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PHY680.01 Quantum Computing and Quantum Information, Spring 2012

MoWe 9:00 - 10:20AM  Physics Building, P122
Instructors:  Vladimir Korepin and Tzu-Chieh Wei
Offices: Physics Building D147 and Math Tower  6-101
Office hour: Vladimir Korepin: TBA.
                   Tzu-Chieh Wei: Mo 10:20AM-11:20AM Math 6-101 or by appointment

For Prof. Korepin's course website, please see here.

Brief content of the course:
The course is divided into three major parts: (1) advanced quantum mechanics, (2) classical and quantum  information theory, (3) quantum algorithms and models of quantum computation. In addition, various physical implementations of quantum computers will be discussed. Additional relevant topics will be included when appropriate.

Textbooks and references:
Quantum Computation and Quantum Information, M. Nielsen and I. Chuang (Cambridge University Press)
An Introduction to Quantum Computing, P. Kaye, R. Laflamme and M. Mosca (Oxford)
Classical and Quantum Computation, A. Yu. Kitaev, A. H. Shen and  M. N. Vyalyi (AMS)
J. Preskill lecture notes (http://www.theory.caltech.edu/~preskill/ph229/#lecture)
Andrew Steane’s paper on Quantum Computing (http://arxiv.org/pdf/quant-ph/9708022 or published version in Report on Progress in Physics)

Homework 15% (2 to 3 sets)
Midterm 40% (March 28, 2012)
Presentation 30% (suggested topics/papers will be given later in the course)
In-class quizzes 15%

Incomplete list of addtional reading:
"Quantum Computing Promises New Insights, Not Just Supermachines" by Scott Aaronson (Professor of Computer Science at MIT) in New York Times

"Quantum Entanglement: A Modern Perspective" by Barbara M. Terhal, Michael M. Wolf and Andrew C. Doherty, Physics Today v56, p46 (2003)

"Recent Progress in Quantum Algorithms" by Dave Bacon and Wim van Dam, Communications of the ACM, Vol. 53, Pp. 84-93(2010)

"Why Now is the Right Time to Study Quantum Computing" by Aram Harrow, The ACM Magazine for Students - The Legacy of Alan Turing: Pushing the Boundaries of Computation archive Volume 18 Issue 3, Spring 2012 Pages 32-37

"The First Quantum Machine" (Breaktrhough of the year 2010), Adrian Cho, Science vol. 330, p. 1604 (2010)

"Silicon Quantum Computer a Possibility" by E.S. Reich, Nature News (2011) doi:10.1038/news.2011.29

"Quantum Entanglement Links 2 Diamonds" by John Matson, Scientific American, Dec. 1, 2011

"Moving Beyond Trust in Quantum Computing" by Vlatko Vedral, Science Vol.335, pp.294-295 (2012).

"Quantum Computers" by Ladd, Jelezko, Laflamme, Nakamura, Monroe & O'Brien, Nature 464, 45-53 (2010)

"Physics: Quantum Computing" by E. Knill, Nature 463, 441-443 (2010)

"Quantifying entanglement in macroscopic systems" by Vlatko Vedral, Nature 453, 1004-1007 (2010)

Student presentation will be on April 30 and May 2.

Announcement, Update and Additional Information

-May 2, 2012: Student Presentation (Abhishodh Prakash and You Quan Chong)
 Slides of You Quan Chong

-April 30, 2012 (Physical Implementations and Student Presentation)
Today, in the first half of the lecture I gave a survey of various physical implementations of quantum information processing.
In the second half, Thomas Fan gave a presentation on adiabatic quantum computation.

Lecture 26
slides (on physical implementations) can be downloaded here.
(1) "Quantum Computers" by Ladd, Jelezko, Laflamme, Nakamura, Monroe & O'Brien, Nature 464, 45-53 (2010)
(2) "Entangled states of trapped atomic ions" by Rainer Blatt & David Wineland, Nature 453, 1008-1015 (2010)
(3) "Quantum coherence and entanglement with ultracold atoms in optical lattices" by Immanuel Bloch, Nature 453, 1016-1022 (2010)
(4) "The quantum internet" by H. J. Kimble, Nature 453, 1023-1030 (2010)
(5) "Superconducting quantum bits" by John Clarke & Frank K. Wilhelm, Nature 453, 1031-1042 (2008)
(6) "Coherent manipulation of single spins in semiconductors" by Ronald Hanson & David D. Awschalom, Nature 453, 1043-1049 (2008)

After the lecture, I noticed a recently paper in Nature that reports a work of quantum simulations using hundreds of trapped ions:
"Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins" by  Joseph W. Britton, Brian C. Sawyer, Adam C. Keith, C.-C. Joseph Wang, James K. Freericks, Hermann Uys, Michael J. Biercuk & John J. Bollinger, Nature 484, 489-492 (2012).

Please fill in the online evaluation if you have not done so.

-April 25, 2012 (Quantum Error Correction)
Today, Prof. Alexander Kirillov finished with Topological Quantum Computation. (The notes of this part have been added to notes in Lecture 24.) Then, I discussed Quantum Error Correction, which is an essential part to make sure quantum computation can be carried out even in the presence of noise.

Lecture 25
notes can be downloaded here.

-April 23, 2012 (Braid group and Topological Quantum Computation)
Today, Prof. Alexander Kirillov discussed Braid group and how Chern-Simons theory helps to enable Topological Quantum Computation.

Lecture 24
notes can be downloaded here.

-April 18, 2012 (Topological Phase and Kitaev Toric Code Model)
Today, Prof. Alexander Kirillov introduced the idea of topological quantum computation and discussed Kitaev's Toric Code Model.

Lecture 23
notes can be downloaded here.

-April 16, 2012 (Quantum Hall Effect, Anyons and TQC)
Today, Prof. Krepin finished discussions on Grover's algorithm and discussed Quantum Hall Effect, anyons, and braid group.

Lecture 22 notes can be downloaded here.

-April 11, 2012 (Complexity Classes and Grover's algorithm)
Today, Prof. Sutherland finished discussions on complexity classes. Prof. Korepin then explained Grover's search algorithm.

Lecture 21 notes can be downloaded here.

-April 9, 2012 (Classical Complexity Classes)
Today, Prof. Sutherland gave a lecture on clssical complexity classes. Currently, there are 495 classes; see Scott Aaronson's Complexity Zoo.

Lecture 20 notes can be downloaded here.

-March 26, 2012 (Crytography and RSA)
Today, Prof. Scott Sutherland gave a lecture on cryptography and explained in detail RSA scheme.

Lecture 19 notes (taken by Tzu-Chieh Wei) can be downloaded here.
Note March 28 is the midterm exam.

-March 21, 2012 (One-way quantum computer)
 I discussed one-way quantum computation (a.k.a. measurement-based computation) today. The original paper by Raussendorf and Briegel is Phys. Rev. Lett. 86, 5188–5191 (2001). There is a review article by Briegel et al. in Nature Physics.Today's lecture was based on a review article Raussendorf and I wrote on "Quantum Computation by Local Measurement" publsihed in the Annual Review of Condensed Matter Physics, Vol. 3: 239-261 (March 2012) . The enrolled students can access this review here.

Lecture 18 notes can be downloaded here.

Note that there will be a mid-term exam on March 28.

-March 19, 2012 (Phase estimation and Shor's algorithm for factoring)
 I discussed phase estimation algorithm today. I also showed the order finding x^r = 1 (mod N) can be solved by phase estimation and the solution can be used to do factoring.

Lecture 17 notes can be downloaded here.

-March 14, 2012 (Quantum Fourier Transform)
 I finished discussions on universal gates and gave motivations of Quantum Fourier Transform (QFT). We actually wrote down the circuit and understood how it works. See Nielsen and Chuang ch. 5 for reference. Next time we shall cover phase estimation, order finding, period finding, and maybe discrete logarithm.
Please start working on Homework 2.

Lecture 16 notes can be downloaded here.

A very nice article by Dave Bacon and Wim van Dam on "Recent Progress in Quantum Algorithms" is highly recommended for reading.

Stephen Jordan's Quantum Algorithm Zoo
Scott Aaronson's Complexity Zoo

 -March 12, 2012 (Universal gates for quantum computation)
 I continued discussion on universal gates. We saw why CNOT and single-qubit gates are universal, in the sense that they can be used to construct arbitrary n-qubit unitary transformation. The main reference is Nielsen and Chuang ch.4.

Lecture 15 notes can be downloaded here.

Homework 2 set is posted and the deadline is March 21.

-March 7, 2012 (Discussions of HW1 and quantum circuit)
 HW1 was graded and returned. Today, I discussed solutions of HW1. I also began discussions on quantum circuit (see Nielsen & Chuang ch.4).

 Lecture 14 notes can be downloaded here.

-March 5, 2012
 Midterm exam will be postponed to March 28, 2012.

-March 5, 2012 (Holevo bound on accessible information)
 Prof. Korepin continued his discussions on quantum information. In particular, he gave an example illustrating the Holevo bound on accessible information.

Lecture 13 notes can be downloaded here.

-March 1, 2012 (still away)
 I believe Prof. Korepin announced there would be a Midterm on March 12, 2012.

-I will not be here next week (Feb. 27- March 2), so there will not be an office hour on Feb. 27.
** Please start working on your homework **
Lecture 11 and 12 (Korepin) notes would be unavailable due to my travel.

One important reference for content covered in Lectures 11 and 12 is Prof. Korepin's paper "Thermodynamic interpretation of quantum error correcting criterion", arXiv: quant-ph/0202054

-Feb. 22, 2012 (Shannon's theory)
 Today Professor Korepin discussed classical information theory, in particular, Shannon's noiseless and noisy channel theory. He also defined conditional entropy, mutual information, and in the quantum case, von Neumann entropy and its various properties.

 Lecture 10 notes (taken by Tzu-Chieh Wei) can be downloaded here.

-Feb. 20, 2012 (Entanglement measures)
 Today, I defined more precisely entangled pure and mixed states. I introduced a few entanglement measures, such as entanglement of distillation, entanglement cost (under the process of entanglement dilution), entanglement of formation (and concurrence). Wootters formula for concurrence, a widely used one in studying qubit systems. I briefly mentioned a measure that my collaborators and I developed, i.e. the geometric measure of entanglement (in short, geometric entanglement) and its application to the XY spin chain in a transverse field.

Lecture 9 notes can be downloaded here.

-Feb. 15, 2012 (Entanglement entropy in physical models)
 Homework Set 1 was distributed today (due March 5th).
 Today Prof. Korepin gave a survey of entanglement in various models (many materials from his own research). I jotted down some notes from his blackboard presentation

Lecture 8 notes can be downloaded here.
A complete list of references from Prof. Korepin is here (not in any particular order):

(1) Universality of Entropy Scaling in 1D Gap-less Models
(2) Entanglement of Valence-Bond-Solid on a Arbitrary Graph
(3) Entanglement and Density Matrix of a Block of Spins in AKLT Model
(4) Negativity for two blocks in the one dimensional Spin 1 AKLT model
(5) Entanglement in an SU(n) Valance-Bond-Solid State
(6) Entanglement in a Valence-Bond-Solid State
(7) Analysis of entropy of XY Spin Chain
(8) Ellipses of Constant Entropy in the XY Spin Chain
(9) Renyi Entropy of the XY Spin Chain
(10) Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture
(11) Entanglement in valence-bond-solid states on symmetric graphs

-Feb. 13, 2012 (Irreversiblity of Superoperators, quantum channels & master equation)
 Today, Prof. Korepin discussed superoperators and showed that if the action of two superoperators gives rise to identity then these two superoperators are unitary operators. He also discussed various quantum channels: depolarizing, phase-damping and amplitude-damping channels. He derived the Linblad or master equation and illustrate with harmonic oscillators.

Lecture 7 notes taken by Tzu-Chieh Wei are here.

** Homework Set 1 will be distributed on Wednesday **

-Feb. 8, 2012 (Schmidt decomposition and superoperators)
 Today, Prof. Korepin discussed further properties of density matrices, Schmidt decomposition, partial trace, partial transpose, and more importantly superoperators (using Kraus operators). I tried to jot down notes, but I do not guarantee they be faithul to the lecture.

Lecture 6 (sketchy) notes can be downloaded here.

-Feb. 6, 2012 (Distinguish nonorthogonal states and POVM)
 I continued on the question posed last time on how we can do to distinguish two nonorthogonal states (we know it won't be with certainty) and consturcted a POVM. I also mentioned the most general measurement postulate can actually be derived from coupling the system to a meter by unitary evolution followed by a projective (von Neumann) measurement.

Prof. Korepin then discussed CHSH-Bell inequality and some properties of density matrices.

Lecture 5 notes can be downloaded here.

-Feb. 1, 2012 (Postulates of QM and POVM)
 I continued on Deutsch-Jozsa algorithm from last lecture. I introduced three basic postulates on quantum mechanics (state vector in Hilbert space, unitary evolution governed by Schroedinger equation, and measurement). We saw that all these can be generalized, which we shall use later in discussions of quantum information and computation. I showed by using POVM we conclude that two nonorthogonal states (e.g., |0> and |+>) can not be distinguished with certainty. But given this "negative" result, how much can we do to distinguish two nonorthogonal states?

Lecture 4 notes can be downloaded here.
-Jan. 30, 2012 (Quantum teleportation, simple quantum algorithms)
 Today, I discussed quantum teleportation, which is one of the important basic quantum information processing protocols and is also listed by PRL as a milestone paper. Teleportation was implemented by many groups, with the earlier ones by Zeilinger's group: Nature 390, 575 (1997) using photon polarizations, Kimble's group: Science 282, 706 (1998) using squeezed states, and De Martini's group: PRL 80, 1121 (1998) using both polarization and momentum. In addition, I also discussed Deutsch and Deutsch-Jozsa algorithms, which demonstrate that quantum algorithms can be better than classical ones (although not much better yet until we discuss Shor's). References: Nielsen and Chuang 1.3 and 1.4.

Lecture 3 notes are contained in last set of notes (below).

-Jan. 25, 2012 (Qubits, density matrices, gates and superdense coding)
  In today's class I discussed quantum bits (qubits), Bloch sphere, mixed one-qubit state, density matrix, one-qubit gates, Controlled-NOT (CNOT) and Conditional Phase (CPhase) gates, and super dense coding. References: Nielsen and Chuang 1.3, 1.4, 2.1 (for Dirac's notation and linear algebra) and 2.3 (for superdense coding).

Superdense coding was invented by Bennett and Wiesner, see the original paper. It was implemented by K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, see "Dense coding in experimental quantum communication," Phys. Rev. Lett. 76, 4656 (1996). However, due to limitation of EPR-Bell-state measurement using photons, it was limited by log_2 (3)=1.58 (with actual transmission being 1.13 bits). My collaborators and I found a way around this limitation and our experiment achieved a transmission of 1.63 bits; see Nature Phys. 4, 282-286 (2008) as well view and news by S.P. Walborn, Nature Phys. 4, 268 (2008).

Lecture 2 notes can be downloaded here (access restricted to enrolled students).

-Jan. 23, 2012 (Overview)
  Prof. Korepin gave an overview of topics to be covered. I summarized them in  (Lecture 1) notes.
  I distributed an article by Scott Aaronson (Professor of Computer Science at MIT) in New York Times on "Quantum Computing Promises New Insights, Not Just Supermachines".

Lecture 1 notes can be downloaded here (access restricted to enrolled students).

For your information.

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