Entanglement in Quantum Proofs
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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/. -
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. -
Entangled Many-Body States As Resources of Quantum Information Processing
ENTANGLED MANY-BODY STATES AS RESOURCES OF QUANTUM INFORMATION PROCESSING LI YING A thesis submitted for the Degree of Doctor of Philosophy CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. LI YING 23 July 2013 Acknowledgments I am very grateful to have spent about four years at CQT working with Leong Chuan Kwek. He always brings me new ideas in science and has helped me to establish good collaborative relationships with other scien- tists. Kwek helped me a lot in my life. I am also very grateful to Simon C. Benjamin. He showd me how to do high quality researches in physics. Simon also helped me to improve my writing and presentation. I hope to have fruitful collaborations in the near future with Kwek and Simon. For my project about the ground-code MBQC (Chapter2), I am thank- ful to Tzu-Chieh Wei, Daniel E. Browne and Robert Raussendorf. In one afternoon, Tzu-Chieh showed me the idea of his recent paper in this topic in the quantum cafe, which encouraged me to think about the ground- code MBQC. Dan and Robert have a high level of comprehension on the subject of the MBQC. And we had some very interesting discussions and communications. I am grateful to Sean D. -
Distribution of Units of Real Quadratic Number Fields
Y.-M. J. Chen, Y. Kitaoka and J. Yu Nagoya Math. J. Vol. 158 (2000), 167{184 DISTRIBUTION OF UNITS OF REAL QUADRATIC NUMBER FIELDS YEN-MEI J. CHEN, YOSHIYUKI KITAOKA and JING YU1 Dedicated to the seventieth birthday of Professor Tomio Kubota Abstract. Let k be a real quadratic field and k, E the ring of integers and the group of units in k. Denoting by E( ¡ ) the subgroup represented by E of × ¡ ¡ ( k= ¡ ) for a prime ideal , we show that prime ideals for which the order of E( ¡ ) is theoretically maximal have a positive density under the Generalized Riemann Hypothesis. 1. Statement of the result x Let k be a real quadratic number field with discriminant D0 and fun- damental unit (> 1), and let ok and E be the ring of integers in k and the set of units in k, respectively. For a prime ideal p of k we denote by E(p) the subgroup of the unit group (ok=p)× of the residue class group modulo p con- sisting of classes represented by elements of E and set Ip := [(ok=p)× : E(p)], where p is the rational prime lying below p. It is obvious that Ip is inde- pendent of the choice of prime ideals lying above p. Set 1; if p is decomposable or ramified in k, ` := 8p 1; if p remains prime in k and N () = 1, p − k=Q <(p 1)=2; if p remains prime in k and N () = 1, − k=Q − : where Nk=Q stands for the norm from k to the rational number field Q. -
QIP 2010 Tutorial and Scientific Programmes
QIP 2010 15th – 22nd January, Zürich, Switzerland Tutorial and Scientific Programmes asymptotically large number of channel uses. Such “regularized” formulas tell us Friday, 15th January very little. The purpose of this talk is to give an overview of what we know about 10:00 – 17:10 Jiannis Pachos (Univ. Leeds) this need for regularization, when and why it happens, and what it means. I will Why should anyone care about computing with anyons? focus on the quantum capacity of a quantum channel, which is the case we understand best. This is a short course in topological quantum computation. The topics to be covered include: 1. Introduction to anyons and topological models. 15:00 – 16:55 Daniel Nagaj (Slovak Academy of Sciences) 2. Quantum Double Models. These are stabilizer codes, that can be described Local Hamiltonians in quantum computation very much like quantum error correcting codes. They include the toric code This talk is about two Hamiltonian Complexity questions. First, how hard is it to and various extensions. compute the ground state properties of quantum systems with local Hamiltonians? 3. The Jones polynomials, their relation to anyons and their approximation by Second, which spin systems with time-independent (and perhaps, translationally- quantum algorithms. invariant) local interactions could be used for universal computation? 4. Overview of current state of topological quantum computation and open Aiming at a participant without previous understanding of complexity theory, we will discuss two locally-constrained quantum problems: k-local Hamiltonian and questions. quantum k-SAT. Learning the techniques of Kitaev and others along the way, our first goal is the understanding of QMA-completeness of these problems. -
Research Statement Bill Fefferman, University of Maryland/NIST
Research Statement Bill Fefferman, University of Maryland/NIST Since the discovery of Shor's algorithm in the mid 1990's, it has been known that quan- tum computers can efficiently solve integer factorization, a problem of great practical relevance with no known efficient classical algorithm [1]. The importance of this result is impossible to overstate: the conjectured intractability of the factoring problem provides the basis for the se- curity of the modern internet. However, it may still be a few decades before we build universal quantum computers capable of running Shor's algorithm to factor integers of cryptographically relevant size. In addition, we have little complexity theoretic evidence that factoring is com- putationally hard. Consequently, Shor's algorithm can only be seen as the first step toward understanding the power of quantum computation, which has become one of the primary goals of theoretical computer science. My research focuses not only on understanding the power of quantum computers of the indefinite future, but also on the desire to develop the foundations of computational complexity to rigorously analyze the capabilities and limitations of present-day and near-term quantum devices which are not yet fully scalable quantum computers. Furthermore, I am interested in using these capabilities and limitations to better understand the potential for cryptography in a fundamentally quantum mechanical world. 1 Comparing quantum and classical nondeterministic computation Starting with the foundational paper of Bernstein and Vazirani it has been conjectured that quantum computers are capable of solving problems whose solutions cannot be found, or even verified efficiently on a classical computer [2]. -
Lecture 16 1 Interactive Proofs
Notes on Complexity Theory Last updated: October, 2011 Lecture 16 Jonathan Katz 1 Interactive Proofs Let us begin by re-examining our intuitive notion of what it means to \prove" a statement. Tra- ditional mathematical proofs are static and are veri¯ed deterministically: the veri¯er checks the claimed proof of a given statement and is either convinced that the statement is true (if the proof is correct) or remains unconvinced (if the proof is flawed | note that the statement may possibly still be true in this case, it just means there was something wrong with the proof). A statement is true (in this traditional setting) i® there exists a valid proof that convinces a legitimate veri¯er. Abstracting this process a bit, we may imagine a prover P and a veri¯er V such that the prover is trying to convince the veri¯er of the truth of some particular statement x; more concretely, let us say that P is trying to convince V that x 2 L for some ¯xed language L. We will require the veri¯er to run in polynomial time (in jxj), since we would like whatever proofs we come up with to be e±ciently veri¯able. A traditional mathematical proof can be cast in this framework by simply having P send a proof ¼ to V, who then deterministically checks whether ¼ is a valid proof of x and outputs V(x; ¼) (with 1 denoting acceptance and 0 rejection). (Note that since V runs in polynomial time, we may assume that the length of the proof ¼ is also polynomial.) The traditional mathematical notion of a proof is captured by requiring: ² If x 2 L, then there exists a proof ¼ such that V(x; ¼) = 1. -
IP=PSPACE. Arthur-Merlin Games
Computational Complexity Theory, Fall 2010 10 November Lecture 18: IP=PSPACE. Arthur-Merlin Games Lecturer: Kristoffer Arnsfelt Hansen Scribe: Andreas Hummelshøj J Update: Ω(n) Last time, we were looking at MOD3◦MOD2. We mentioned that AND required size 2 MOD3◦ MOD2 circuits. We also mentioned, as being open, whether NEXP ⊆ (nonuniform)MOD2 ◦ MOD3 ◦ MOD2. Since 9/11-2010, this is no longer open. Definition 1 ACC0 = class of languages: 0 0 ACC = [m>2ACC [m]; where AC0[m] = class of languages computed by depth O(1) size nO(1) circuits with AND-, OR- and MODm-gates. This is in fact in many ways a natural class of languages, like AC0 and NC1. 0 Theorem 2 NEXP * (nonuniform)ACC . New open problem: Is EXP ⊆ (nonuniform)MOD2 ◦ MOD3 ◦ MOD2? Recap: We defined arithmetization A(φ) of a 3-SAT formula φ: A(xi) = xi; A(xi) = 1 − xi; 3 Y A(l1 _ l2 _ l3) = 1 − (1 − A(li)); i=1 m Y A(c1 ^ · · · ^ cm) = A(cj): j=1 1 1 X X ]φ = ··· P (x1; : : : ; xn);P = A(φ): x1=0 xn=0 1 Sumcheck: Given g(x1; : : : ; xn), K and prime number p, decide if 1 1 X X ··· g(x1; : : : ; xn) ≡ K (mod p): x1=0 xn=0 True Quantified Boolean Formulae: 0 0 Given φ ≡ 9x18x2 ::: 8xnφ (x1; : : : ; xn), where φ is a 3SAT formula, decide if φ is true. Observation: φ true , P1 Q1 P ··· Q1 P (x ; : : : ; x ) > 0, P = A(φ0). x1=0 x2=0 x3 x1=0 1 n Protocol: Can't we just do it analogous to Sumcheck? Id est: remove outermost P, P sends polynomial S, V checks if S(0) + S(1) ≡ K, asks P to prove Q1 P1 ··· Q1 P (a) ≡ S(a), where x2=0 x3=0 xn=0 a 2 f0; 1; : : : ; p − 1g is chosen uniformly at random. -
Manipulating Entanglement Sudden Death in Two Coupled Two-Level
Manipulating entanglement sudden death in two coupled two-level atoms interacting off-resonance with a radiation field: an exact treatment Gehad Sadieka,1, 2 Wiam Al-Drees,3 and M. Sebaweh Abdallahb4 1Department of Applied Physics and Astronomy, University of Sharjah, Sharjah 27272, UAE 2Department of Physics, Ain Shams University, Cairo 11566, Egypt 3Department of Physics and Astronomy, King Saud University, Riyadh 11451, Saudi Arabia 4Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia arXiv:1911.08208v1 [quant-ph] 19 Nov 2019 a Corresponding author: [email protected] b Deceased. 1 Abstract We study a model of two coupled two-level atoms (qubits) interacting off-resonance (at non-zero detuning) with a single mode radiation field. This system is of special interest in the field of quan- tum information processing (QIP) and can be realized in electron spin states in quantum dots or Rydberg atoms in optical cavities and superconducting qubits in linear resonators. We present an exact analytical solution for the time evolution of the system starting from any initial state. Uti- lizing this solution, we show how the entanglement sudden death (ESD), which represents a major threat to QIP, can be efficiently controlled by tuning atom-atom coupling and non-zero detuning. We demonstrate that while one of these two system parameters may not separately affect the ESD, combining the two can be very effective, as in the case of an initial correlated Bell state. However in other cases, such as a W-like initial state, they may have a competing impacts on ESD. Moreover, their combined effect can be used to create ESD in the system as in the case of an anti-correlated initial Bell state. -
Quantum Information Processing II: Implementations
Lecture Quantum Information Processing II: Implementations spring term (FS) 2017 Lectures & Exercises: Andreas Wallraff, Christian Kraglund Andersen, Christopher Eichler, Sebastian Krinner Please take a seat offices:in HPFthe frontD 8/9 (AW),center D part14 (CKA), of the D lecture2 (SK), D hall 17 (SK) ifETH you Hoenggerberg do not mind. email: [email protected] web: www.qudev.ethz.ch moodle: https://moodle-app2.let.ethz.ch/course/view.php?id=3151 What is this lecture about? Introduction to experimental realizations of systems for quantum information processing (QIP) Questions: • How can one use quantum physics to process and communicate information more efficiently than using classical physics only? ► QIP I: Concepts • How does one design, build and operate physical systems for this purpose? ► QIP II: Implementations Is it really interesting? Goals of the Lecture • generally understand how quantum physics is used for – quantum information processing – quantum communication – quantum simulation – quantum sensing • know basic implementations of important quantum algorithms – Deutsch-Josza Algorithm – searching in a database (Grover algorithm) Concept Question You are searching for a specific element in an unsorted database with N entries (e.g. the phone numbers in a regular phone book). How many entries do you have to look at (on average) until you have found the item you were searching for? 1. N^2 2. N Log(N) 3. N 4. N/2 5. Sqrt(N) 6. 1 Goals of the Lecture • generally understand how quantum physics is used for – quantum information