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Testing the Quantum-Classical Boundary and Dimensionality of Quantum Systems
Testing the Quantum-Classical Boundary and Dimensionality of Quantum Systems Poh Hou Shun Centre for Quantum Technologies National University of Singapore This dissertation is submitted for the degree of Doctor of Philosophy September 2015 Acknowledgements No journey of scientific discovery is ever truly taken alone. Every step along the way, we encounter people who are a great source of encouragement, guidance, inspiration, joy, and support to us. The journey I have embarked upon during the course of this project is no exception. I would like to extend my gratitude to my project partner on many occasion during the past 5 years, Ng Tien Tjeun. His humorous take on various matters ensures that there is never a dull moment in any late night lab work. A resounding shout-out to the ‘elite’ mem- bers of 0205 (our office), Tan Peng Kian, Shi Yicheng, and Victor Javier Huarcaya Azanon for their numerous discourses into everything under the sun, some which are possibly work related. Thank for tolerating my borderline hoarding behavior and the frequently malfunc- tioning door? I would like to thank Alessandro Ceré for his invaluable inputs on the many pesky problems that I had with data processing. Thanks for introducing me to world of Python programming. Now there is something better than Matlab? A big thanks also goes out to all of my other fellow researchers and colleagues both in the Quantum Optics group and in CQT. They are a source of great inspiration, support, and joy during my time in the group. Special thanks to my supervisor, Christian Kurtsiefer for his constant guidance on and off the project over the years. -
Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems*
Electronic Colloquium on Computational Complexity, Report No. 75 (2019) Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems* Lijie Chen Dylan M. McKay Cody D. Murray† R. Ryan Williams MIT MIT MIT Abstract A frontier open problem in circuit complexity is to prove PNP 6⊂ SIZE[nk] for all k; this is a neces- NP sary intermediate step towards NP 6⊂ P=poly. Previously, for several classes containing P , including NP NP NP , ZPP , and S2P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove k C 6⊂ SIZE[n ] for all k, we show that C ⊂ P=poly implies a “collapse” D = C for some larger class D, where we already know D 6⊂ SIZE[nk] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for PNP that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for PNP are equivalent to fixed-polynomial size circuit lower NP NP k NP bounds for P . That is, P 6⊂ SIZE[n ] for all k if and only if (NP ⊂ P=poly implies PH ⊂ i.o.-P=n ). Next, we present new consequences of the assumption NP ⊂ P=poly, towards proving similar re- sults for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. -
Report to Industry Canada
Report to Industry Canada 2013/14 Annual Report and Final Report for 2008-2014 Granting Period Institute for Quantum Computing University of Waterloo June 2014 1 CONTENTS From the Executive Director 3 Executive Summary 5 The Institute for Quantum Computing 8 Strategic Objectives 9 2008-2014 Overview 10 2013/14 Annual Report Highlights 23 Conducting Research in Quantum Information 23 Recruiting New Researchers 32 Collaborating with Other Researchers 35 Building, Facilities & Laboratory Support 43 Become a Magnet for Highly Qualified Personnel in the Field of Quantum Information 48 Establishing IQC as the Authoritative Source of Insight, Analysis and Commentary on Quantum Information 58 Communications and Outreach 62 Administrative and Technical Support 69 Risk Assessment & Mitigation Strategies 70 Appendix 73 2 From the Executive Director The next great technological revolution – the quantum age “There is a second quantum revolution coming – which will be responsible for most of the key physical technological advances for the 21st Century.” Gerard J. Milburn, Director, Centre for Engineered Quantum Systems, University of Queensland - 2002 There is no doubt now that the next great era in humanity’s history will be the quantum age. IQC was created in 2002 to seize the potential of quantum information science for Canada. IQC’s vision was bold, positioning Canada as a leader in research and providing the necessary infrastructure for Canada to emerge as a quantum industry powerhouse. Today, IQC stands among the top quantum information research institutes in the world. Leaders in all fields of quantum information science come to IQC to participate in our research, share their knowledge and encourage the next generation of scientists to continue on this incredible journey. -
The Computational Complexity of Nash Equilibria in Concisely Represented Games∗
Electronic Colloquium on Computational Complexity, Report No. 52 (2005) The Computational Complexity of Nash Equilibria in Concisely Represented Games∗ Grant R. Schoenebeck y Salil P. Vadhanz May 4, 2005 Abstract Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1 Introduction In recent years, there has been a surge of interest at the interface between computer science and game theory. On one hand, game theory and its notions of equilibria provide a rich framework for modelling the behavior of selfish agents in the kinds of distributed or networked environments that often arise in computer science, and offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior. -
Towards Non-Black-Box Lower Bounds in Cryptography
Towards Non-Black-Box Lower Bounds in Cryptography Rafael Pass?, Wei-Lung Dustin Tseng, and Muthuramakrishnan Venkitasubramaniam Cornell University, {rafael,wdtseng,vmuthu}@cs.cornell.edu Abstract. We consider average-case strengthenings of the traditional assumption that coNP is not contained in AM. Under these assumptions, we rule out generic and potentially non-black-box constructions of various cryptographic primitives (e.g., one-way permutations, collision-resistant hash-functions, constant-round statistically hiding commitments, and constant-round black-box zero-knowledge proofs for NP) from one-way functions, assuming the security reductions are black-box. 1 Introduction In the past four decades, many cryptographic tasks have been put under rigorous treatment in an eort to realize these tasks under minimal assumptions. In par- ticular, one-way functions are widely regarded as the most basic cryptographic primitive; their existence is implied by most other cryptographic tasks. Presently, one-way functions are known to imply schemes such as private-key encryp- tion [GM84,GGM86,HILL99], pseudo-random generators [HILL99], statistically- binding commitments [Nao91], statistically-hiding commitments [NOVY98,HR07] and zero-knowledge proofs [GMW91]. At the same time, some other tasks still have no known constructions based on one-way functions (e.g., key agreement schemes or collision-resistant hash functions). Following the seminal paper by Impagliazzo and Rudich [IR88], many works have addressed this phenomenon by demonstrating black-box separations, which rules out constructions of a cryptographic task using the underlying primitive as a black-box. For instance, Impagliazzo and Rudich rule out black-box con- structions of key-agreement protocols (and thus also trapdoor predicates) from one-way functions; Simon [Sim98] rules out black-box constructions of collision- resistant hash functions from one-way functions. -
Implementation of Elliptic Curve Cryptography in DNA Computing
International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 49 ISSN 2229-5518 Implementation of Elliptic Curve Cryptography in DNA computing 1Sourav Sinha, 2Shubhi Gupta 1Student: Department of Computer Science, 2Assistant Professor Amity University (dit school of Engineering) Greater Noida, India Abstract— DNA computing is the recent and powerful aspect of computer science Iin near future DNA computing is going to replace today’s silicon-based computing. In this paper, we are going to propose a method to implement Elliptic Curve Cryptography in DNA computing. Keywords—DNA computing; Elliptic Curve Cryptography 1. INTRODUCTION (HEADING 1) 2.2 DNA computer The hardware limitation of today’s computer is a barrier The DNA computer is different from Modern day’s in a development in technology. DNA computers, also classic computers. The DNA computer is nothing just a test known as molecular computer, have proven beneficial in tube containing a DNA and solvents for better mobility. such cases. Recent developments have seen massive The operations are done by chemical process and protein progress in technologies that enables a DNA computer to synthesis. DNA does not have any operational capacities, solve Hamiltonian Path Problem [1]. Cryptography and but it can be used as a hard drive to store and transfer data. network security is the most important section in development. Data Encryption Standard(DES) can also be 3. DNA-BASED ELLIPTIC CURVE ALGORITHM broken in a DNA computer, due to its ability to process The Elliptic curve cryptography makes use of an Elliptic parallel[2]. Curve to get the value of Its variable coefficients. -
Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines
Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines Zachary Remscrim Department of Computer Science, The University of Chicago, IL, USA [email protected] Abstract The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the ∗ O(n) language Lpal = {w ∈ {a, b} : w is a palindrome} with bounded error in expected time 2 . We prove that their result cannot be improved upon: a 2QCFA (of any size) cannot recognize o(n) Lpal with bounded error in expected time 2 . This is the first example of a language that can be recognized with bounded error by a 2QCFA in exponential time but not in subexponential time. Moreover, we prove that a quantum Turing machine (QTM) running in space o(log n) and expected n1−Ω(1) time 2 cannot recognize Lpal with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or o(log n)-space QTM that recognizes any language L in terms of a natural “hardness measure” of L. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Theory of computation → Quantum computation theory Keywords and phrases Quantum computation, Lower bounds, Finite automata Digital Object Identifier 10.4230/LIPIcs.ITCS.2021.39 Related Version Full version of the paper https://arxiv.org/abs/2003.09877. -
Many-Body Entanglement in Classical & Quantum Simulators
Many-Body Entanglement in Classical & Quantum Simulators Johnnie Gray A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Physics & Astronomy University College London January 15, 2019 2 3 I, Johnnie Gray, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the work. Abstract Entanglement is not only the key resource for many quantum technologies, but es- sential in understanding the structure of many-body quantum matter. At the interface of these two crucial areas are simulators, controlled systems capable of mimick- ing physical models that might escape analytical tractability. Traditionally, these simulations have been performed classically, where recent advancements such as tensor-networks have made explicit the limitation entanglement places on scalability. Increasingly however, analog quantum simulators are expected to yield deep insight into complex systems. This thesis advances the field in across various interconnected fronts. Firstly, we introduce schemes for verifying and distributing entanglement in a quantum dot simulator, tailored to specific experimental constraints. We then confirm that quantum dot simulators would be natural candidates for simulating many-body localization (MBL) - a recently emerged phenomenon that seems to evade traditional statistical mechanics. Following on from that, we investigate MBL from an entanglement perspective, shedding new light on the nature of the transi- tion to it from a ergodic regime. As part of that investigation we make use of the logarithmic negativity, an entanglement measure applicable to many-body mixed states. -
Circuit Lower Bounds for Merlin-Arthur Classes
Electronic Colloquium on Computational Complexity, Report No. 5 (2007) Circuit Lower Bounds for Merlin-Arthur Classes Rahul Santhanam Simon Fraser University [email protected] January 16, 2007 Abstract We show that for each k > 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA AM ZPPNP , and k . We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the average-case setting, i.e., we show that MA/1 cannot be solved on more than 1/2+1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudo-random generators computable using O(1) bits of advice. 1 Introduction Proving circuit lower bounds within uniform complexity classes is one of the most fundamental and challenging tasks in complexity theory. Apart from clarifying our understanding of the power of non-uniformity, circuit lower bounds have direct relevance to some longstanding open questions. Proving super-polynomial circuit lower bounds for NP would separate P from NP. The weaker result that for each k there is a language in NP which doesn’t have circuits of size nk would separate BPP from NEXP, thus answering an important question in the theory of derandomization. -
Nondeterministic Direct Product Reducations and the Success
Nondeterministic Direct Product Reductions and the Success Probability of SAT Solvers [Preliminary version] Andrew Drucker∗ May 6, 2013 Last updated: May 16, 2013 Abstract We give two nondeterministic reductions which yield new direct product theorems (DPTs) for Boolean circuits. In these theorems one assumes that a target function f is mildly hard against nondeterministic circuits, and concludes that the direct product f ⊗t is extremely hard against (only polynomially smaller) probabilistic circuits. The main advantage of these results compared with previous DPTs is the strength of the size bound in our conclusion. As an application, we show that if NP is not in coNP=poly then, for every PPT algorithm attempting to produce satisfying assigments to Boolean formulas, there are infinitely many instances where the algorithm's success probability is nearly-exponentially small. This furthers a project of Paturi and Pudl´ak[STOC'10]. Contents 1 Introduction 2 1.1 Hardness amplification and direct product theorems..................2 1.2 Our new direct product theorems.............................4 1.3 Application to the success probability of PPT SAT solvers...............6 1.4 Our techniques.......................................7 2 Preliminaries 9 2.1 Deterministic and probabilistic Boolean circuits..................... 10 2.2 Nondeterministic circuits and nondeterministic mappings............... 10 2.3 Direct-product solvers................................... 12 2.4 Hash families and their properties............................ 13 2.5 A general nondeterministic circuit construction..................... 14 ∗School of Mathematics, IAS, Princeton, NJ. Email: [email protected]. This material is based upon work supported by the National Science Foundation under agreements Princeton University Prime Award No. CCF- 0832797 and Sub-contract No. 00001583. -
CURRICULUM VITA N. David Mermin Laboratory of Atomic and Solid State Physics Clark Hall, Cornell University, Ithaca, NY 14853-2501
CURRICULUM VITA N. David Mermin Laboratory of Atomic and Solid State Physics Clark Hall, Cornell University, Ithaca, NY 14853-2501 Born: 30 March 1935, New Haven, Connecticut, USA Education: 1956 A.B., Harvard (Mathematics, summa cum laude) 1957 A.M., Harvard (Physics) 1961 Ph.D., Harvard (Physics) Positions: 1961 - 1963 NSF Postdoctoral Fellow, University of Birmingham, England 1963 - 1964 Postdoctoral Associate, University of California, San Diego 1964 - 1967 Assistant Professor, Cornell University 1967 - 1972 Associate Professor, Cornell University 1972 - 1990 Professor, Cornell University 1984 - 1990 Director, Laboratory of Atomic and Solid State Physics 1990 - 2006 Horace White Professor of Physics, Cornell University 2006 - Horace White Professor of Physics Emeritus, Cornell University Visiting Positions and Lecturerships: 1970 - 1971 Visiting Professor, Instituto di Fisica \G. Marconi," Rome 1978 - 1979 Senior Visiting Fellow, University of Sussex 1980 Morris Loeb Lecturer, Harvard University 1981 Emil Warburg Professor, University of Bayreuth 1982 Phillips Lecturer, Haverford College 1982 Japan Association for the Advancement of Science Fellow, Nagoya 1984 Walker-Ames Professor, University of Washington 1987 Welch Lecturer, University of Toronto 1990 Sargent Lecturer, Queens University, Kingston Ontario 1991 Joseph Wunsch Lecturer, the Technion, Haifa 1993 Feenberg Lecturer, Washington University, St. Louis 1993 Guptill Lecturer, Dalhousie University, Halifax 1994 Chesley Lecturer, Carleton College 1995 Lorentz Professor, University -
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Arxiv:1708.06101V1 [Quant-Ph] 21 Aug 2017 High-Dimensional OAM States 10 B
Twisted Photons: New Quantum Perspectives in High Dimensions Manuel Erhard,1, 2, ∗ Robert Fickler,3, y Mario Krenn,1, 2, z and Anton Zeilinger1, 2, x 1Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. 2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. 3Department of Physics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada. CONTENTS E. Stronger Violations of Quantum Mechanical vs. Local Realistic Theories 11 I. General Introduction 1 VI. Final remark 11 II. OAM of photons 2 References 11 III. Advantages of Higher Dimensional Quantum Systems 3 A. Higher Information Capacity 3 I. GENERAL INTRODUCTION B. Enhanced robustness against eavesdropping and quantum cloning 3 C. Quantum Communication without Quantum information science and quantum in- monitoring signal disturbance 4 formation technology have seen a virtual explosion D. Larger Violation of Local-Realistic world-wide. It is all based on the observation that Theories and its advantages in Quantum fundamental quantum phenomena on the individual Communication 4 particle or system-level lead to completely novel ways E. Quantum Computation with QuDits 5 of encoding, processing and transmitting information. Quantum mechanics, a child of the first third of the IV. Recent Developments in High Dimensional 20th century, has found numerous realizations and Quantum Information with OAM 5 technical applications, much more than was thought A. Creation of High Dimensional at the beginning. Decades later, it became possible Entanglement 5 to do experiments with individual quantum particles B. Unitary Transformations 6 and quantum systems.