Grover Algorithm Marek Perkowski

Slides:



Advertisements
Similar presentations
Analysis of Algorithms
Advertisements

Quantum Computation and Quantum Information – Lecture 3
Greedy Algorithms.
1 ECE734 VLSI Arrays for Digital Signal Processing Chapter 3 Parallel and Pipelined Processing.
22C:19 Discrete Math Graphs Fall 2010 Sukumar Ghosh.
Max Cut Problem Daniel Natapov.
Combinational Circuits. Analysis Diagram Designing Combinational Circuits In general we have to do following steps: 1. Problem description 2. Input/output.
Great Theoretical Ideas in Computer Science.
CPE702 Complexity Classes Pruet Boonma Department of Computer Engineering Chiang Mai University Based on Material by Jenny Walter.
1 Tree Searching Strategies Updated: 2010/12/27. 2 The procedure of solving many problems may be represented by trees. Therefore the solving of these.
Solving Graph Problems with Boolean Methods Randal E. Bryant Carnegie Mellon University.
Management Science 461 Lecture 2b – Shortest Paths September 16, 2008.
Computability and Complexity 23-1 Computability and Complexity Andrei Bulatov Search and Optimization.
Complexity 15-1 Complexity Andrei Bulatov Hierarchy Theorem.
Midterm Exam for Quantum Computing Class Marek Perkowski.
Designing Oracles for Grover Algorithm
KEG PARTY!!!!!  Keg Party tomorrow night  Prof. Markov will give out extra credit to anyone who attends* *Note: This statement is a lie.
Optimal Layout of CMOS Functional Arrays ECE665- Computer Algorithms Optimal Layout of CMOS Functional Arrays T akao Uehara William M. VanCleemput Presented.
Grover. Part 2. Components of Grover Loop The Oracle -- O The Hadamard Transforms -- H The Zero State Phase Shift -- Z O is an Oracle H is Hadamards H.
Grovers Search Jacob D. Biamonte Portland Quantum Logic Group Jake Biamonte Should be “one in four search”
High-Performance Simulation of Quantum Computation using QuIDDs George F. Viamontes, Manoj Rajagopalan, Igor L. Markov, and John P. Hayes Advanced Computer.
Grover’s Algorithm: Single Solution By Michael Kontz.
Ternary Deutsch’s, Deutsch-Jozsa and Affine functions Problems All those problems are not published yet.
Grover. Part 2 Anuj Dawar. Components of Grover Loop The Oracle -- O The Hadamard Transforms -- H The Zero State Phase Shift -- Z.
Review of basic quantum and Deutsch-Jozsa. Can we generalize Deutsch-Jozsa algorithm? Marek Perkowski, Department of Electrical Engineering, Portland State.
1 Tree Searching Strategies. 2 The procedure of solving many problems may be represented by trees. Therefore the solving of these problems becomes a tree.
Anuj Dawar.
Grovers Search Jacob D. Biamonte Portland Quantum Logic Group Jake Biamonte Should be “one in four search”
Black-box (oracle) Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
Grover’s Algorithm in Machine Learning and Optimization Applications
Chapter 4 Graphs.
Grover Algorithm Anuj Dawar. Another example.
Quantum Error Correction Jian-Wei Pan Lecture Note 9.
Complexity Classes (Ch. 34) The class P: class of problems that can be solved in time that is polynomial in the size of the input, n. if input size is.
Spring 2015 Mathematics in Management Science Network Problems Networks & Trees Minimum Networks Spanning Trees Minimum Spanning Trees.
Lecture 16 Maximum Matching. Incremental Method Transform from a feasible solution to another feasible solution to increase (or decrease) the value of.
Combinational Problems: Unate Covering, Binate Covering, Graph Coloring and Maximum Cliques Example of application: Decomposition.
Penn ESE535 Spring DeHon 1 ESE535: Electronic Design Automation Day 24: April 18, 2011 Covering and Retiming.
CSE 589 Part VI. Reading Skiena, Sections 5.5 and 6.8 CLR, chapter 37.
Lecture 14: Graph Theory I Discrete Mathematical Structures: Theory and Applications.
Great Theoretical Ideas In Computer Science Anupam GuptaCS Fall 2006 Lecture 28Dec 5th, 2006Carnegie Mellon University Complexity Theory: A graph.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 1 4 Graph Theory (Networks) The Mathematics of Relationships 4.
1.5 Graph Theory. Graph Theory The Branch of mathematics in which graphs and networks are used to solve problems.
EMIS 8373: Integer Programming Combinatorial Relaxations and Duals Updated 8 February 2005.
Technology Mapping. 2 Technology mapping is the phase of logic synthesis when gates are selected from a technology library to implement the circuit. Technology.
Pipelining and Retiming
Optimality Study of Logic Synthesis for LUT-Based FPGAs Jason Cong and Kirill Minkovich.
Chapter 13 Backtracking Introduction The 3-coloring problem
NP-Completeness Note. Some illustrations are taken from (KT) Kleinberg and Tardos. Algorithm Design (DPV)Dasgupta, Papadimitriou, and Vazirani. Algorithms.
Sequential Circuit Design Section State Machines Design Procedure 1.Specification- obtain (produce) problem description 2.Formulation - Obtain.
Spanning Trees Dijkstra (Unit 10) SOL: DM.2 Classwork worksheet Homework (day 70) Worksheet Quiz next block.
NP-completeness Ch.34.
Introduction to Discrete Mathematics
Some Great Theoretical Ideas in Computer Science for.
EXAMPLE 1 Represent relations
Example of application: Decomposition
! Review for euler graphs WIN a FREE HW PASS or bonus point for FRIDAY’s quiz!!!!!!
Lecture 16 Maximum Matching
Quantum Computing Dorca Lee.
ESE535: Electronic Design Automation
Great Theoretical Ideas in Computer Science
Solution to problem 4. a) Control flow graph art Start k = i + 2 * j
A path that uses every vertex of the graph exactly once.
Grover. Part 2 Anuj Dawar.
Richard Anderson Lecture 26 NP-Completeness
Richard Anderson Lecture 27 Survey of NP Complete Problems
! Review for euler graphs WIN a FREE HW PASS or bonus point for FRIDAY’s quiz!!!!!!
Tree Searching Strategies
Grover Algorithm Anuj Dawar.
Presentation transcript:

Grover Algorithm Marek Perkowski I used slides of Anuj Dawar, Jake Biamonte, Julian Miller and Orlin Grabbe but I am to be blamed for extensions and (possible) mistakes

From Grover to general search

The Graph Coloring Problem Building oracle for graph coloring is a better example. This is not an optimal way to do graph coloring but explains well the principle of building oracles. The Graph Coloring Problem 1 Color every node with a color. Every two nodes that share an edge should have different colors. Number of colors should be minimum 1 3 2 3 2 5 4 5 4 6 7 6 7 This graph is 3-colorable

Simpler Graph Coloring Problem 1 3 2 We need to give all possible colors here 4 Two wires for color of node 1 Two wires for color of node 2 Two wires for color of node 3 Two wires for color of node 4      Gives “1” when nodes 1 and 2 have different colors 13 23 24 34 12 Value 1 for good coloring

Simpler Graph Coloring Problem We need to give all possible colors here Give Hadamard for each wire to get superposition of all state, which means the set of all colorings H |0> |0> H |0> H H H      13 23 24 34 12 Value 1 for good coloring

Oracle with Comparators, All good colorings are encoded by negative phase Block Diagram Think about this as a very big Kmap with -1 for every good coloring Vector Of Basic States |0> Vector Of Hadamards Oracle with Comparators, Global AND gate Output of AND Work bits

What Grover algorithm does? Grover algorithm looks to a very big Kmap and tells where is the -1 in it. Here is -1 1 -1

What “Grover for multiple solutions” algorithm does? Grover algorithm looks to a very big Kmap and tells where is the -1 in it. “Grover for many solutions” will tell all solutions. Here is -1, and here is -1, and here 1 -1

Variants of Grover With this oracle the “Grover algorithm for many solutions” will find all good colorings of the graph. If we want to find the coloring, that is good and in addition has less than K colors, we need to add the cost comparison circuit to the oracle. Then the oracle’s answers will be “one” only if the coloring is good and has less colors than K. The oracle thus becomes more complicated but the Grover algorithm can be still used.

Another example

This is a better example of problems for Grover

Advantage of Grover, although not “tractable”

Example of oracle

Another example of Oracle, more realistic

Answer bit or oracle bit tour Answer bit or oracle bit Working bits

Possible Applications Formulate an oracle (reversible circuit) for the following problems: 1. Graph coloring with of a planar map. 2. Graph coloring with the minimum number of colors of an arbitrary graph. 3. Satisfiability. 4. Set Covering. 5. Euler Path in a graph. 6. Hamiltonian Path in a graph. 7. Cryptographic Puzzle like SEND+MORE=MONEY.