Quantum Interactive Proofs and the Complexity of Separability Testing
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Randomised Computation 1 TM Taking Advices 2 Karp-Lipton Theorem
INFR11102: Computational Complexity 29/10/2019 Lecture 13: More on circuit models; Randomised Computation Lecturer: Heng Guo 1 TM taking advices An alternative way to characterize P=poly is via TMs that take advices. Definition 1. For functions F : N ! N and A : N ! N, the complexity class DTime[F ]=A consists of languages L such that there exist a TM with time bound F (n) and a sequence fangn2N of “advices” satisfying: • janj ≤ A(n); • for jxj = n, x 2 L if and only if M(x; an) = 1. The following theorem explains the notation P=poly, namely “polynomial-time with poly- nomial advice”. S c Theorem 1. P=poly = c;d2N DTime[n ]=nd . Proof. If L 2 P=poly, then it can be computed by a family C = fC1;C2; · · · g of Boolean circuits. Let an be the description of Cn, andS the polynomial time machine M just reads 2 c this description and simulates it. Hence L c;d2N DTime[n ]=nd . For the other direction, if a language L can be computed in polynomial-time with poly- nomial advice, say by TM M with advices fang, then we can construct circuits fDng to simulate M, as in the theorem P ⊂ P=poly in the last lecture. Hence, Dn(x; an) = 1 if and only if x 2 L. The final circuit Cn just does exactly what Dn does, except that Cn “hardwires” the advice an. Namely, Cn(x) := Dn(x; an). Hence, L 2 P=poly. 2 Karp-Lipton Theorem Dick Karp and Dick Lipton showed that NP is unlikely to be contained in P=poly [KL80]. -
Quantum Cryptography: from Theory to Practice
Quantum cryptography: from theory to practice by Xiongfeng Ma A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto Copyright °c 2008 by Xiongfeng Ma Abstract Quantum cryptography: from theory to practice Xiongfeng Ma Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto 2008 Quantum cryptography or quantum key distribution (QKD) applies fundamental laws of quantum physics to guarantee secure communication. The security of quantum cryptog- raphy was proven in the last decade. Many security analyses are based on the assumption that QKD system components are idealized. In practice, inevitable device imperfections may compromise security unless these imperfections are well investigated. A highly attenuated laser pulse which gives a weak coherent state is widely used in QKD experiments. A weak coherent state has multi-photon components, which opens up a security loophole to the sophisticated eavesdropper. With a small adjustment of the hardware, we will prove that the decoy state method can close this loophole and substantially improve the QKD performance. We also propose a few practical decoy state protocols, study statistical fluctuations and perform experimental demonstrations. Moreover, we will apply the methods from entanglement distillation protocols based on two-way classical communication to improve the decoy state QKD performance. Fur- thermore, we study the decoy state methods for other single photon sources, such as triggering parametric down-conversion (PDC) source. Note that our work, decoy state protocol, has attracted a lot of scienti¯c and media interest. The decoy state QKD becomes a standard technique for prepare-and-measure QKD schemes. -
Pilot Quantum Error Correction for Global
Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Theses and Dissertations Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes Thesis by James William Harrington Advisor John Preskill In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 17, 2004) ii c 2004 James William Harrington All rights Reserved iii Acknowledgements I can do all things through Christ, who strengthens me. Phillipians 4:13 (NKJV) I wish to acknowledge first of all my parents, brothers, and grandmother for all of their love, prayers, and support. Thanks to my advisor, John Preskill, for his generous support of my graduate studies, for introducing me to the studies of quantum error correction, and for encouraging me to pursue challenging questions in this fascinating field. Over the years I have benefited greatly from stimulating discussions on the subject of quantum information with Anura Abeyesinge, Charlene Ahn, Dave Ba- con, Dave Beckman, Charlie Bennett, Sergey Bravyi, Carl Caves, Isaac Chenchiah, Keng-Hwee Chiam, Richard Cleve, John Cortese, Sumit Daftuar, Ivan Deutsch, Andrew Doherty, Jon Dowling, Bryan Eastin, Steven van Enk, Chris Fuchs, Sho- hini Ghose, Daniel Gottesman, Ted Harder, Patrick Hayden, Richard Hughes, Deborah Jackson, Alexei Kitaev, Greg Kuperberg, Andrew Landahl, Chris Lee, Debbie Leung, Carlos Mochon, Michael Nielsen, Smith Nielsen, Harold Ollivier, Tobias Osborne, Michael Postol, Philippe Pouliot, Marco Pravia, John Preskill, Eric Rains, Robert Raussendorf, Joe Renes, Deborah Santamore, Yaoyun Shi, Pe- ter Shor, Marcus Silva, Graeme Smith, Jennifer Sokol, Federico Spedalieri, Rene Stock, Francis Su, Jacob Taylor, Ben Toner, Guifre Vidal, and Mas Yamada. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Arxiv:1506.08857V1 [Quant-Ph] 29 Jun 2015
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices Dardo Goyeneche National Quantum Information Center of Gda´nsk,81-824 Sopot, Poland and Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk,80-233 Gda´nsk,Poland Daniel Alsina Dept. Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Spain. Jos´e I. Latorre Dept. Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Spain. and Center for Theoretical Physics, MIT, USA Arnau Riera ICFO-Institut de Ciencies Fotoniques, Castelldefels (Barcelona), Spain Karol Zyczkowski_ Institute of Physics, Jagiellonian University, Krak´ow,Poland and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland (Dated: June 29, 2015) Absolutely Maximally Entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible partitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME, namely their relation to tensors that can be understood as unitary transformations in every of its bi-partitions. We call this property multi-unitarity. I. INTRODUCTION entropy S(ρ) = −Tr(ρ log ρ) ; (1) A complete characterization, classification and it is possible to show [1, 2] that the average entropy quantification of entanglement for quantum states re- of the reduced state σ = TrN=2j ih j to N=2 qubits mains an unfinished long-term goal in Quantum Infor- reads: N mation theory. -
Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians
Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians Na¨ıri Usher,1, ∗ Matty J. Hoban,2, 3 and Dan E. Browne1 1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom. 2University of Oxford, Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. 3School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK A central result in the study of Quantum Hamiltonian Complexity is that the k-local hamilto- nian problem is QMA-complete [1]. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models based on measurement instead of unitary evolution, further- more we can use post-selected measurement as an additional computational tool. In this work, we generalise Kitaev's construction to allow for non-unitary evolution including post-selection. Fur- thermore, we consider a type of post-selection under which the construction is consistent, which we call tame post-selection. We consider the computational complexity consequences of this construc- tion and then consider how the probability of an event upon which we are post-selecting affects the gap between the ground state energy and the energy of the first excited state of its corresponding Hamiltonian. -
Lecture 10: Learning DNF, AC0, Juntas Feb 15, 2007 Lecturer: Ryan O’Donnell Scribe: Elaine Shi
Analysis of Boolean Functions (CMU 18-859S, Spring 2007) Lecture 10: Learning DNF, AC0, Juntas Feb 15, 2007 Lecturer: Ryan O’Donnell Scribe: Elaine Shi 1 Learning DNF in Almost Polynomial Time From previous lectures, we have learned that if a function f is ǫ-concentrated on some collection , then we can learn the function using membership queries in poly( , 1/ǫ)poly(n) log(1/δ) time.S |S| O( w ) In the last lecture, we showed that a DNF of width w is ǫ-concentrated on a set of size n ǫ , and O( w ) concluded that width-w DNFs are learnable in time n ǫ . Today, we shall improve this bound, by showing that a DNF of width w is ǫ-concentrated on O(w log 1 ) a collection of size w ǫ . We shall hence conclude that poly(n)-size DNFs are learnable in almost polynomial time. Recall that in the last lecture we introduced H˚astad’s Switching Lemma, and we showed that 1 DNFs of width w are ǫ-concentrated on degrees up to O(w log ǫ ). Theorem 1.1 (Hastad’s˚ Switching Lemma) Let f be computable by a width-w DNF, If (I, X) is a random restriction with -probability ρ, then d N, ∗ ∀ ∈ d Pr[DT-depth(fX→I) >d] (5ρw) I,X ≤ Theorem 1.2 If f is a width-w DNF, then f(U)2 ǫ ≤ |U|≥OX(w log 1 ) ǫ b O(w log 1 ) To show that a DNF of width w is ǫ-concentrated on a collection of size w ǫ , we also need the following theorem: Theorem 1.3 If f is a width-w DNF, then 1 |U| f(U) 2 20w | | ≤ XU b Proof: Let (I, X) be a random restriction with -probability 1 . -
On the Power of Quantum Fourier Sampling
On the Power of Quantum Fourier Sampling Bill Fefferman∗ Chris Umansy Abstract A line of work initiated by Terhal and DiVincenzo [TD02] and Bremner, Jozsa, and Shepherd [BJS10], shows that restricted classes of quantum computation can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov [AA13] take this further by considering a distribution that can be sampled ef- ficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled classically within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled exactly by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson-Arkhipov conjectures. In particular, we show a general class of quantumly sampleable distributions each of which is based on an “Efficiently Specifiable” polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, as well as many other familiar #P-hard polynomials. Since our distribution likely requires the full power of universal quantum computation, while the Aaronson-Arkhipov distribution uses only linear optical quantum computation with noninteracting bosons, why is this result interesting? We can think of at least three reasons: 1. Since the conjectures required in [AA13] have not yet been proven, it seems worthwhile to weaken them as much as possible. -
Interactive Proofs 1 1 Pspace ⊆ IP
CS294: Probabilistically Checkable and Interactive Proofs January 24, 2017 Interactive Proofs 1 Instructor: Alessandro Chiesa & Igor Shinkar Scribe: Mariel Supina 1 Pspace ⊆ IP The first proof that Pspace ⊆ IP is due to Shamir, and a simplified proof was given by Shen. These notes discuss the simplified version in [She92], though most of the ideas are the same as those in [Sha92]. Notes by Katz also served as a reference [Kat11]. Theorem 1 ([Sha92]) Pspace ⊆ IP. To show the inclusion of Pspace in IP, we need to begin with a Pspace-complete language. 1.1 True Quantified Boolean Formulas (tqbf) Definition 2 A quantified boolean formula (QBF) is an expression of the form 8x19x28x3 ::: 9xnφ(x1; : : : ; xn); (1) where φ is a boolean formula on n variables. Note that since each variable in a QBF is quantified, a QBF is either true or false. Definition 3 tqbf is the language of all boolean formulas φ such that if φ is a formula on n variables, then the corresponding QBF (1) is true. Fact 4 tqbf is Pspace-complete (see section 2 for a proof). Hence to show that Pspace ⊆ IP, it suffices to show that tqbf 2 IP. Claim 5 tqbf 2 IP. In order prove claim 5, we will need to present a complete and sound interactive protocol that decides whether a given QBF is true. In the sum-check protocol we used an arithmetization of a 3-CNF boolean formula. Likewise, here we will need a way to arithmetize a QBF. 1.2 Arithmetization of a QBF We begin with a boolean formula φ, and we let n be the number of variables and m the number of clauses of φ. -
Interactive Proofs for Quantum Computations
Innovations in Computer Science 2010 Interactive Proofs For Quantum Computations Dorit Aharonov Michael Ben-Or Elad Eban School of Computer Science, The Hebrew University of Jerusalem, Israel [email protected] [email protected] [email protected] Abstract: The widely held belief that BQP strictly contains BPP raises fundamental questions: Upcoming generations of quantum computers might already be too large to be simulated classically. Is it possible to experimentally test that these systems perform as they should, if we cannot efficiently compute predictions for their behavior? Vazirani has asked [21]: If computing predictions for Quantum Mechanics requires exponential resources, is Quantum Mechanics a falsifiable theory? In cryptographic settings, an untrusted future company wants to sell a quantum computer or perform a delegated quantum computation. Can the customer be convinced of correctness without the ability to compare results to predictions? To provide answers to these questions, we define Quantum Prover Interactive Proofs (QPIP). Whereas in standard Interactive Proofs [13] the prover is computationally unbounded, here our prover is in BQP, representing a quantum computer. The verifier models our current computational capabilities: it is a BPP machine, with access to few qubits. Our main theorem can be roughly stated as: ”Any language in BQP has a QPIP, and moreover, a fault tolerant one” (providing a partial answer to a challenge posted in [1]). We provide two proofs. The simpler one uses a new (possibly of independent interest) quantum authentication scheme (QAS) based on random Clifford elements. This QPIP however, is not fault tolerant. Our second protocol uses polynomial codes QAS due to Ben-Or, Cr´epeau, Gottesman, Hassidim, and Smith [8], combined with quantum fault tolerance and secure multiparty quantum computation techniques.