Math 41001/51001: Introduction to Quantum Computing, Fall 2018
Instructor: ChiKwong Li
Meeting time and place: TR 12:30  1:50 p.m. Morton 342
Office: Jones 128, Tel: 2212042
Email: ckli@math.wm.edu,
http://cklixx.people.wm.edu
Office hours: TWR 9:00  10:30 a.m., or by
appointments
Course description:
Quantum computation and quantum information science is a rapidly growing area. Quantum cryptology
is in commercial use, and the construction of practical quantum compter still require a lot of research
from different disciplines including mathematics, physics, computer science, chemistry, engineering,
material science, etc. In this course,
nn introduction of the mathematics background of quantum computing will be given based on
the
first 12 chapters of the following required textbook.
 M. Nakahara and T. Ohmi, Quantum computing: From Linear Algebra to Physical Realizations,
CRC Press, Taylor and Francis Group, New York, 2008.
http://www.amazon.com/QuantumComputingAlgebraPhysicalRealizations/dp/0750309830
Additional references:
 M. Hirvensalo, Quantum Computing, (2nd edition), Springer, New York, 2004.
 G. Johnson, A Shortcut Through Time: The Path to the Quantum Computer, Alfred A. Knopf, New york,
2003.
 P. Kaye, R. Laflamme and M. Mosca, An introduction to quantum computing, Oxford University Press,
Oxford, 2007.
 A. Yu Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and Quantum Computation (Graduate Studies in
Mathematics), AMS, Rhode Island, 2002.
 D. McMahon, Quantum Computing Explained, Wiley and Sons, 2008, New York.
 M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University
Press, Cambridge, 2000.
 A. O. Pittenger, An introduction to quantum computing algorithms, Birkhauser, Boston 1999.
Homework
 Problems will be assigned every lecture and due on Thursday in the following week.
 Challenging problems will be assigned from time to time,
extracredits will be given to successful (or partially successful)
attempts.
 Homework help sessions will be conducted on Wednesday, 2:003:00 p.m. Jones 113 or 131.
Of course, help is also available at office hours.
 Math 51001 students are required to write term papers.

You have to use LaTex to typset mathematical document,
an excellent skill to acquire. You may get the programs for free.
 For windows users,
(a) download the MikTex program from http://miktex.org/download;
(b) then download the Texmaker program from http://www.xm1math.net/texmaker/download.html;
(c) then open the program "texmaker";
(d) copy (or download and then open) the "homework01.tex" file to the texmaker window;
(e) select "LaTex" from the "Quick Build" menu;
(f) click the "=>" arrow on the left of "Quick Buil" to get the pdf output.
 For Mac users,
(a) download MacTex from http://tug.org/mactex/;
(b) open the "texshop" program,
(c) copy (or download and then open) the "homework01.tex" file to the texshop window;
(d) change "PlainTex" to "LaTex" at "Typeset" menu;
(e) click "Typeset" icon to get the pdf output.
 You may also use the online editor
Write LaTeX .
 Here is
a list of TeX commands for mathematics symbols.
Assessment
Homework Midterm Final
30% 30% 40%
(Extra credit problems may add up to another 5%)
Exams: Midterm Oct. 12 Due noon Take home
Final Dec. 17 3 hrs ( 9:0012:00)
Grades (for homework, exams, final grade, etc.):
%: 0  60  65  70  75  80  83  87  90  93  100
F D C C C+ B B B+ A A
Class notes (Under continuous revision)
Homework list

Homework 1. Exercise 1.1  1.7. Due Sept. 6, 5:00 p.m.
 Homework 2. Due Sept. 20, 5:00 p.m.
[Sample solution.
Tex file and
pdf file.]
Exercise 1.8. 1.9, 1.10 (a), 1.11, 1.12, 1.13, 1.14.
Additional Optional Problem. Show that for every $2\times 2$ Hermitian matrix $A$ has the form
$u_0 I_2 + u_1 \sigma_x + u_2 \sigma_y + u_3 \sigma_z$ for some real numbers
$u_0, u_1, u_2, u_3$. Moreover, the matrix $A$ has
is a postive rank one projection, i.e., $A^2 = A$ has rank 1, if and only if
$u_0 = 1/2$ and $u_1^2 + u_2^2 + u_3^2 = 1/4$.
Hints: (1.8) Assume $A = (A_{ij})$, $B = (B_{ij})$. Show that the matrix inequalities
by comparing the $(i,j)$ entries on both sides.
(1.12) Write $H = VDV^\dag$ for some unitary $V$ and real diagonal matrix $D$. Explain why
$(IiH)$ is invertible and that $U = VFV^\dag$ so that $V$ is a diagonal matrix with unimodular
diagonal entries.
For the additional problem. You may use the fact that
$A$ is a rank one orthogonal projection if and only if $trace(A) = 1$ and $\det(A) = 0$.
 Homework 3. Due Sept. 27, 5:00 p.m. (extended to Setp. 28, 5:00 p.m.)
[Sample solution.
Tex file and
pdf file.]
Exercise 1.15, 1.16, 1.17, 1.22, 2.1 (1)(4).
Additonal problem. Show that if A \in M_m, B \in M_p, then there are unitary
U \in M_m and V \in M_p such that
A \otimes I_p + I_m \otimes B = (U\otimes V)(S \otimes I_p + I_m \otimes T)(U\otimes V)^\dag
for some upper triagular S \in M_m and T \in M_p.
Hint: Ex. 1.16. See the proof of Prop. 1.2 for the meaning of A = n. \sigma.
Show that A = P1  P2 =  v1 > < v1    v2 > < v2  where v1> are v2> are the eigenvectors
of A corresponding to the eigenvalues 1 and 1. Then apply spetral theorem, and rearrange.
Ex. 1.17. Note that SVD also works for rectangular matrices. See Example 1.7.
Ex. 2.1. Note that = * if A, B are Hermitian.
 Homework 4. Due: Oct 5, noon.
[Sample solution.
Tex file and
pdf file.]
Exercise 2.2 2.6. For problems 2.5 and 2.6, determine the eigenvalues of the parial
transposes of the matrices, and show that the negativity vanises when there is no negative
eigenvalues for the partial transposes.
 Take Home Exam. Due: Oct 12, noon.
Question paper
[
Tex file and
pdf file.]
[ Solution.]
 Homework 5. Due: Oct. 19, noon (extends to Oct. 20, noon.)
Exercise 2.72.11. Ex.3.1, 3.3, 3.5.
[Sample solution.
pdf file.]
 Homework 6. Due: Oct. 26, noon. (extends to Oct. 29 noon.)
Exercise 4.1  4.10.
[Sample solution.
pdf file.]
Hint: Ex. 4.1. Agrue U_{CNOT} is not A \otimes B for any 2x2 A, B.
Ex. 4.4 Show W_nW_n^\dag = I. Ex. 4.5. Show that the two gates have same values for 00>, 10>
01>, 11>. Ex. 4.7. Show that U_swap (xy>) = yx>. Ex. 4.8 and 4.9.
Check the cases xy> = 00>, 10>, 01>, 11>. Ex. 4.10 (1) \Psi> has two forms, \Phi>
has two forms. So <\Psi\Phi> has 4 possible forms.
 Homework 7. Due: Nov. 6, (Tuesday).
Exercise 4.12, 4.13, 4.14. 4.15, 4.16, 4.17, 5.1, 5.2.
[Sample solution.
pdf file.]
 Homework 8. Due: Nov 13, (Tuesday).
Exercise 6.1, 6.2, 6.3, 7.1, 7.2, 7.3, 7.4.
 Homework 9. Due: Nov. 27, (Tuesday).
Exercise 8.1, 8.3, 8.4, (cf. Example 8.2), 9.1, 9.2. Note that the channel should be
defined by {\cal E}(\rho_S) = (1p)\rho_S + p Y\rho_S Y^\dag.
Optional 8.2.