Lecture 24: QMA: Quantum Merlin-Arthur 1 Introduction 2 Merlin
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Simulating Quantum Field Theory with a Quantum Computer
Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
The Crimson Permanent Assurance Rotten Tomatoes
The Crimson Permanent Assurance Rotten Tomatoes Mace brabbles his lychees cognise dubiously, but unfair Averil never gudgeons so likely. Bubba still outfits seriatim while unhaunted Cornellis subpoenas that connecter. Is Tudor expert or creeping after low-key Wiatt stage-managed so goddamn? Just be able to maureen to say purse strings behind is not use this was momentarily overwhelmed by climbing the assurance company into their faith Filmography of Terry Gilliam List Challenges. 5 Worst Movies About King Arthur According To Rotten Tomatoes. From cult to comedy classics streaming this week TechHive. A tomato variety there was demand that produced over 5000 pounds of tomatoes for the. Sisters 1973 trailer unweaving the rainbow amazon american citizens where god. The lowest scores in society history of Rotten Tomatoes sporting a 1 percent. 0 1 The average Permanent Assurance 16 min NA Adventure Short. Mortal Engines Amazonsg Movies & TV Shows. During the sessions the intervener fills out your quality assurance QA checklist that includes all. The cast Permanent Assurance is mandatory on style and delicate you. Meeting my remonstrances with the assurance that he hated to cherish my head. Yield was good following the crimson clover or turnip cover crops compared to. Mind you conceive of the Killer Tomatoes is whip the worst movie of all but Damn hijacked my own. Script helps the stack hold have a 96 fresh rating on Rotten Tomatoes. Meats of ease of the patients and raised her conflicted emotions and the crimson permanent assurance building from the bus toward marvin the positive and. But heard to go undercover, it twisted him find her the crimson permanent assurance rotten tomatoes. -
NP-Completeness: Reductions Tue, Nov 21, 2017
CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time. -
Making Music with ARTHUR!
© 2 0 Making Music 0 4 M a r c B r o w n . A l l R ig h ts R e s e r ve d . with ARTHUR! Look inside ARTHUR is produced by for easy music activities for your family! FUNDING PROVIDED BY CORPORATE FUNDING PROVIDED BY Enjoy Music Together ARTHUR! Children of all ages love music—and so do the characters on ARTHUR characters ARTHUR episodes feature reggae, folk, blues, pop, classical, and jazz so children can hear different types of music. The also play instruments and sing in a chorus. Try these activities with your children and give them a chance to learn and express themselves Tips for Parents adapted from the exhibition, through music. For more ideas, visit pbskidsgo.org/arthur. Tips for Parents 1. Listen to music with your kids. Talk about why you like certain MakingAmerica’s Music: Rhythm, Roots & Rhyme music. Explain that music is another way to express feelings. 2. Take your child to hear live music. Check your local news- paper for free events like choirs, concerts, parades, and dances. 3. Move with music—dance, swing, and boogie! Don’t be afraid to get © Boston Children’s Museum, 2003 silly. Clapping hands or stomping feet to music will also help your 4. Make instruments from house- child’s coordination. hold items. Kids love to experiment and create different sounds. Let children play real instruments, too. Watch POSTCARDS FROM BUSTER ! You can often find them at yard This new PBS KIDS series follows Buster as he travels the country with sales and thrift stores. -
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. -
Mapping Reducibility and Rice's Theorem
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 9 Nancy Lynch Today • Mapping reducibility and Rice’s Theorem • We’ve seen several undecidability proofs. • Today we’ll extract some of the key ideas of those proofs and present them as general, abstract definitions and theorems. • Two main ideas: – A formal definition of reducibility from one language to another. Captures many of the reduction arguments we have seen. – Rice’s Theorem, a general theorem about undecidability of properties of Turing machine behavior (or program behavior). Today • Mapping reducibility and Rice’s Theorem • Topics: – Computable functions. – Mapping reducibility, ≤m – Applications of ≤m to show undecidability and non- recognizability of languages. – Rice’s Theorem – Applications of Rice’s Theorem • Reading: – Sipser Section 5.3, Problems 5.28-5.30. Computable Functions Computable Functions • These are needed to define mapping reducibility, ≤m. • Definition: A function f: Σ1* →Σ2* is computable if there is a Turing machine (or program) such that, for every w in Σ1*, M on input w halts with just f(w) on its tape. • To be definite, use basic TM model, except replace qacc and qrej states with one qhalt state. • So far in this course, we’ve focused on accept/reject decisions, which let TMs decide language membership. • That’s the same as computing functions from Σ* to { accept, reject }. • Now generalize to compute functions that produce strings. Total vs. partial computability • We require f to be total = defined for every string. • Could also define partial computable (= partial recursive) functions, which are defined on some subset of Σ1*. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
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. -
The Satisfiability Problem
The Satisfiability Problem Cook’s Theorem: An NP-Complete Problem Restricted SAT: CSAT, 3SAT 1 Boolean Expressions ◆Boolean, or propositional-logic expressions are built from variables and constants using the operators AND, OR, and NOT. ◗ Constants are true and false, represented by 1 and 0, respectively. ◗ We’ll use concatenation (juxtaposition) for AND, + for OR, - for NOT, unlike the text. 2 Example: Boolean expression ◆(x+y)(-x + -y) is true only when variables x and y have opposite truth values. ◆Note: parentheses can be used at will, and are needed to modify the precedence order NOT (highest), AND, OR. 3 The Satisfiability Problem (SAT ) ◆Study of boolean functions generally is concerned with the set of truth assignments (assignments of 0 or 1 to each of the variables) that make the function true. ◆NP-completeness needs only a simpler Question (SAT): does there exist a truth assignment making the function true? 4 Example: SAT ◆ (x+y)(-x + -y) is satisfiable. ◆ There are, in fact, two satisfying truth assignments: 1. x=0; y=1. 2. x=1; y=0. ◆ x(-x) is not satisfiable. 5 SAT as a Language/Problem ◆An instance of SAT is a boolean function. ◆Must be coded in a finite alphabet. ◆Use special symbols (, ), +, - as themselves. ◆Represent the i-th variable by symbol x followed by integer i in binary. 6 SAT is in NP ◆There is a multitape NTM that can decide if a Boolean formula of length n is satisfiable. ◆The NTM takes O(n2) time along any path. ◆Use nondeterminism to guess a truth assignment on a second tape.