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Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 51 (11 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Lower bounds for Quantum Oblivious Transfer
 IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), volume 8 of Leibniz International Proceedings in Informatics (LIPIcs
, 2010
"... Oblivious transfer is a fundamental primitive in cryptography. While perfect information theoretic security is impossible, quantum oblivious transfer protocols can limit the dishonest players ’ cheating. Finding the optimal security parameters in such protocols is an important open question. In this ..."
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Cited by 5 (4 self)
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Oblivious transfer is a fundamental primitive in cryptography. While perfect information theoretic security is impossible, quantum oblivious transfer protocols can limit the dishonest players ’ cheating. Finding the optimal security parameters in such protocols is an important open question. In this paper we show that every 1outof2 oblivious transfer protocol allows a dishonest party to cheat with probability bounded below by a constant strictly larger than 1/2. Alice’s cheating is defined as her probability of guessing Bob’s index, and Bob’s cheating is defined as his probability of guessing both input bits of Alice. In our proof, we relate these cheating probabilities to the cheating probabilities of a coin flipping protocol and conclude by using Kitaev’s coin flipping lower bound. Then, we present an oblivious transfer protocol with two messages and cheating probabilities at most 3/4. Last, we extend Kitaev’s semidefinite programming formulation to more general primitives, where the security is against a dishonest player trying to force the outcome of the other player, and prove optimal lower and upper bounds for them. Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2010.157 1
Least span program witness size equals the general adversary lower bound on quantum query complexity
 Electronic Colloquium on Computational Complexity
, 2010
"... Abstract Span programs form a linearalgebraic model of computation, with span program "size" used in proving classical lower bounds. Quantum query complexity is a coherent generalization, for quantum algorithms, of classical decisiontree complexity. It is bounded below by a semidefinit ..."
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Abstract Span programs form a linearalgebraic model of computation, with span program "size" used in proving classical lower bounds. Quantum query complexity is a coherent generalization, for quantum algorithms, of classical decisiontree complexity. It is bounded below by a semidefinite program (SDP) known as the general adversary bound. We connect these classical and quantum models by proving that for any boolean function, the optimal "witness size" of a span program for that function coincides exactly with the general adversary bound. A consequence is an optimal quantum algorithm for evaluating "balanced," readonce formulas over any finite boolean gate set. For example, the gate set may be taken to be all functions {0, 1} k → {0, 1} with k ≤ 1000. A formula is a tree whose nodes are associated to functions from the gate set. The notion of balance is technical, but it includes layered formulas. A previous quantum algorithm optimally evaluates formulas for which an optimal span program is given for each constantsize gate. However, span programs have been found only by hand. The SDP automates this procedure, and its value surprisingly always matches the lower bound. Other implications of the SDP include an exact composition rule for the general adversary bound, and that the general adversary bound upperbounds the sign degree. The connection can also be seen as half of a universality result for span programs. For any boolean function, there exists a span program with witness size at most the function's quantum query complexity. Conversely, solutions to the SDP give span programs, and therefore also new quantum algorithmsbeyond evaluating formulas. Subsequent work has bounded the query complexity by the witness size, thus implying that the general adversary bound is tight.
How low can approximate degree and quantum query complexity be for total boolean functions
, 2012
"... Abstract—It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Ω(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and boundederror quantum query comple ..."
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Abstract—It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Ω(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and boundederror quantum query complexity. We show that for these measures the correct lower bound is Ω(log n / log log n), and we exhibit quantum algorithms for two functions where this bound is achieved. KeywordsAnalysis of Boolean functions, approximate degree, quantum algorithms, query complexity, lower bounds I.
Uniform Approximation by (Quantum) Polynomials
, 2010
"... We show that quantum algorithms can be used to reprove a classical theorem in approximation theory, Jackson’s Theorem, which gives a nearlyoptimal quantitative version of Weierstrass’s Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectivel ..."
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We show that quantum algorithms can be used to reprove a classical theorem in approximation theory, Jackson’s Theorem, which gives a nearlyoptimal quantitative version of Weierstrass’s Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation. 1
Research Statement
"... Quantum information and computation, a novel field at the intersection of two most significant discoveries of the last century (i.e, quantum mechanics and computer science), has become increasingly relevant. Recent developments of quantum computers have gradually brought the theoretically appealing ..."
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Quantum information and computation, a novel field at the intersection of two most significant discoveries of the last century (i.e, quantum mechanics and computer science), has become increasingly relevant. Recent developments of quantum computers have gradually brought the theoretically appealing model to a reality, raising serious concerns about the safety and the functionality of our current cryptographic and computational systems [1]. Although fully fledged quantum machines are still a long way off, specialpurpose quantum devices have already been commercialized and used in practice (e.g., the Quantis generators of ID Quantique). It is thus an imperative task for our generation to come up with cryptographic and computational systems in the world with potential quantum computers. My research aims to contribute solutions to such a fundamental task. Not only do I want to investigate the power of quantum computers so as to build systems that are secure against potentially quantum adversaries, but also I am interested in making good use of such quantum features to achieve security and functionality beyond the reach of classical devices. In particular, I focus on specialpurpose quantum devices, which makes my research not purely theoretical but include significant nearterm practical applications. The original motivation of studying quantum computation is to simulate physical systems that occur in Nature. Nowadays, such simulations have taken a significant fraction of classical supercomputer time.
Quantum Computing: Lecture Notes
, 2011
"... These lecture notes were formed in small chunks during my “Quantum computing ” course at the University of Amsterdam, FebMay 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 × 45 minutes, with an additional 45minute lecture for exercises and homework. The fir ..."
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These lecture notes were formed in small chunks during my “Quantum computing ” course at the University of Amsterdam, FebMay 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 × 45 minutes, with an additional 45minute lecture for exercises and homework. The first half of the course (Chapters 1–7) covers quantum algorithms, the second half covers quantum complexity (Chapters 8–9), stuff involving Alice and Bob (Chapters 10–13), and errorcorrection (Chapter 14). A 15th lecture about physical implementations and general outlook was more sketchy, and I didn’t write lecture notes for it. These chapters may also be read as a general introduction to the area of quantum computation and information from the perspective of a theoretical computer scientist. While I made an effort to make the text selfcontained and consistent, it may still be somewhat rough around the edges; I hope to continue polishing and adding to it. Comments & constructive criticism are very welcome, and can be sent to rdewolf@cwi.nl
On the Computational Power of Locally Random Reductions
, 2010
"... We study the computational complexity of languages L that have a nonadaptive (t(n), k(n), ɛ(n))locally random reduction (lrr) to any target function g. Such a reduction is a probabilistic polynomialtime oracle algorithm for deciding L using nonadaptive queries to g, and has the following propertie ..."
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We study the computational complexity of languages L that have a nonadaptive (t(n), k(n), ɛ(n))locally random reduction (lrr) to any target function g. Such a reduction is a probabilistic polynomialtime oracle algorithm for deciding L using nonadaptive queries to g, and has the following properties: the number of queries is k(n), the error probability is 1/2 − ɛ(n), and the distribution on t(n)element subsets of queries depends only on n. We show that 1. If a language L has a nonadaptive (1, 2, 1/poly(n))lrr to any target function g whose answers are O(log n)bit long, then L is in PP NP /poly. 2. If a language L has a nonadaptive (n/2 + √ n, n, 1/4)lrr to any boolean target function g, then L is in PP NP /poly. The first result extends and improves a result of Pavan and Vinodchandran [PV08], and provides a partial answer to a question posed in [FS92]. Our proofs use quantum computational techniques which are inspired from those in [KdW04].