday 14:00-16:00, otherwise you may arrange a meeting by appointment.

Instructor:

Students educate and peer-review each other under supervision of the instructor.

Teaching Assistant:

M. A. Nielsen and I. L. Chuang, "Quantum Computation and Quantum Information" (optional)

J. Preskill, "Quantum Information and Computation," available free online (optional)

A.Y. Kitaev, A.H. Shen, and M.N. Vyalyi,"Classical and Quantum Computation" (optional)

Textbooks:

## Course overview

The main subjects of this advanced graduate course are quantum information, quantum computation, and quantum advantage over classical information processing. While basic knowledge about quantum mechanics and related mathematics like linear algebra is assumed, the course will be taught in a self-contained manner. We adopt modern pedagogy of active learning and engagement in the class. In particular, students are encouraged to educate and peer-review each other, so that they can also learn valuable skills as researchers to explain and communicate well.

The course has two objectives. One is to enlighten the basic concepts of quantum information, which are expected to be useful regardless of research fields everyone chooses. The other is to help preparing the ground to work in quantum information if one is interested in contributing to its research frontiers. It is expected that comprehensive understanding is obtained, when another graduate course PHYC 572 "Quantum Information Theory," in Spring 2017 was taken together. However, even if one has missed the prequel, it would be still possible to follow the lectures by reading the Chapter 2 of the textbook by Nielsen and Chuang supplementarily. There is a timely introduction to the subject, "Why now is the right time to study quantum computing," written by Harrow particularly for students of computer science department.

Students adopting the graded track will be graded on their attendance and performance on the assignments and projects. To receive a grade of CR on the ungraded track, students need to attend the lectures and show interest. Students who plan to work in quantum information or relevant research fields are highly encouraged to be in the graded track.

## Syllabus

The course is organized as follows. More information will be provided during the first class.

### 0. Review of quantum information theory   [ Lecture note, Sec.0 ]

0.1 Qubit

0.2 Pauli operators and their eigenstates

0.3 Density operator formalism

0.4 Quantum operations

0.5 Two qubits: tensor product

0.6 Entanglement theory 101

0.7 Purification and complete positivity

0.8 Choi-Jamiołkowski isomorphism

### 1. Quantum computation and algorithms   [ Lecture note, Sec.1]

1.1 Church-Turing thesis

1.2 Quantum circuit model

1.3 Oracle problems: exponential speed-up over classical computer

1.4 Grover's algorithm: amplitude amplification

1.5 Shor's integer factorization algorithm

1.6 Quantum complexity theory 101

### 2. Quantum error correction and fault-tolerance   [ Lecture note, Sec.2 ]

2.1 Classical repetition code for error correction

2.2 Quantum 3-qubit bit-flip code

2.3 Quantum 9-qubit error correction code

2.4 Stabilizer codes

2.5 Fault-tolerance of quantum computation

### 3. Many-body entangled states and applications to quantum information   [ Lecture note, Sec.3 ]

3.1 Stabilizer states and classical simulation of stabilizer circuits

3.2 Cluster states, teleportation, and measurement-based quantum computation

3.3 Matrix product states and simulations using MPS

3.4 Tensor network states and topologically ordered states

## Tentative schedule

Last updated on May 2, 2018

 Dates Subjects Assignments Jan. 15  Mon Martin Luther King Jr. Holiday Jan. 17  Wed Course overview Jan. 22  Mon 0.1-0.2 Jan. 24  Wed 0.2 Jan. 29  Mon 0.2-0.3 Jan. 31  Wed 0.3-0.4 Assignment 1 is due Feb. 5  Mon 0.4 Feb. 7  Wed 0.4-0.6 Assignment 2 is due Feb. 12  Mon 0.6 Feb. 14  Wed 0.6-0.7 Assignment 3 and 4 are due Feb. 19  Mon 1.1 Assignment 5 is due Feb. 21  Wed no class due to SQuInT Feb. 26  Mon 1.1-1.2 Feb. 28  Wed 1.2 Assignments 6 and 7 are due Mar. 5  Mon 1.2 Mar. 7  Wed 1.2-1.3 Assignments 8 is due Mar. 12  Mon spring break Mar. 14  Wed spring break Mar. 19  Mon 1.3 Assignments 9 is due Mar. 21  Wed 1.3-1.4 Mar. 26  Mon 1.4-1.5 Assignments 10 and 11 are due Mar. 28  Wed 1.5 Apr. 2  Mon 1.5 Assignments 12 and 13 are due Apr. 4  Wed 2.1-2.2 Apr. 9  Mon 2.2 Assignments 14 and 15 are due Apr. 11  Wed 2.2-2.3 Apr. 16  Mon 2.3 Assignment 16 is due Apr. 18  Wed 2.3-2.4 Apr. 23  Mon 2.4 Assignment 17 is due Apr. 25  Wed 2.4-2.5 Assignment 18 is due Apr. 30  Mon 3.2 May 2  Wed 3.2 Advanced project is due on May 4 May 7-11 no exam

## Assignments

Assignments are as follows.

1. Assignment due on Jan. 31  [Problems: What are algebraic relations enough to derive a matrix representation of Pauli operators? What are the freedoms which remain in the representation?   Solutions]

2. Assignment due on Feb. 7   [Problems: Any quantum pure state of 1 qubit should be able to be characterized as an eigenstate of a Pauli operator along an arbitrary axis. By parameterizing the axis in terms of the spherical coordinate, describe its 2 eigenstates.   Solutions]

3. Assignment due on Feb. 14   [Peer-review of Assignment 2   Solutions]

4. Assignment due on Feb. 14   [Problem: Prove $$\tfrac{1}{2}(|\Phi^+\rangle \langle\Phi^+|+|\Psi^-\rangle \langle\Psi^-|)$$ is separable.   Solutions]

5. Assignment due on Feb. 19   [Peer-review of Assignment 4   Solutions]

6. Assignment due on Feb. 28   [Prove the necessary and sufficient condition for a Bell diagonal state to be separable. You may use the positive partial transposition (PPT) criterion without a proof if you like.   Solutions]

7. Assignment due on Feb. 28   [Essay: Explain to what extent reversible computation is possible?   Solutions]

8. Assignment due on Mar. 7   [Peer-review of Assignments 6 and 7   Solutions]

9. Challenging assignment due on Mar. 19   [Problem: Prove the exact universality of the gate set by all 1-qubit rotations and the 2-qubit Controlled-NOT gate for a 2-qubit register.   Solutions]

10. Assignment due on Mar. 26   [Peer-review of Assignment 9   Solutions]

11. Assignment due on Mar. 26   [Problem: Analyze Deutsch's algorithm in the Heisenberg picture instead of the Schrödinger picture.   Solutions]

12. Assignment due on Apr. 2   [Peer-review of Assignment 11   Solutions]

13. Assignment due on Apr. 2   [Problem: Generalize Grover's algorithm to the case of multiple target states, and analyze its performance.   Solutions]

14. Assignment due on Apr. 9   [Peer-review of Assignment 13   Solutions]

15. Assignment due on Apr. 9   [Problem: Analyze in details Shor's algorithm to factorize $$N=15$$, with 2 different choices of coprime integers $$a=11$$ and $$a=7$$.   Solutions]

16. Assignment due on Apr. 16   [Peer-review of Assignment 15   Solutions]

17. Assignment due on Apr. 23   [Problem: Consider a noisy rotational error channel $$\Lambda'(\rho)=(1-p)\rho + p R^x (\epsilon) \rho R^{x \dagger}(\epsilon)$$, where $$R^x(\epsilon)= \cos \epsilon 1 + i\sin \epsilon X = e^{i\epsilon X}$$. Analyze the performance of the 3-qubit repetition code on it, and obtain the corresponding encoded error probability.   Solutions]

18. Assignment due on Apr. 25   [Problem: Prove that any single-qubit Pauli error operator is correctable for (1) the Steane 7-qubit code (2) the 5-qubit code.   Solutions]

19. Assignment due on Apr. 30   [Peer-review of Assignment 17   Solutions]

20. Assignment due on May 2   [Peer-review of Assignment 18   Solutions]