QuIDD-Optimised Quantum Algorithms
|
|
- Silvia Marshall
- 5 years ago
- Views:
Transcription
1 QuIDD-Optimised Quantum Algorithms by S K University of York Computer science 3 rd year project Supervisor: Prof Susan Stepney 03/05/2004 1
2 Project Objectives Investigate the QuIDD optimisation techniques Implement QFT transform step of Shor s factorisation algorithm using these techniques 03/05/2004 2
3 Topics The QuIDD Optimisation techniques Simulation of the QFT step of Shor s Algorithm Design Issues Implementation Results and Analysis Conclusions Future Work 03/05/2004 3
4 What is QuIDD anyway? It s an extension of Algebraic Decision Diagram (ADD), which is itself an extension of Binary Decision Diagram (BDD) More on next slides. BDD MTBDD ROBDD ADD 03/05/2004 4
5 Binary Decision Diagrams (BDD) Direct Acyclic Graph (DAG) Widely used in design and verification of digital systems Powerful representation of Boolean functions of the form f = x0.x1 + x1 Nodes represent function variables Terminals represent function values from the set {0, 1} 03/05/2004 5
6 BDD contd DAG Nodes variables Terminals values = 0 and 1 Solid edge is called the then edge and traversing this edge is same as assigning 1 to a variable represented by that edge Broken edge is known as the else edge, traversing this edge = assign 0 03/05/2004 6
7 ROBDD The original BDD was not canonical Lots of redundancy Bryant introduced reduction rules and variable ordering to make BDDs canonical Reduction rule1: Merge Isomorphic sub graphs. Applying this rule to the graph on slide 6 we get this ==> 03/05/2004 7
8 ROBDD Reduction rule 2 Collapse nodes whose then and else edges point to the same node Apply this rule to the graph on the right; to get the graph shown on the next slide 03/05/2004 8
9 ROBDD One gets canonical representation of f after the reductions The value of the function f depends only on the value of X1 03/05/2004 9
10 MTBDD Adopted the BDD; introduced multi terminal nodes Instead of two nodes representing 0 and 1 we can have multi-terminals terminals representing numbers from 0 to (2^n)- 1 Introduced algorithms for arithmetic symbolic computation on BDDs But limited to special class of matrices which arise in Walsh transform 03/05/
11 Algebraic Decision Diagrams (ADD) Bahar et all took the MTBDD and introduced broad symbolic algorithms Called it ADDs More efficient algorithms, however, remains exponential in worst case Efficiency dependent on the seize of ROBDD, which in turn depends on Variable ordering Finding optimal variable ordering is NP-complete However, efficient heuristics have been developed for many cases of interest Robust and well documented ADD based package - CUDD 03/05/
12 Quantum Information Decision Diagrams (QuIDDs) Viamontes et al observed that tensor products induces regularity into matrices representing quantum operators, which lend themselves to the ADD/BDD structure Restricted the values of terminals to set of complex numbers Used clever technique for implementing complex terminals => terminal nodes as array indices Designed a package called QUIDD Pro to simulate Quantum computations classically Its underlying package is a package called CUDD which implements ADDs 03/05/
13 Complexity Analysis ( A B) ), where a binary operator Tensor product O(ab), a and b are number of ADD nodes of A and B respectively Matrix multiplication O(ab)^2 03/05/
14 What has been done with QuIDDs so far Simulated Grover s search algorithm On quantum computers Ο(N^1/2) Classically O(N/2) = quadratic speed up 03/05/
15 Grover s algorithm decomposed Quantum circuit representation involves Five major components: An Oracle Controlled phase shift gates Sets of Hadamard gates The data qubits An Oracle Qubit 03/05/
16 Hadamard operator Tensor product of Hadamard operators => lots of repetition of equivalent blocks Recursive block sub-structure, structure, which means huge compression in the QuIDDs representation 03/05/
17 Complexity- Grover s Init = > 1> = implemented by n-n tensor products O(n) memory and space complexity N Hadamard gates applied to the above state vector => matrix vector multiplication =>O(n^4) N-data qubits => O(1) By Similar analysis on the rest of the gates they O(A^16 n^14), A the size of the oracle 03/05/
18 Comments This is just after one Grover s iteration But we need O(N^1/2) iteration to boost the probability of finding the key N = 2^n I don t see a polynomial algorithm here, but then this is the worst case!? 03/05/
19 Can QuIDDs simulate Shor s algorithm efficiently? Factorisation can be reduced to Phase Estimation Phase estimation needs: The Quantum Fourier Transform 03/05/
20 The bottlenecks of SFA Quantum Fourier Transform (QFT) Modular Exponentiation The rest can be done efficiently classically If the QFT step can be simulated classically, efficiently then we can implement Shor s algorithm efficiently on classical computers 03/05/
21 Focus on QFT Naïve approach direct implementation of the 2^n x 2^n matrix O(2^2n) Attempted to simulate the QFT circuit by mimicking the circuit on QuIDD Pro Because of QuIDD pro s s limited primitive programming facilities, code become too large; moreover, couldn t t scale more than 7 qubits Investigated the definition for a better design 03/05/
22 Implementation Found a better way of implementing the QFT Now QuIDD Pro s primitive operators could be used to program the design in a simple and scalable way Managed to simulate up to 13 qubits of the QFT step of Shor s algorithm 32 qubits QFT step simulator is still running for over 45 hours 03/05/
23 Emperical results I QFT step simulator run-time in seconds qubits (n) output1 output2 output3 output4 output5 output6 output7 output8 output9 average /05/
24 Emperical Result II QFT step simulator Peak Memory Usage (MB) qubits (n) output1 output2 output3 output4 output5 output6 output7 output8 output9 average /05/
25 Clearly exponential RUNTIME runtime in seconds Series number of quibits 03/05/
26 Fairly good peak memory usage less than 0.5 MB! Mean Peak Memory Usage peak memory usage in MB Series number of quibits 03/05/
27 Current Work Working to be able to factor 77 Finding A way around a bug causing a measurement operator crush 03/05/
28 Conclusions Simulation of QFT on QuIDD Pro still requires exponential runtime resources Up to 13-qubits, peak memory usage of 0.5 MB is not bad at all! The constantly changing phase factor of the control phase shift gate appearing in the QFT circuit makes compression difficult for QuIDDs Don t know yet for what memory complexity will be for qubits >13 03/05/
29 Future Works QuIDD Pro implementation provides modest memory complexity for the QFT step of Shor s Simulation of the QFT Pro on a single basis state of 21-qubits are relatively faster than other implementation based (e.g. on MATLAB ) May be finding a way of parallelising this QFT step simulation and executing them in parallel, perhaps over clusters of processors, may reduce run time for large qubits significantly 03/05/
30 Future Work QuIDD Pro precision for larger qubits is not reliable, this is worsen by division by increasingly small numbers in the controlled phase shift gate of the QFT circuit It appeared to have impeded reductions of QuIDD nodes Approximate QFT implementation could ease this problem 03/05/
Binary Decision Diagrams
Binary Decision Diagrams Logic Circuits Design Seminars WS2010/2011, Lecture 2 Ing. Petr Fišer, Ph.D. Department of Digital Design Faculty of Information Technology Czech Technical University in Prague
More informationImproving Gate-Level Simulation of Quantum Circuits 1
Quantum Information Processing, Vol. 2, No. 5, October 2003 (# 2004) Improving Gate-Level Simulation of Quantum Circuits 1 George F. Viamontes, 2 Igor L. Markov, 2 and John P. Hayes 2 Received September
More informationGate-Level Simulation of Quantum Circuits
Gate-Level Simulation of Quantum ircuits George F. Viamontes, Manoj Rajagopalan, gor L. Markov and John P. ayes The University of Michigan, Advanced omputer Architecture Laboratory Ann Arbor, M 489-, USA
More informationGraph-based simulation of quantum computation in the density matrix representation
Graph-based simulation of quantum computation in the density matrix representation George F. Viamontes, Igor L. Markov, John P. Hayes University of Michigan, Advanced Computer Architecture Laboratory,
More informationChecking Equivalence of Quantum Circuits and States
Checking Equivalence of Quantum Circuits and States George F Viamontes, Igor L Markov, and John P Hayes Lockheed Martin ATL University of Michigan 3 Executive Campus Advanced Computer Architecture Lab
More informationBinary Decision Diagrams and Symbolic Model Checking
Binary Decision Diagrams and Symbolic Model Checking Randy Bryant Ed Clarke Ken McMillan Allen Emerson CMU CMU Cadence U Texas http://www.cs.cmu.edu/~bryant Binary Decision Diagrams Restricted Form of
More informationEECS 219C: Computer-Aided Verification Boolean Satisfiability Solving III & Binary Decision Diagrams. Sanjit A. Seshia EECS, UC Berkeley
EECS 219C: Computer-Aided Verification Boolean Satisfiability Solving III & Binary Decision Diagrams Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: Lintao Zhang Announcement Project proposals due
More informationIs Quantum Search Practical?
DARPA Is Quantum Search Practical? George F. Viamontes Igor L. Markov John P. Hayes University of Michigan, EECS Outline Motivation Background Quantum search Practical Requirements Quantum search versus
More informationBinary Decision Diagrams
Binary Decision Diagrams Sungho Kang Yonsei University Outline Representing Logic Function Design Considerations for a BDD package Algorithms 2 Why BDDs BDDs are Canonical (each Boolean function has its
More informationQUANTUM APPROACHES TO LOGIC CIRCUIT SYNTHESIS AND TESTING
AFRL-IF-RS-TR-6-6 Final Technical Report June 6 QUANTUM APPROACHES TO LOGIC CIRCUIT SYNTHESIS AND TESTING University of Michigan Sponsored by Defense Advanced Research Projects Agency DARPA Order No. L486
More informationBinary Decision Diagrams. Graphs. Boolean Functions
Binary Decision Diagrams Graphs Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Part II Emma Strubell http://cs.umaine.edu/~ema/quantum_tutorial.pdf April 13, 2011 Overview Outline Grover s Algorithm Quantum search A worked example Simon s algorithm
More informationBinary Decision Diagrams
Binary Decision Diagrams Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant in 1986.
More informationFast Equivalence-checking for Quantum Circuits
Fast Equivalence-checking for Quantum Circuits Shigeru Yamashita Ritsumeikan University 1-1-1 Noji igashi, Kusatsu, Shiga 525-8577, Japan Email: ger@cs.ritsumei.ac.jp Igor L. Markov University of Michigan
More informationBoolean decision diagrams and SAT-based representations
Boolean decision diagrams and SAT-based representations 4th July 200 So far we have seen Kripke Structures 2 Temporal logics (and their semantics over Kripke structures) 3 Model checking of these structures
More informationThe quantum threat to cryptography
The quantum threat to cryptography Ashley Montanaro School of Mathematics, University of Bristol 20 October 2016 Quantum computers University of Bristol IBM UCSB / Google University of Oxford Experimental
More informationSimon s algorithm (1994)
Simon s algorithm (1994) Given a classical circuit C f (of polynomial size, in n) for an f : {0, 1} n {0, 1} n, such that for a certain s {0, 1} n \{0 n }: x, y {0, 1} n (x y) : f (x) = f (y) x y = s with
More informationCOMPRESSED STATE SPACE REPRESENTATIONS - BINARY DECISION DIAGRAMS
QUALITATIVE ANALYIS METHODS, OVERVIEW NET REDUCTION STRUCTURAL PROPERTIES COMPRESSED STATE SPACE REPRESENTATIONS - BINARY DECISION DIAGRAMS LINEAR PROGRAMMING place / transition invariants state equation
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 1: Quantum circuits and the abelian QFT
Quantum algorithms (CO 78, Winter 008) Prof. Andrew Childs, University of Waterloo LECTURE : Quantum circuits and the abelian QFT This is a course on quantum algorithms. It is intended for graduate students
More informationChapter 1. Introduction
Chapter 1 Introduction Symbolical artificial intelligence is a field of computer science that is highly related to quantum computation. At first glance, this statement appears to be a contradiction. However,
More informationBinary Decision Diagrams Boolean Functions
Binary Decision Diagrams Representation of Boolean Functions BDDs, OBDDs, ROBDDs Operations Model-Checking over BDDs 72 Boolean functions:b = {0,1}, f :B B B Boolean Functions Boolean expressions: t ::=
More informationSymbolic Model Checking with ROBDDs
Symbolic Model Checking with ROBDDs Lecture #13 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de December 14, 2016 c JPK Symbolic
More informationMulti-Terminal Multi-Valued Decision Diagrams for Characteristic Function Representing Cluster Decomposition
22 IEEE 42nd International Symposium on Multiple-Valued Logic Multi-Terminal Multi-Valued Decision Diagrams for Characteristic Function Representing Cluster Decomposition Hiroki Nakahara, Tsutomu Sasao,
More informationCS 310 Advanced Data Structures and Algorithms
CS 310 Advanced Data Structures and Algorithms Runtime Analysis May 31, 2017 Tong Wang UMass Boston CS 310 May 31, 2017 1 / 37 Topics Weiss chapter 5 What is algorithm analysis Big O, big, big notations
More informationReduced Ordered Binary Decision Diagrams
Reduced Ordered Binary Decision Diagrams Lecture #13 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de June 5, 2012 c JPK Switching
More informationFault Collapsing in Digital Circuits Using Fast Fault Dominance and Equivalence Analysis with SSBDDs
Fault Collapsing in Digital Circuits Using Fast Fault Dominance and Equivalence Analysis with SSBDDs Raimund Ubar, Lembit Jürimägi (&), Elmet Orasson, and Jaan Raik Department of Computer Engineering,
More informationReduced Ordered Binary Decision Diagrams
Reduced Ordered Binary Decision Diagrams Lecture #12 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de December 13, 2016 c JPK
More informationBinary Decision Diagrams
Binary Decision Diagrams An Introduction and Some Applications Manas Thakur PACE Lab, IIT Madras Manas Thakur (IIT Madras) BDDs 1 / 25 Motivating Example Binary decision tree for a truth table Manas Thakur
More informationA High Level Programming Language for Quantum Computing
QUARK QUantum Analysis and Realization Kit A High Level Programming Language for Quantum Computing Team In lexicographical order: Name UNI Role Daria Jung djj2115 Verification and Validation Jamis Johnson
More informationQuantum Phase Estimation using Multivalued Logic
Quantum Phase Estimation using Multivalued Logic Agenda Importance of Quantum Phase Estimation (QPE) QPE using binary logic QPE using MVL Performance Requirements Salient features Conclusion Introduction
More informationAnalysis of Algorithms
October 1, 2015 Analysis of Algorithms CS 141, Fall 2015 1 Analysis of Algorithms: Issues Correctness/Optimality Running time ( time complexity ) Memory requirements ( space complexity ) Power I/O utilization
More information- Why aren t there more quantum algorithms? - Quantum Programming Languages. By : Amanda Cieslak and Ahmana Tarin
- Why aren t there more quantum algorithms? - Quantum Programming Languages By : Amanda Cieslak and Ahmana Tarin Why aren t there more quantum algorithms? there are only a few problems for which quantum
More informationParallelization of the QC-lib Quantum Computer Simulator Library
Parallelization of the QC-lib Quantum Computer Simulator Library Ian Glendinning and Bernhard Ömer September 9, 23 PPAM 23 1 Ian Glendinning / September 9, 23 Outline Introduction Quantum Bits, Registers
More informationShort Course in Quantum Information Lecture 5
Short Course in Quantum Information Lecture 5 Quantum Algorithms Prof. Andrew Landahl University of New Mexico Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html
More information1188 IEEE TRANSACTIONS ON COMPUTERS, VOL. 55, NO. 9, SEPTEMBER 2006
1188 IEEE TRANSACTIONS ON COMPUTERS, VOL. 55, NO. 9, SEPTEMBER 2006 Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs Maciej Ciesielski, Senior Member, IEEE, Priyank
More informationQuantum Searching. Robert-Jan Slager and Thomas Beuman. 24 november 2009
Quantum Searching Robert-Jan Slager and Thomas Beuman 24 november 2009 1 Introduction Quantum computers promise a significant speed-up over classical computers, since calculations can be done simultaneously.
More informationCompute the Fourier transform on the first register to get x {0,1} n x 0.
CS 94 Recursive Fourier Sampling, Simon s Algorithm /5/009 Spring 009 Lecture 3 1 Review Recall that we can write any classical circuit x f(x) as a reversible circuit R f. We can view R f as a unitary
More informationIntroduction The Search Algorithm Grovers Algorithm References. Grovers Algorithm. Quantum Parallelism. Joseph Spring.
Quantum Parallelism Applications Outline 1 2 One or Two Points 3 4 Quantum Parallelism We have discussed the concept of quantum parallelism and now consider a range of applications. These will include:
More informationShor s Prime Factorization Algorithm
Shor s Prime Factorization Algorithm Bay Area Quantum Computing Meetup - 08/17/2017 Harley Patton Outline Why is factorization important? Shor s Algorithm Reduction to Order Finding Order Finding Algorithm
More informationQuantum Computing. 6. Quantum Computer Architecture 7. Quantum Computers and Complexity
Quantum Computing 1. Quantum States and Quantum Gates 2. Multiple Qubits and Entangled States 3. Quantum Gate Arrays 4. Quantum Parallelism 5. Examples of Quantum Algorithms 1. Grover s Unstructured Search
More informationAutomatic Parallelisation of Quantum Circuits Using the Measurement Based Quantum Computing Model
Automatic Parallelisation of Quantum Circuits Using the Measurement Based Quantum Computing Model Einar Pius August 26, 2010 MSc in High Performance Computing The University of Edinburgh Year of Presentation:
More informationQuantum Circuits and Algorithms
Quantum Circuits and Algorithms Modular Arithmetic, XOR Reversible Computation revisited Quantum Gates revisited A taste of quantum algorithms: Deutsch algorithm Other algorithms, general overviews Measurements
More informationLinear Algebra, Boolean Rings and Resolution? Armin Biere. Institute for Formal Models and Verification Johannes Kepler University Linz, Austria
Linear Algebra, Boolean Rings and Resolution? Armin Biere Institute for Formal Models and Verification Johannes Kepler University Linz, Austria ACA 08 Applications of Computer Algebra Symbolic Computation
More informationQuantum Multiple-Valued Decision Diagrams Containing Skipped Variables
Quantum Multiple-Valued Decision Diagrams Containing Skipped Variables DAVID Y. FEINSTEIN 1, MITCHELL A. THORNTON 1 Innoventions, Inc., 1045 Bissonnet Street, Houston, TX, USA Dept. of Computer Science
More informationFormal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
More informationOverview. Discrete Event Systems Verification of Finite Automata. What can finite automata be used for? What can finite automata be used for?
Computer Engineering and Networks Overview Discrete Event Systems Verification of Finite Automata Lothar Thiele Introduction Binary Decision Diagrams Representation of Boolean Functions Comparing two circuits
More informationα x x 0 α x x f(x) α x x α x ( 1) f(x) x f(x) x f(x) α x = α x x 2
Quadratic speedup for unstructured search - Grover s Al- CS 94- gorithm /8/07 Spring 007 Lecture 11 01 Unstructured Search Here s the problem: You are given an efficient boolean function f : {1,,} {0,1},
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationQuantum Error Correcting Codes and Quantum Cryptography. Peter Shor M.I.T. Cambridge, MA 02139
Quantum Error Correcting Codes and Quantum Cryptography Peter Shor M.I.T. Cambridge, MA 02139 1 We start out with two processes which are fundamentally quantum: superdense coding and teleportation. Superdense
More informationPolynomial Methods for Component Matching and Verification
Polynomial Methods for Component Matching and Verification James Smith Stanford University Computer Systems Laboratory Stanford, CA 94305 1. Abstract Component reuse requires designers to determine whether
More informationIntroduction to Quantum Algorithms Part I: Quantum Gates and Simon s Algorithm
Part I: Quantum Gates and Simon s Algorithm Martin Rötteler NEC Laboratories America, Inc. 4 Independence Way, Suite 00 Princeton, NJ 08540, U.S.A. International Summer School on Quantum Information, Max-Planck-Institut
More informationBinary Decision Diagrams
Binary Decision Diagrams Literature Some pointers: H.R. Andersen, An Introduction to Binary Decision Diagrams, Lecture notes, Department of Information Technology, IT University of Copenhagen Tools: URL:
More informationTopic 17. Analysis of Algorithms
Topic 17 Analysis of Algorithms Analysis of Algorithms- Review Efficiency of an algorithm can be measured in terms of : Time complexity: a measure of the amount of time required to execute an algorithm
More information)j > Riley Tipton Perry University of New South Wales, Australia. World Scientific CHENNAI
Riley Tipton Perry University of New South Wales, Australia )j > World Scientific NEW JERSEY LONDON. SINGAPORE BEIJING SHANSHAI HONG K0N6 TAIPEI» CHENNAI Contents Acknowledgments xi 1. Introduction 1 1.1
More informationIntroduction to Quantum Information Processing
Introduction to Quantum Information Processing Lecture 6 Richard Cleve Overview of Lecture 6 Continuation of teleportation Computation and some basic complexity classes Simple quantum algorithms in the
More informationComplexity. Complexity Theory Lecture 3. Decidability and Complexity. Complexity Classes
Complexity Theory 1 Complexity Theory 2 Complexity Theory Lecture 3 Complexity For any function f : IN IN, we say that a language L is in TIME(f(n)) if there is a machine M = (Q, Σ, s, δ), such that: L
More informationMind the gap Solving optimization problems with a quantum computer
Mind the gap Solving optimization problems with a quantum computer A.P. Young http://physics.ucsc.edu/~peter Work supported by Talk at Saarbrücken University, November 5, 2012 Collaborators: I. Hen, E.
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 7: Complexity & Algorithms I 13th October 016 School of Informatics, University of Edinburgh Complexity - Computational Complexity: Classification
More informationLogic gates. Quantum logic gates. α β 0 1 X = 1 0. Quantum NOT gate (X gate) Classical NOT gate NOT A. Matrix form representation
Quantum logic gates Logic gates Classical NOT gate Quantum NOT gate (X gate) A NOT A α 0 + β 1 X α 1 + β 0 A N O T A 0 1 1 0 Matrix form representation 0 1 X = 1 0 The only non-trivial single bit gate
More informationQuantified Synthesis of Reversible Logic
Quantified Synthesis of Reversible Logic Robert Wille 1 Hoang M. Le 1 Gerhard W. Dueck 2 Daniel Große 1 1 Group for Computer Architecture (Prof. Dr. Rolf Drechsler) University of Bremen, 28359 Bremen,
More informationAnalysis of Algorithm Efficiency. Dr. Yingwu Zhu
Analysis of Algorithm Efficiency Dr. Yingwu Zhu Measure Algorithm Efficiency Time efficiency How fast the algorithm runs; amount of time required to accomplish the task Our focus! Space efficiency Amount
More informationLecture note 8: Quantum Algorithms
Lecture note 8: Quantum Algorithms Jian-Wei Pan Physikalisches Institut der Universität Heidelberg Philosophenweg 12, 69120 Heidelberg, Germany Outline Quantum Parallelism Shor s quantum factoring algorithm
More informationOptimization Bounds from Binary Decision Diagrams
Optimization Bounds from Binary Decision Diagrams J. N. Hooker Joint work with David Bergman, André Ciré, Willem van Hoeve Carnegie Mellon University ICS 203 Binary Decision Diagrams BDDs historically
More informationDesigning Oracles for Grover Algorithm
Designing Oracles for Grover Algorithm Homework 1. You have time until December 3 to return me this homework. 2. Please use PPT, Word or some word processor. You may send also PDF. The simulation should
More informationMind the gap Solving optimization problems with a quantum computer
Mind the gap Solving optimization problems with a quantum computer A.P. Young http://physics.ucsc.edu/~peter Work supported by Talk at the London Centre for Nanotechnology, October 17, 2012 Collaborators:
More informationLecture 22: Counting
CS 710: Complexity Theory 4/8/2010 Lecture 22: Counting Instructor: Dieter van Melkebeek Scribe: Phil Rydzewski & Chi Man Liu Last time we introduced extractors and discussed two methods to construct them.
More information13th International Conference on Relational and Algebraic Methods in Computer Science (RAMiCS 13)
13th International Conference on Relational and Algebraic Methods in Computer Science (RAMiCS 13) Relation Algebras, Matrices, and Multi-Valued Decision Diagrams Francis Atampore and Dr. Michael Winter
More informationBDD Based Upon Shannon Expansion
Boolean Function Manipulation OBDD and more BDD Based Upon Shannon Expansion Notations f(x, x 2,, x n ) - n-input function, x i = or f xi=b (x,, x n ) = f(x,,x i-,b,x i+,,x n ), b= or Shannon Expansion
More informationIntroduction to Quantum Computation
Introduction to Quantum Computation Ioan Burda Introduction to Quantum Computation Copyright 2005 Ioan Burda All rights reserved. Universal Publishers Boca Raton, Florida USA 2005 ISBN: 1-58112- 466-X
More informationDiscrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST
Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and
More informationExtended Superposed Quantum State Initialization Using Disjoint Prime Implicants
Extended Superposed Quantum State Initialization Using Disjoint Prime Implicants David Rosenbaum, Marek Perkowski Portland State University, Department of Computer Science Portland State University, Department
More informationSums of Products. Pasi Rastas November 15, 2005
Sums of Products Pasi Rastas November 15, 2005 1 Introduction This presentation is mainly based on 1. Bacchus, Dalmao and Pitassi : Algorithms and Complexity results for #SAT and Bayesian inference 2.
More informationImprovements for Implicit Linear Equation Solvers
Improvements for Implicit Linear Equation Solvers Roger Grimes, Bob Lucas, Clement Weisbecker Livermore Software Technology Corporation Abstract Solving large sparse linear systems of equations is often
More informationAnalysis of Trivium Using Compressed Right Hand Side Equations
5.3 Analysis of Trivium Using Compressed Right Hand Side Equations 65 Analysis of Trivium Using Compressed Right Hand Side Equations Thorsten Ernst Schilling, Håvard Raddum thorsten.schilling@ii.uib.no,havard.raddum@ii.uib.no
More informationLecture 22: Quantum computational complexity
CPSC 519/619: Quantum Computation John Watrous, University of Calgary Lecture 22: Quantum computational complexity April 11, 2006 This will be the last lecture of the course I hope you have enjoyed the
More informationClassical simulations of non-abelian quantum Fourier transforms
Classical simulations of non-abelian quantum Fourier transforms Diploma Thesis Juan Bermejo Vega December 7, 2011 Garching First reviewer: Prof. Dr. J. Ignacio Cirac Second reviewer: Prof. Dr. Alejandro
More informationChapter 10. Quantum algorithms
Chapter 10. Quantum algorithms Complex numbers: a quick review Definition: C = { a + b i : a, b R } where i = 1. Polar form of z = a + b i is z = re iθ, where r = z = a 2 + b 2 and θ = tan 1 y x Alternatively,
More informationIs Quantum Search Practical?
Is Quantum Search Practical? George F Viamontes, Igor L Markov, and John P Hayes {gviamont, imarkov, jhayes}@eecsumichedu The University of Michigan, Advanced Computer Architecture Laboratory Ann Arbor,
More informationParallelization of the QC-lib Quantum Computer Simulator Library
Parallelization of the QC-lib Quantum Computer Simulator Library Ian Glendinning and Bernhard Ömer VCPC European Centre for Parallel Computing at Vienna Liechtensteinstraße 22, A-19 Vienna, Austria http://www.vcpc.univie.ac.at/qc/
More informationTitle. Citation Information Processing Letters, 112(16): Issue Date Doc URLhttp://hdl.handle.net/2115/ Type.
Title Counterexamples to the long-standing conjectur Author(s) Yoshinaka, Ryo; Kawahara, Jun; Denzumi, Shuhei Citation Information Processing Letters, 112(16): 636-6 Issue Date 2012-08-31 Doc URLhttp://hdl.handle.net/2115/50105
More informationCSE332: Data Structures & Parallelism Lecture 2: Algorithm Analysis. Ruth Anderson Winter 2019
CSE332: Data Structures & Parallelism Lecture 2: Algorithm Analysis Ruth Anderson Winter 2019 Today Algorithm Analysis What do we care about? How to compare two algorithms Analyzing Code Asymptotic Analysis
More informationBasing Decisions on Sentences in Decision Diagrams
Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence Basing Decisions on Sentences in Decision Diagrams Yexiang Xue Department of Computer Science Cornell University yexiang@cs.cornell.edu
More informationEGFC: AN EXACT GLOBAL FAULT COLLAPSING TOOL FOR COMBINATIONAL CIRCUITS
EGFC: AN EXACT GLOBAL FAULT COLLAPSING TOOL FOR COMBINATIONAL CIRCUITS Hussain Al-Asaad Department of Electrical & Computer Engineering University of California One Shields Avenue, Davis, CA 95616-5294
More informationDRAFT. Algebraic computation models. Chapter 14
Chapter 14 Algebraic computation models Somewhat rough We think of numerical algorithms root-finding, gaussian elimination etc. as operating over R or C, even though the underlying representation of the
More informationIntroduction into Quantum Computations Alexei Ashikhmin Bell Labs
Introduction into Quantum Computations Alexei Ashikhmin Bell Labs Workshop on Quantum Computing and its Application March 16, 2017 Qubits Unitary transformations Quantum Circuits Quantum Measurements Quantum
More informationLimitations of Algorithm Power
Limitations of Algorithm Power Objectives We now move into the third and final major theme for this course. 1. Tools for analyzing algorithms. 2. Design strategies for designing algorithms. 3. Identifying
More information1 Algebraic Methods. 1.1 Gröbner Bases Applied to SAT
1 Algebraic Methods In an algebraic system Boolean constraints are expressed as a system of algebraic equations or inequalities which has a solution if and only if the constraints are satisfiable. Equations
More informationProbabilistic Transfer Matrices in Symbolic Reliability Analysis of Logic Circuits
Probabilistic Transfer Matrices in Symbolic Reliability Analysis of Logic Circuits SMITA KRISHNASWAMY, GEORGE F. VIAMONTES, IGOR L. MARKOV, and JOHN P. HAYES University of Michigan, Ann Arbor We propose
More informationGraph structure in polynomial systems: chordal networks
Graph structure in polynomial systems: chordal networks Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationThe Separation Problem for Binary Decision Diagrams
The Separation Problem for Binary Decision Diagrams J. N. Hooker Joint work with André Ciré Carnegie Mellon University ISAIM 2014 Separation Problem in Optimization Given a relaxation of an optimization
More informationCrash course Verification of Finite Automata Binary Decision Diagrams
Crash course Verification of Finite Automata Binary Decision Diagrams Exercise session 10 Xiaoxi He 1 Equivalence of representations E Sets A B A B Set algebra,, ψψ EE = 1 ψψ AA = ff ψψ BB = gg ψψ AA BB
More informationResource: color-coded sets of standards cards (one page for each set)
Resource: color-coded sets of standards cards (one page for each set) Fluency With Operations: on blue cardstock Expressions and Equations: on yellow cardstock Real-World Applications: on green cardstock
More informationProblem. Problem Given a dictionary and a word. Which page (if any) contains the given word? 3 / 26
Binary Search Introduction Problem Problem Given a dictionary and a word. Which page (if any) contains the given word? 3 / 26 Strategy 1: Random Search Randomly select a page until the page containing
More informationarxiv:quant-ph/ v1 15 Jan 2006
Shor s algorithm with fewer (pure) qubits arxiv:quant-ph/0601097v1 15 Jan 2006 Christof Zalka February 1, 2008 Abstract In this note we consider optimised circuits for implementing Shor s quantum factoring
More informationComplex numbers: a quick review. Chapter 10. Quantum algorithms. Definition: where i = 1. Polar form of z = a + b i is z = re iθ, where
Chapter 0 Quantum algorithms Complex numbers: a quick review / 4 / 4 Definition: C = { a + b i : a, b R } where i = Polar form of z = a + b i is z = re iθ, where r = z = a + b and θ = tan y x Alternatively,
More informationLecture 2. Fundamentals of the Analysis of Algorithm Efficiency
Lecture 2 Fundamentals of the Analysis of Algorithm Efficiency 1 Lecture Contents 1. Analysis Framework 2. Asymptotic Notations and Basic Efficiency Classes 3. Mathematical Analysis of Nonrecursive Algorithms
More informationComputational Boolean Algebra. Pingqiang Zhou ShanghaiTech University
Computational Boolean Algebra Pingqiang Zhou ShanghaiTech University Announcements Written assignment #1 is out. Due: March 24 th, in class. Programming assignment #1 is out. Due: March 24 th, 11:59PM.
More informationCryptographic Protocols Notes 2
ETH Zurich, Department of Computer Science SS 2018 Prof. Ueli Maurer Dr. Martin Hirt Chen-Da Liu Zhang Cryptographic Protocols Notes 2 Scribe: Sandro Coretti (modified by Chen-Da Liu Zhang) About the notes:
More informationDetecting Support-Reducing Bound Sets using Two-Cofactor Symmetries 1
3A-3 Detecting Support-Reducing Bound Sets using Two-Cofactor Symmetries 1 Jin S. Zhang Department of ECE Portland State University Portland, OR 97201 jinsong@ece.pdx.edu Malgorzata Chrzanowska-Jeske Department
More information/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: NP-Completeness I Date: 11/13/18
601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: NP-Completeness I Date: 11/13/18 20.1 Introduction Definition 20.1.1 We say that an algorithm runs in polynomial time if its running
More information