Physics-inspired computer algorithms - PowerPoint Presentation

1. Physics-inspired computer algorithms Liliana Teodorescu

2. Physics vs Computer Science/EngineeringPhysics and Computer Science/Engineering have long established mutually beneficial linksComputer Science/Engineering to PhysicsTools to support progress in physics (and science, in general) research Physics to Computer Science/Engineeringdevelopment of semiconductor-based computer technology based on solid-state physicschallenging environments/data – (extreme) test beds for hardware/software pushing up their developments (e.g. CERN OpenLab activities)advanced algorithms benefit from mathematical models of physical phenomena - topic of this lecture2

3. OutlineBasics of combinatorial optimisation Examples of algorithms inspired by physics - fundamental physics concepts used in the algorithm - the basics of the algorithm - some applicationsConcluding remarks on the benefits and concerns related to these algorithms 3

4. Optimisation problemCombinatorial optimisation – finding min or max value of a function of many independent variablesThis function is a quantitative measure of the “quality” of the system  cost function or objective functionCost function - depends of the configuration of the many parts of the system- exact solution can be found only for systems with relatively small number of components- for practical solutions – use heuristic methods (provide satisfactory solution, no guarantee to be perfect or optimal) 4

5. HeuristicsHeuristic methods - find near-optimal solutions - are problem-specific – no guarantee that a heuristics procedure that gives good results for one problem will do the same for another problem - iterative methods (mostly)Common first approaches – hill climbing 5Starts with system in a known configurationApply change to the system => new configurationCalculate cost function of new configurationContinue until new configuration has lower costNew configuration becomes starting configurationRepeat 2-5 until no improvement in cost

6. DifficultiesWhat to do? – look to physics for inspirationPhysical systems with large number of components - domain of statistical mechanics.Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by Simulated Annealing,” Science, Volume 220, Number 4598, 13 May 1983, pp. 671-680.6Search procedure can be stuck in local minimaCommon solution – process carried a few times with randomly generated initial solutions and retain the best result Not satisfactorily for large optimisation problems

7. Statistical mechanicsStatistical mechanicsA set of methods that describe the behaviour of a physical system with many components ( atoms in a mole of substance – Avogadro number) Provides a connection between - the macroscopic properties of a material (e.g. temperature and pressure), and - the microscopic behaviour (states) of the systems’ components (e.g. atoms’ motion)Example - System in equilibrium at a give temperatureEquilibrium - steady state – state in which the system as a whole (the ensemble of atoms) does not evolve anymore; but the atoms themselves can still evolve/moveObserve experimentally - only the most probable behaviour of the system (not the individual states of the atoms) characterised by - the average behaviour of the atoms, and fluctuations around this average The average is taken over the ensemble of atoms (ensemble of identical systems introduced by mathematical physicist, Willard Gibbs) 7

8. Boltzmann/Gibbs distribution - set of atomic positions of the system - define a configuration in the ensemble - energy of the configurationProbability the system is in thermal equilibrium at temperature is : Boltzmann/Gibbs distribution Boltzmann factor - Boltzmann constant - normalization factorAs T decreases, the Boltzmann distribution concentrates on states with the lowest energy, as high energy states become increasingly unlikely.  8Lower T

9. AnnealingIn practice – finding the ground state requires annealing (not just lowering the temperature)first melt the substancethen lower the temperature slowlyat each temperature the system is allowed to reach thermal equilibriumspend a long time at temperatures close to the freezing pointWithout annealing, the substance will go into a metastable stateExample for a crystal lowering the temperature without annealing => a crystal with many defects9

10. Statistical mechanics vs Combinatorial optimisation10Statistical mechanicsDescribe the behaviour of a physical system with many components Describes the process of reaching the lowestenergy level Annealing process generates stable ground states System state Energy Temperature Change of state Ground stateCombinatorial optimisationDescribes the search for the optimal solution of a function with many variablesDescribes the algorithm to search for the minimum of the cost functionAlgorithm that simulates the annealing process likely to find an optimal minimum Simulated Annealing Algorithm Candidate solution Cost function value Control parameter (same units as cost function) Neighbouring solution Optimal solution

11. Metropolis algorithmFrom early days of scientific computing - a simple algorithm to simulation a collection of atoms in equilibrium at a given temperatureMetropolis,N., A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, "Equation of State Calculations by Fast Computing Machines", J. Chem. Phys.,21, 6, 1087-1092, 1953.Metropolis algorithm - based on statistical mechanics - generalization of the iterative improvement of the solution by incorporating uphill steps in the search process => better solutions11

12. Metropolis algorithm (cont.)in each step of the algorithm an atom is given a small random displacement => new configurationdisplacement of atoms => change in the energy of the system if , the new configuration is accepted as the starting point of the next stepif the new configuration is accepted with the probability P => the system evolves in a Boltzmann/Gibbs distribution => - at high T, all states have equal probability of occurring - as T → 0, only states with minimum energy have a non-zero probability of occurrence.Novelty of the approach – algorithm accepts states with higher energyIn optimisation terms – solutions with higher function costs are accepted with some probability => allows going out of a local minimum This ideas was incorporated in Simulated Annealing for the first time 12

13. Annealing simulationAnnealinguse initially high temperaturesreduce temperatures in slow stages until there are no further changes in the systemat each temperature the simulation proceed long enough for the system to reach a steady stateAnnealing schedule – the sequence of temperatures and the number of rearrangements of the configurations at each temperature T = αT, where α is a constant slightly less than 1 (e.g 0.8 - 0.99)High T => the algorithm discover the gross features of the search spaceLow T => the algorithm discover the fine details of the search space13

14. Simulated Annealing Algorithm1. Choose a random configuration , select the initial temperature , and specify the annealing schedule (choose α)2. Evaluate 3. Perturb } to obtain a neighbouring configuration 4. Evaluate 5. If , is the new configuration6. If , accept as the new configuration with probability P , where -7. Repeat 3-6 until termination criteria met8. Reduce the system temperature according to the cooling schedule, 9. Repeat 3-8 until termination criteria metMany versions of the algorithm available, with different procedures for each step proposed 14

15. ApplicationsSome applications in particle physicsA software implementation available in TMVAMeasurement of |V_ub| using inclusive B -> X_u l nu decays with a novel X_u-reconstruction method, H. Kakuno, Belle Coll., arXiv:hep-ex/0311048 and H. Kakuno’s Ph.D. thesis X_u reconstraction – inspired by Simulated AnnealingSupersymmetry Parameter Analysis with Fittino P. Bechtle, K. Desch, W. Porod, P. Wienemann, arXiv:hep-ph/0506244Simulated annealing for generalized Skyrme models, J.-P. Longpre, L. Marleau, arXiv:hep-ph/0502253 15

16. Other way of describing the world – quantum physicsCan it help?16

17. Quantum Annealinginspired by Simulated Annealing based on quantum mechanics’ formalismalso used to solve optimisation problemsfirst version proposed inKadowaki T, Nishimori H. Quantum annealing in the transverse Ising model. Phys. Rev. E 1998;58(5):5355-5363.17

18. Quantum systemSchrödinger equation – describe the evolution in time of the system - Hamiltonian (operator) generates the time evolution of quantum states Given the state at an initial time (t = 0), we obtain the state at any subsequent time by solving this equation. If Hamiltonian is independent of time18State vector – state of the system at time ttime evolution operator, or propagator of the quantum system

19. Quantum mechanics vs optimisation problem 19Candidate solution – the analogue of the quantum state vectorAn optimisation problem can be treated as a quantum systemThe candidate solution evolves in time in agreement with the Schrodinger equationTransition from one candidate solution to another is made with the operator U

20. HamiltonianHamiltonian – operator corresponding to the total energy of the system (sum of the kinetic and potential energy)Free particle The particle is not bound by any potential energy, so the potential energy is zero For one dimension: 20Particle in a region of constant potential (no dependence of space or time)For one dimension

21. Ising model 21Mathematical model of ferromagnetism (magnets) in statistical mechanics.Spins of the atoms arranged in a latticeEach spin interacts with its nearest neighboursand with an external magnetic field Parameter describing the interaction between spinsMagnetic momentExternal magnetic fieldHamiltonianPauli matrices

22. Quantum Annealing - Hamiltonian that encodes the function to be optimised (cost/objective function) - another Hamiltonian that introduces an external transverse field Г - transverse field coefficient that is used to control the intensity of the external field Different Hamiltonians mean different versions of the algorithm Commonly used – version based on Ising model 22

23. Quantum Annealing23- is a function of time => the intensity of the external field will change in timeis intense at the beginning and gradually decrease as the algorithm is executed Quantum Computers - computer systems that implement a quantum annealing algorithm in hardware  not discussed in this lectureIf the system evolves very slowly, it will eventually reach a final ground state that, with a certain probability, will correspond to the optimal value of the function encoded (cost/objective function)Г

24. Simulated Quantum AnnealingExact implementation of a quantum annealing process on a digital computer is costly.Use simulations instead => Simulated quantum annealing (SQA) SQA - classical algorithm that uses Quantum Monte Carlo methods to simulate quantum annealing Hamiltonians (to perform calculations necessary for solving the Schrödinger equation)SQA can be exponentially faster than SA for some problems24

25. Quantum tunneling25Schrödinger equation predicts a quantum system can penetrate a potential barrier, even if system's energy is lower than the height of the barrier  tunneling effectsmall probability effect but importanthttps://commons.wikimedia.org/wiki/File:Quantum_Tunnelling_animation.gif

26. 26Schrodinger equation predicts a quantum system can penetrate a potential barrier, even if system's energy is lower than the height of the barrier  tunneling effectsmall probability effect but importantQuantum tunnelinghttps://commons.wikimedia.org/wiki/File:Quantum_Tunnelling_animation.gifQA designed with intense quantum tunnelling at the beginning and then decreased as the algorithm runs

27. SQA vs SA27Theor. Int. Appl,., 45 (2011) 99-119SQA can be exponentially faster than SA for some problemsSA - thermal transition probabilities depend only on the height of the barriers -> proportional to => difficult to get the system out of local minima when there are very high barriersSQA - quantum tunneling probability depends on the height and width of the barrier -> approximately given by - if very thin barriers, system can go out of local minima through tunneling SQA outperforms SA when the cost function landscape has very high but thin barriers surrounded by shallow local minima

28. ApplicationsA few very recent applications in particle physicsCharged particle tracking with quantum annealing-inspired optimization, A. Zlokapa et. al., arXiv:1908.04475Unfolding as Quantum Annealing , K. Cormier, R. Di Sipio, P. Wittek, arXiv:1908.08519Quantum Algorithms for Jet Clustering, A. Y. Wei, P. Naik, A. W. Harrow, J. Thaler, arXiv:1908.08949 28

29. Other algorithms - Many… too many29A. Biswas, K.K. Mishra, S. Tiwari, A.K. Misra, “Physics-Inspired Optimization Algorithms: A Survey”Journal of Optimisation, Vol. 2013, article ID 438152

30. Concerns and criticisms30

31. Originality!31

32. ConclusionsPhysics - is a source of inspiration for computer science algorithms for a long time- has generated very competitive algorithmsSimulated Annealing – one of the most studied algorithmsQuantum Annealing – at the origin of a different computing paradigm, Quantum ComputingOther interesting (and some not that interesting !) algorithms Be inspired to explore some of these algorithms and …why not … develop your own (a good one!) 32