# Quantum Computing (5314QUCO6Y)

### University of Amsterdam and Mastermath course, Spring 2017 semester Lecturer: Ronald de Wolf (CWI and ILLC) Teaching assistants: Andras Gilyen (CWI) and Joran van Apeldoorn (CWI)

#### Contents of the course:

Today's computers---both in theory (Turing machines) and practice (PCs and smart phones)---are based on classical physics. However, modern quantum physics tells us that the world behaves quite differently. A quantum system can be in a superposition of many different states at the same time, and can exhibit interference effects during the course of its evolution. Moreover, spatially separated quantum systems may be entangled with each other and operations may have non-local'' effects because of this. Quantum computation is the field that investigates the computational power and other properties of computers based on quantum-mechanical principles. Its main building block is the qubit which, unlike classical bits, can take both values 0 and 1 at the same time, and hence affords a certain kind of parallelism. The laws of quantum mechanics constrain how we can perform computational operations on these qubits, and thus determine how efficiently we can solve a certain computational problem. Quantum computers generalize classical ones and hence are at least as efficient. However, the real aim is to find computational problems where a quantum computer is much more efficient than classical computers. For example, Peter Shor in 1994 found a quantum algorithm that can efficiently factor large integers into their prime factors. This problem is generally believed to take exponential time on even the best classical computers, and its assumed hardness forms the basis of much of modern cryptography (particularly the widespread RSA system). Shor's algorithm breaks all such cryptography. A second important quantum algorithm is Grover's search algorithm, which searches through an unordered search space quadratically faster than is possible classically. In addition to such algorithms, there is a plethora of other applications: quantum cryptography, quantum communication, simulation of physical systems, and many others. The course is taught from a mathematical and theoretical computer science perspective, but should be accessible for physicists as well.

#### Prerequisites:

Familiarity with basic linear algebra, probability theory, discrete math, algorithms and complexity theory

#### Material:

Ronald's lecture notes.

Those who want to read more (much more...) can consult the standard textbook in this area:
Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

#### Lectures and location:

February 6 - May 22 2017. Every Monday 14:00-16:45. At Amsterdam Science Park, building 904 (UvA), room F1.02 (until March 20) and G4.15 (from March 27 on).
Each Monday block consists of 2 hours of lectures followed by an exercise session.

This is an 8 ECTS course over 15 weeks, plus a final exam, which comes to roughly 13 hours of work per week. There will be homework exercises for each lecture, to be handed in before the start of the next lecture, i.e., next Monday 14:00. You can either hand in a paper version right before the start of the lecture (if you come in late, you're too late for handing in homework), or email a readable file to Andras Gilyen: gilyen at cwi dot nl, with cc to rdewolf at cwi dot nl. Please start the email subject with [QCHW]. If you have questions about the homework exercises, email Ronald. The answers should be in English. Handwritten solutions (or emailed scans thereof) are fine, as long as they are clearly readable. Cooperation among the students is allowed, but everyone has to hand in their own solution set in their own words. Do NOT post the homework questions and/or anwers online. If you use LaTeX and want to draw circuits, you could consider using qasm2circ, which is the package used for the Nielsen-Chuang book.

Each homework set will get a grade between 1 and 10; if you don't hand it in you'll score a 1 for that week. When determining the average grade for the homework, we will ignore your two lowest scores. The final exam (June 12) will be open book, meaning you can bring the lecture notes, your own notes, homework, and any other papers you want, but no electronic devices (there's the possibility for a re-examination on July 3). The final grade is determined 40% by the homework-grade and 60% by the final exam. In accordance with the Mastermath rules, the final grade will be rounded to the nearest integer (also for Master of Logic students).

#### Preliminary course schedule:

1. Monday February 6, 14:00-16:45
Introduction to quantum mechanics and qubits, overview of the course
Chapter 1 of lecture notes
Homework: Exercises 2,3,4,6 of Chapter 1 (to be handed in by Monday Feb 13, before 14:00)

2. Monday February 13, 14:00-16:45
The circuit model, Deutsch-Jozsa algorithm
Chapter 2 of lecture notes
Homework: Exercises 2,4,5,7,11 of Chapter 2 (to be handed in by Monday Feb 20, before 14:00)
Exercise 7 is a bit tricky; the solution is given in the 2015 exam, near the bottom of this page

3. Monday February 20, 14:00-16:45
Simon's algorithm
Chapter 3 of lecture notes
Homework: Exercises 1,2,3 of Chapter 3 (to be handed in by Monday Feb 27, before 14:00)
In Ex 2: for consistency with the notation of the chapter, you can assume the input is numbered (x_0,...,x_{N-1}) instead of (x_1,...,x_N)
There's a typo in 3.a: "n-qubit state" should be "N-qubit state" (sorry!)

4. Monday February 27, 14:00-16:45
Quantum Fourier transform
Chapter 4 of lecture notes
Homework: Exercises 3,4,5 of Chapter 4 (to be handed in by Monday March 6, before 14:00)

5. Monday March 6, 14:00-16:45
Shor's factoring algorithm
Chapter 5 of lecture notes
Homework: Exercises 1,2,3 of Chapter 5 (to be handed in by Monday March 13, before 14:00)

6. Monday March 13, 14:00-16:45
Grover's search algorithm
Chapter 6 of lecture notes
Homework: Exercises 2,4,5 of Chapter 6 (to be handed in by Monday March 20, before 14:00)
For fun: Grover search in action

7. Monday March 20, 14:00-16:45
Quantum walk algorithms
Chapter 7 of lecture notes
To avoid the degenerate case delta=0, assume the graph on which the algorithm walks is not bipartite
Homework: Exercises 1,2,3 of Chapter 7 (to be handed in by Monday March 27, before 14:00)
NB: you can solve Ex 3 without actually using *quantum* walks; you may assume the number of clauses of the formula is at most some polynomial in n

8. Monday March 27, 14:00-16:45 (NB: as of today, we'll be in room G4.15)
Quantum query lower bounds
Chapter 8 of lecture notes
Homework: Exercises 2,6,7,8 of Chapter 8 (to be handed in by Monday April 3, before 14:00)

9. Friday April 3, 14:00-16:45
Quantum complexity theory
Chapter 9 of lecture notes
Homework: Exercises 1,2,3 of Chapter 9 (to be handed in by Monday April 10, before 14:00)
In Ex 2, "efficient" means using time that's at most polynomial in S.
Today we will also vote about the topic for the last lecture. Possible topics are: (1) Hidden Subgroup Problem, (2) Hamiltonian simulation and the HHL-algorithm for systems of linear equations, (3) QMA and the local Hamiltonian problem, (4) the general adversary bound. Winner: Hidden Subgroup Problem

10. Monday April 10, 14:00-16:45
Quantum encodings, with a non-quantum application
Chapter 10 of lecture notes
Homework: Exercises 1,2,4 of Chapter 10 (to be handed in by Monday April 24, before 14:00)

Monday April 17, no class (Easter Monday)

11. Monday April 24, 14:00-16:45
Quantum communication complexity
Chapter 11 of lecture notes
Homework: Exercises 3,6,8 of Chapter 11 (to be handed in by Monday May 1, before 14:00)

12. Monday May 1, 14:00-16:45
Entanglement and non-locality
Chapter 12 of lecture notes
Homework: Exercises 2,3,4 of Chapter 12 (to be handed in by Monday May 8, before 14:00)
The 2nd part of the hint at 4d is wrong. Instead you can use:
for all Hermitian matrices D and E, and unit vector |\psi>, we have <\psi| D\otimes E |\psi> <= ||D\otimes E || = ||D||\cdot ||E||, where ||D|| denotes the operator norm of D.

13. Monday May 8, 14:00-16:45
Quantum cryptography
Chapter 13 of lecture notes
Homework: Exercises 1,2,3,4 of Chapter 13 (to be handed in Monday May 15, before 14:00)
To clarify Ex 1.a: the first basis vector corresponds to outcome 0, the second basis vector to outcome 1. And 1.c: if two angles are possible (for instance because we treat -|-> and |-> as the same state), then take the one that's smallest in absolute value.
The online Quantum Cryptography course by Vidick and Wehner
Christian Schaffner is offering a MoL June project on quantum cryptography

14. Monday May 15, 14:00-16:45
Error-correction and fault-tolerance
Chapter 14 of lecture notes
Homework: Exercises 3,5,6 of Chapter 14 (to be handed in Monday May 22, before 14:00)

15. Monday May 22, 14:00-16:45
Hidden Subgroup Problem
Extra chapter of the lecture notes
NB: if you're not familiar with basic group theory, then I recommend you read the new chapter before the lecture.
No homework for this lecture, but you're likely to get an exam-question about it.

16. Monday June 12, 14:00-17:00
Final exam (open book: all paper is allowed, no electronics)
IWO-Tentamenzalen (exam-rooms) 4.04B (Geel), Meibergdreef 29, 1105AZ Amsterdam-Zuidoost
If you want to practice, here's the exam from 2015, with solutions.
Update June 13: here's yesterday's exam, with solutions.

17. Monday July 3, 14:00-17:00
Re-sit of the exam (open book: all paper is allowed, no electronics)
SP A1.10
If you want to take the resit: let Ronald know by email, at least one day in advance. If you take the resit, your final grade will be 40% homework-grade + 60% resit-grade (rounded to the nearest integer).