CMSC 38410-1: Quantum Computing
Course description
Spring 13: Tuesday, Thursday 1:30pm-2:50pm (Ry 251)
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Office hours. By appointment.
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Homeworks
Assignment #1, due May 9.
Assignment #2, due May 23.
Assignment #3, due June 5.
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Lecture notes are available here. Any comments, corrections and
suggestions are most welcome.
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Literature. Most of the material pertained to this course can be found in the monographs [1-3]. References to more specific sources covering individual topics will be posted here as we go along.
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M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
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A. Kitaev, A. Shen, M. Vyalyi, Classical and Quantum Computation, Graduate Studies in Mathematics, Vol. 47, American Mathematical Society, 2002.
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P. Kaye, R. Laflamme, M. Mosca, An Introduction to Quantum Computing, Oxford University Press, 2007.
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M. Sipser, Introduction to the Theory of Computation, second edition, Course Technology, 2005.
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S. Arora, B. Barak, Computational Complexity: a modern approach, Cambridge University Press, 2009.
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R. B. Bapat, T. E. S. Raghavan, Nonnegative matrices and applications, Cambridge University Press, 1997.
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C. Lomont, The Hidden Subgroup Problem - Review and Open Problems, 2004.
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H. Buhrman and R. de Wolf. Complexity Measures and Decision Tree Complexity: A Survey.
Theoretical Computer Science, 288(1):21-43, 2002.
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P. Hatamii, R. Kulkarni and D. Pankratov. Variations on the Sensitivity Conjecture, Theory of Computing Library, Graduate Surveys Number 4 (2011) pp. 1-27.
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Y. Shi. Approximate polynomial degree of Boolean functions and its applications. Proceedings of the 4th International Congress of Chinese Mathematicians, 2007.
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M. Bun and J. Thaler. Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities, 2013.
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A. Sherstov, Approximating the AND-OR Tree, 2013.
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S. Laplante, T. Lee, M. Szegedy. The Quantum Adversary Method and Classical Formula Size Lower Bounds. Computational Complexity, 15(2): 163-196 (2006).
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A. Ambainis, A. M. Childs, B. Reichardt, R. Spalek, S. Zhang. Any AND-OR Formula of Size N can be Evaluated in time N1/2+o(1) on a Quantum Computer. FOCS 2007: 363-372.
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A. Ambainis, Quantum Walk Algorithm for Element Distinctness. SIAM Journal on Computing, 37(1): 210-239 (2007).
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E. Kushilevitz and N.Nisan, Communication Complexity, Cambridge University Press, 1997.
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A. Razborov, Communication Complexity, in the International Mathematical Olympiad 50th Anniversary Book.
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A.C. Yao, Quantum circuit complexity, Proceedings of the 34th Annual IEEE
Symposium on Foundations of Computer Science, November 1993, pp. 352 - 361.
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H. Buhrman, R. Cleve and A. Wigderson, Quantum vs. classical communication
and computation, Proceedings of the 30th Annual ACM Symposium on Theory of
Computing, May 1998, pp. 63 - 68.
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S. Aaronson and A. Ambainis, Quantum Search of Spatial Regions, Theory of Computing, 1:47-79, 2005.
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I. Kremer, Quantum communication, Master's thesis, Hebrew University, Jerusalem 1995.
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A. Razborov, Quantum Communication Complexity of Symmetric Predicates, Izvestiya: Mathematics, Vol. 67, No 1, 2003, pages 145-159.
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A. Sherstov, The pattern matrix method for lower bounds on quantum communication, 2008.
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Progress and references.
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First week: A crash course in computational complexity that should be sufficient for our purposes can be found in [2, Part 1] or [1, Ch. 3.1-3.2]. For more detailed treatment see excellent textbooks [4] (undergraduate level) and [5] (a little bit advanced). Reversible circuits: [3, Ch. 1.5] or [2, Ch, 7]; simulation of any classical computation with a reversible one (aka Garbage Removal Lemma): [3, Fig. 1.6] or [2, Lemma 7.2]. The physics of Landauer's principle is discussed in [1, Ch. 3.2.5]. Matrix form of circuit computations: [3, Ch. 1.4]. Probabilistic computations: [2, Ch. 4.3] (as always, for an in-depth treatment see [4,5]). For doubly stochastic matrices,
Birkhoff-von Neumann theorem (and much more) see e.g. [6]. Rudimentary notation from linear algebra, including Dirac's notation can be found in any of our textbooks; I particularly recommend [1, Ch. 2.1].
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Second week: Universal (aka complete) bases: [2, Ch, 8.2] or [1, Ch. 4.5]. Equivalence of universal bases: [2, Cor. 8.5.1]. Discussion of probabilistic vs. quantum: [3, Ch. 6.1].
Deutsch algorithm: [3, Ch. 6.3]. Our exposition of Deutsch-Jozsa and Simon's algorithms closely follows the one given in [3, Ch. 6.3] and [3, Ch. 6.4]. We will thoroughly discuss
the Hidden Subgroup Problem later, but if you are already curious, you can check out [7]. $BQP\subspace PP$.
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Third week: $BQP\subset PP$ cntd.; see [1, Ch. 4.5.5] for a proof of a slightly weaker result. Grover's search algorithm: [3, Ch. 8.1]. The reduction of factoring to order-finding goes back to Miller (1975); our exposition is close to [2, Ch. 13.3]. Continued fractions: [3, Thm. 7.1.7]. Eigenvalue estimation problem and construction of the operator $c-U^x$ (we finally decided to call it Shor-U) [3, Ch 7.2 ]. Quantum Fourier Transform and its inverse [3, Ch 7.1].
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Fourth week: Shor's factoring algorithm cntd. Our exposition of the eigenvalue estimation includes elements of phase estimation [3, Ch. 7.1]; we did not discuss the latter problem separately. And the exposition in [3, Ch. 7.3] also includes an efficient realization of $QFT_{2^h}$ that we left as an exercise in class. Finally, the analytical expression for the error can be found in [3, Ch. 7.1.1]. Discrete Logarithm: [3, Ch. 7.4] (note, however, that we did not use the fancy notation with $\widetilde$). For the Hidden Subgroup Problem in Abelian groups, I recommend [2, Ch. 13.8], and [7] is a good survey on the non-Abelian case. Our exposition of the hybrid method follows [3, Chp. 9.3] (although, as in many other cases, we somewhat re-arranged the material). Mega-theorem about polynomial equivalence of various complexity measures for total functions.
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Fifth week: The proof of mega-theorem cntd. It can be found e.g. (in pieces) in the survey [8]: see theorems 10, 11, 2, 7, 17 and 18 there. A good survey on the sensitivity vs. block sensitivity is [9], and a good survey on the approximate polynomial degree is [10]. The recent breakthrough (approximate polynomial degree of the tribes function) was achieved independently in [11] and [12]. Ambainis's adversary method (we only did a sketch without proof): [3, Ch. 9.7 and A5]. For the relations between quantum query complexity and classical formula size see [13,14]. Ambainis's quantum walk (sketch): [15].
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Sixth week: Ambainis's quantum walk cntd. The standard textbook in classical communication complexity is [16] (or check out my survey [17] at a very introductory level). The quantum communication complexity was introduced in [18]. The relation between quantum decision tree and communication complexities was observed in [19], and the improvement of the upper bound for DISJOINTNESS is in [20]. The decomposition theorem for quantum communication protocols also appeared in [18], and it was further refined in [21,22]. The discrepancy method is the standard tool in communication complexity (not only in quantum!), see again [18, 21] for examples. Lower bound in terms of the spectral norm of the communication matrix is from [21].
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Seventh week: Approximate trace norm was introduced in [22], and the same paper proved tight lower bounds for symmetric predicates. The proof based on generalized discrepancy was given in [23]. Quantum Probability: [3, Ch. 3.5] or [2, Ch. 10]. An excellent overview of various noise models (some of them are quite sophisticated) can be found in [1, Ch. 8.3]. The axiomatic definition of physically realizable operators (aka superoperatotrs) is in [1, Ch. 8.2.4]; note that they consider a slightly more general case when the trace is allowed to decrease. Operator-sum representation: [1, Ch. 8.2.3.]. For another account see [2, Ch, 11.1]; in particular, [2, Problems 11.1, 11.2] are precisely the two parts of our theorem on different characterizations of superoperators. [1, Theorem 8.1] provides a complete proof of the part that we sketched in class.
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Eighth week: Trace distance [1, Ch. 9.2.1]. Superoperators do not increase trace distance [1, Theorem 9.2]. No cloning theorem: [1, Box 12.1]. Three qubit bit/phase flip codes are in [1, Ch. 10.1], and the Shor code is in [1, Ch. 10.2]. General theory of quantum error-correction: [1, Ch. 10.3]; again, in class we considered only the trace-preserving case. Unitary equivalence for mixed states: [1, Theorem 2.6]. Unitary equivalence for operator-sum representations: [1, Theorem 8.2]. Criterium of quantum recovery: [1, Theorem 10.1].
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Ninth week: Criterium of quantum recovery cntd. Stabilizer codes [1, Ch. 10.5]. Error-correction conditions for stabilizer codes [1, Theorem 10.8]. Quantum teleportation [1, Ch. 1.3.7]. Superdense coding [1, Ch. 2.3]. What we did not have time to do and left for another course: Shannon and von Neumann entropies [1, Ch. 11].
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Tenth week (short): projects. Charles Forgy: "A Summary of Current Implementation of Adiabatic Quantum Computing" and Young Kun Ko: "Extended Formulations".