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Quantum Robots

Quantum Robots. QUANTUM ROBOTICS IN ROBOT THEATRE. Quantum Signals and Automata. 0, 1. Binary Logic. Fuzzy Logic. [0,1]. Quantum Logic. Hilbert Space, Bloch Sphere. Quantum Signals and Automata. Quantum array. Logic circuit. Quantum state Machine. Finite State Machine. Algorithm.

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Quantum Robots

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  1. Quantum Robots

  2. QUANTUM ROBOTICS IN ROBOT THEATRE

  3. Quantum Signals and Automata 0, 1 Binary Logic Fuzzy Logic [0,1] Quantum Logic Hilbert Space, Bloch Sphere

  4. Quantum Signals and Automata Quantum array Logic circuit Quantum state Machine Finite State Machine Algorithm Quantum Algorithm

  5. A “Quantum Robot” Concept Quantum Computing Intelligent Robotics Since 1999 Since 1969 Quantum Robotics Since 2004

  6. Quantum Robotics Constraint Satisfaction Model for Grover Algorithm Collaboration: Martin Lukac, Michitaka Kameyama, Tohoku University, Quantum Robot Vision Vamsi Parasa, Erik Paul Quantum Fuzzy Logic Arushi Raghuvanshi Michael Miller, Univ. Victoria, BC Quantum Braitenberg Vehicles Siddhar Manoj Arushi Raghuvanshi Quantum Emotions Martin Lukac Quantum Initialization and Neural Networks David Rosenbaum

  7. The first quantum robot in the world?

  8. Our concept of quantum robot based on reducing all problems to constraint satisfaction solved on a quantum computer

  9. The whole proposed PSU Quantum Robot system Orion Quantum Adiabatic Computer in Vancouver BC, Canada Orion interface PC Personal computer Bluetooth connection CUDA/GPU supercomputer

  10. QUBOT-1 – the world’s first quantum robot

  11. Constraints Satisfaction Problems S E N D + M O R E M O N E Y Cryptographic Problems Graph coloring

  12. Grover Algorithm Reminder in new light

  13. Graph Coloring • Building oracle for graph coloring is a better explanation of Grover than database search. • 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

  14. 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 F(x) Value 1 for good coloring

  15. Oracle for Quantum Map of Europe Coloring Germany France Switzerland Spain Spain France Germany Quaternary qudits Switzerland     Multiple-Valued Quantum Circuits Good coloring

  16. Constraint Satisfaction Problem is to satisfy the constraints and minimize the energy Germany France Switzerland Spain Spain France Germany Switzerland     Good coloring Constraints to be satisfied Energy to be minimized Count number of ones Adiabatic Quantum Computer

  17. 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 Discuss naïve non-quantum circuit with a full counter of minterms f(x) Value 1 for good coloring Now we will generate whole Kmap at once using quantum properties - Hadamard

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

  19. Simplified schematic of our Graph Coloring Oracle. “Classical” Quantum Computer Circuit Model for Graph Coloring We designed 35 oracles

  20. Hadamard Transform Single qubit H =  H H = 1/2 Parallel connection of two Hadamard gates is calculated by Kronecker Product (tensor product) Here I calculated Kronecker product of two Hadamards

  21. Motivating calculations for 3 variables |0> H |0> + |1> |1> H |0> - |1> • As we remember, these are transformations of Hadamard gate: In general: |x> H |0> + (-1) x |1> For 3 bits, vector of 3 Hadamards works as follows: From multiplication |abc>  (|0>+(-1)a|1>) (|0>+(-1)b|1>) (|0>+(-1)c|1>) = |000> +(-1)c |001> +(-1)b |001>+(-1)b+c |001>000> +(-1)a |001> + (-1)a+c |001> + (-1)a+b |001> (-1)a+b+c |001> If a = b = c =0 then all phases positive

  22. |0> oracle |000> +|001> + |010>+|011> +|100> + |101> +|110> + |111> This is like a Kmap with every true minterm (1) encoded by -1 And every false minterm (0) encoded by 1 |0> f(x) We can say that Hadamard gates before the oracle create the Kmap of the function, setting the function in each of its possible minterms (cells) in parallel

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

  24. A practical Example • This presentation shows clearly how to perform a so called 1 in 4 search • We start out with the basics 1 in 4 search

  25. Pick your needle and I will find you a haystack The point of this slide is to show examples of 4 different oracles. Grovers search can tell between these oracles in a single iteration, classically we would need 3 iterations.

  26. Properties of the oracle Let f : {0,1}2 {0,1} have the property that there is exactly one x {0,1}2 for which f(x) = 1 Goal: find x {0,1}2 for which f(x) = 1 Classically: 3 queries are necessary Quantumly: ? Only after 3 tests can we determine with certainty that the oracles is 1 for only a single input value x

  27. x1 x1 f f x2 x2 y y  f(x1,x2) Output state: 0 H ((–1)f(00)00 +(–1)f(01)01 + (–1)f(10)10 +(–1)f(11)11)(0 –1) 0 H 1 H A 1-4 search can chose between 4 oracles in one iteration Black box for 1-4 search: Start by creating phases in superposition of all inputs to f: Input state to query: (00 + 01 + 10 + 11)(0 –1) Here we clearly see the Kmap encoded in phase – the main property of many quantum algorithms

  28. This slide illustrates how the state of the system is changed as it propagates through the quantum network implementation of Grovers Search algorithm. state =       0       1       0       0       0       0       0       0 state =0.353-0.3530.353-0.3530.353-0.3530.353-0.353 state =0.353-0.3530.353-0.3530.353-0.353-0.3530.353 state =       0       0    -0.5     0.5       0       0     0.5    -0.5 state =0.353-0.3530.353-0.3530.353-0.353-0.3530.353 state =-0.3530.3530.353-0.3530.353-0.3530.353-0.353 state =       0       0    -0.5     0.5     0.5    -0.5       0       0 ab c 0 1 1 00 01 11 10 Time 0 X H f X H M H 0 X X M H H H H H 1 M H H ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 0.3 –0,3 00 01 11 10 0.3 –0,3 00 01 11 10 00 01 11 10 00 01 11 10 00 01 11 10 00 01 11 10 0.3 –0,3 - 0.3 0,3 0 0 0 0 0.3 –0,3 0.3 –0,3 0.3 –0,3 0.3 –0,3 - 0.5 0,5 - 0.5 0,5 0.3 –0,3 - 0.3 0,3 - 0.3 0,3 0.3 - 0,3 0 0 0.5 - 0.5 0.3 –0,3 0.3 –0,3 0.3 –0,3 0.3 – 0,3 0.5 – 0,5 0 0

  29. state =       0       0       0       0       0       0       0      -1 state =-0.3530.3530.353-0.3530.353-0.353-0.3530.353 state =-0.3530.3530.353-0.3530.353-0.353-0.3530.353 state =       0       0    -0.5     0.5       0       0     0.5    -0.5 state =0.353-0.3530.353-0.3530.353-0.353-0.3530.353 state =-0.3530.3530.353-0.3530.353-0.3530.353-0.353 state =       0       0    -0.5     0.5     0.5    -0.5       0       0 state =       0       0       0       0       0       0       0      1 Time 0 X H f X H M H 0 X X M H H H H H 1 M H H Hadamard of affine function Ibverters flip between 00 and 11 Inverter flips second bit when first is 1 Hadamard addis in 00 and 11 Ibverters flip between 00 and 11 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 ab c 0 1 00 01 11 10 00 01 11 10 00 01 11 10 -0.3 0.3 - 0.3 0.3 00 01 11 10 00 01 11 10 00 01 11 10 00 01 11 10 0.3 –0,3 - 0.3 0,3 0 0 0 0 0.3 -0.3 0.3 - 0.3 0.3 –0,3 0.3 –0,3 - 0.5 0,5 - 0.5 0,5 -1 -0.3 0.3 - 0.3 0.3 - 0.3 0,3 0.3 - 0,3 0 0 0.5 - 0.5 0.3 -0.3 0.3 - 0.3 0.3 –0,3 0.3 – 0,3 0.5 – 0,5 0 0

  30. Grover Loop Time 0 X H f X H M H 0 X X M H H H H H 1 M H H Inversion about the mean ψ00 =–00 + 01 + 10 + 11 After Hadamard the solution is “known” in Hilbert space by having value -1. But it is hidden from us ψ01 =+00 – 01 + 10 + 11 ψ10 =+00 + 01 –10 + 11 ψ11 =+00 + 01 + 10 – 11 This was a special case where we could transform the state vector without repeating the oracle. In general we have to repeat the oracle – general Grover Loop The state corresponding to the input to the oracle that has a output result of 1 is ‘tagged’ with a negative 1. We need to repeat the Grover Loop N times

  31. Grover Search Future work Grover Loop Constants Hadamards Worst case quadratic speedup on every problem that you can build an oracle! Measurements Oracle or Quantum Circuit Quantum Braitenberg Vehicle Inputs- sensors Outputs - actuators Measurements New Concept of Real-time Quantum Search Grover Loop Constants Controlled Hadamards Outputs - actuators Measurements Inputs- sensors LEARNING Control

  32. Our concept of quantum robot based on reducing all problems to constraint satisfaction solved on a quantum computer

  33. Robot Obstacle Avoidance Problem New Approach to Quantum Robotics Robot Reasoning Problem Robot Communication Problem Robot Vision Problem Constraint Satisfaction Problem Adiabatic Quantum Computer Classical quantum computing

  34. The whole proposed PSU Quantum Robot system Orion Quantum Adiabatic Computer in Vancouver BC, Canada Real quantum computer Orion interface Complete multiprocessor classical robot PC Personal computer CUDA/GPU supercomputer Powerful quantum computer simulator

  35. SAT as a constraint satisfaction problem (a + b’ + c) * (b + d’) … = 1 Highly parallel system of updating nodes c =1 =0 =1 a b d =1 (a + b’ + c) (b + d’) Yes , do nothing to nodes No , update nodes

  36. SAT as a constraint satisfaction problem (a + b’ + c) * (b + d’) … = 1 Constraints: (a + b’ + c) = 1 (b + d’) = 1 ….. Orion programming is just writing equations for constraints and equations for energy Energy optimization: (a + b’ + c) = f1 (b + d’) = f2 …. Min ( f1’ + f2’ + …..)

  37. Constraint Satisfaction for Robotics • Insufficient speed of robot image processing and pattern recognition. • This can be solved by special processors, DSP processors, FPGA architectures and parallel computing. • Prolog allows to write CSP programs very quickly. • An interesting approach is to formulate many problems using the same general model. • This model may be predicate calculus, Satisfiability, Artificial Neural Nets or Constraints Satisfaction Model. • Constraints to be satisfied (complex formulas in general) • Energies to be minimized (complex formulas)

  38. Constraint satisfaction model in robotics Used in main areas of robotics: • vision, • knowledge acquisition, • knowledge usage. • In particular the following: • planning, scheduling, allocation, motion planning, gesture planning, assembly planning, graph problems including graph coloring, graph matching, floor-plan design, temporal reasoning, spatial and temporal planning, assignment and mapping problems, resource allocation in AI, combined planning and scheduling, arc and path consistency, general matching problems, belief maintenance, experiment planning, satisfiability and Boolean/mixed equationsolving, machine design and manufacturing, diagnostic reasoning, qualitative and symbolic reasoning, decision support, computational linguistics, hardware design and verification, configuration, real-time systems, and robot planning, implementation of non-conflicting sensor systems, man-robot and robot-robot communication systems and protocols,contingency-tolerant motion control, multi-robot motion planning, multi-robot task planning and scheduling, coordination of a group of robots, and many others

  39. End of this part

  40. Additional Slides on Quantum and emotional robotics

  41. Constraint Satisfaction Image Analysis by Waltz • Huffman and Clowes created an approach to polyhedral scene analysis, scenes with opaque, trihedral solids, next improved significantly by Waltz • Popularized the concept of constraints satisfaction and its use in problem solving, especially image interpretation. • Objects in this approach had always three plane surfaces intersecting in every vertex.

  42. Constraint Satisfaction Image Analysis by Waltz • There are only four ways to label a line in this blocks world model. • The line can be convex, concave, a boundary line facing up and a boundary line facing down (left, or right). • The direction of the boundary line depends on the side of the line corresponding to the face of the causing it object. • Waltz created a famous algorithm which for this world model which always finds the unique correct labeling if a figure is correct.

  43. AC-3: State 2 • Queue: (2,3)(3,2)(3,4)(4,3)(4,1)(1,4) (1,3)(3,1) • Removing (2,3). • L3 on 2 inconsistent with 3, so it is removed. • Of arcs (k,2), (1,2) is not on queue, so it is added.

  44. Examples of CSP in robotics • Scene recognition • Motion generation in presence of constraints • internal (low power, don’t hit itself) • external (shape of racing track, wolf-man-cabbage-goat) • Gesture under emotions • Communication in a swarm of robots (graph coloring) • Robot guard (set covering)

  45. Adiabatic Quantum Computingto solve Constraint Satisfaction Problems efficiently

  46. Adiabatic Quantum Computing to solve Constraint Satisfaction Problem efficiently. • Will February 13th 2007 be remembered in annals of computing.? • DWAVE company demonstrated their Orion quantum computing system in Computer History Museum in Mountain View, California. • The first time in history a commercial quantum computer was presented. • The Orion system is a hardware accelerator designed to solve in principle a particular NP-complete problem called the two-dimensional Ising model in a magnetic field (for instance quadratic programming). • It is built around a 16-qubit superconducting adiabatic quantum computer (AQC) processor.

  47. 7 1 5 3 2 8 9 2 8 2 3 9 8 3 7 4 5 7 6 4 6 5 4 6 2 Orion computer from DWAVE • Conventional front end • The solution of an NP-complete problem: • 1. Pattern matching applied to searching databases of molecules. • 2. Planning/scheduling application for assigning people to seats subject to constraints. • 3. Sudoku

  48. Orion Is the Constraint Satisfaction Solver • The company promises to provide free access by Internet to one of their systems to those researchers who want to develop their own applications. Does it have quadratic speed-up?

  49. Orion computer from DWAVE • The plans are that by the end of year 2008 the Orion systems will be scaled to more than 1000 qubits. • Company plans to build in 2009 processors specifically designed for quantum simulation, which represents a big commercial opportunity. • These problems include: protein folding, drug design and many other in chemistry, biology and material science. • Thus the company claims to dominate enormous markets of NP-complete problems and quantum simulation.

  50. We plan to concentrate on robotic applications of the Constraint Satisfaction Model. • Adiabatic Quantum Computing was proved equivalent to standard QC circuit model. • Each of the developed by us methods can be transformed to an adiabatic quantum program and run on Orion. • We developed logic minimization methods to reduce the graph that is created in AQC to program problems such as Maximum Clique or SAT. • This programming is like on “assembly level” but with time more efficient methods will be developed in our group. • This is also similar to programming current Field-Programmable Gate Arrays.

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