Craps principle: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
→‎Application: lowercased "Then"
 
en>Fyrael
Disambiguated: draw → draw (tie)
Line 1: Line 1:
The writer is known by [http://muzikoman.com/profile-682/info/ over the counter std test] title of Numbers Wunder. Years ago we moved to North Dakota. I am a meter reader. One of the extremely very best issues in the world for me is to do aerobics and now I'm trying to earn money with it.
{{Other uses|Annealing (disambiguation)}}
 
Quantum annealing (QA) is a general method for finding the [[global minimum]] of a given [[objective function]] over a given set of candidate solutions (candidate states), by a process using [[quantum fluctuation]]s.<ref>T. Kadowaki and H. Nishimori, "Quantum annealing in the transverse Ising model" [http://pre.aps.org/abstract/PRE/v58/i5/p5355_1 Phys. Rev. E '''58''', 5355 (1998)]</ref><ref>A. B. Finilla, M. A. Gomez, C. Sebenik and D. J. Doll, "Quantum annealing: A new method for minimizing multidimensional functions" [http://www.sciencedirect.com/science/article/pii/0009261494001170 Chem. Phys. Lett. '''219''', 343 (1994)]</ref>  It is used mainly for problems where the search space is discrete ([[combinatorial optimization]] problems) with many [[local minimum|local minima]]; such as finding the [[ground state]] of a [[spin glass]].
 
Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights.  Then the system evolves following the time-dependent [[Schrödinger equation]], a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse-field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian, i.e., [[adiabatic quantum computation]].<ref>E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Ludgren and D. Preda, "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem" [http://www.sciencemag.org/content/292/5516/472 Science '''292''', 472 (2001)]</ref> The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical [[Ising model]] that corresponds to the solution to the original optimization problem.  An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal.<ref>J. Brooke, D. Bitko, T. F. Rosenbaum and G. Aeppli, "Quantum annealing of a disordered magnet", [http://www.sciencemag.org/content/284/5415/779 Science '''284''' 779 (1999)]</ref>
 
==Comparison to simulated annealing==
Quantum annealing can be compared to [[simulated annealing]], whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state.  In quantum annealing, the strength of transverse field  determines the quantum-mechanical probability to change the amplitudes of all states in parallel.  Analytical <ref>S. Morita and H. Nishimori, "Mathematical foundation of quantum annealing", [http://scitation.aip.org/content/aip/journal/jmp/49/12/10.1063/1.2995837 J.Math. Phys. '''49''', 125210 (2008)]</ref> and numerical <ref>G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" [http://iopscience.iop.org/0305-4470/39/36/R01 J. Phys. A '''39''', R393 (2006)]</ref> evidence suggests that quantum annealing outperforms simulated annealing under certain conditions.
 
== Quantum mechanics: Analogy & advantage ==
[[Image:quant-annl.jpg|right|thumb|300px]]
The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using [[quantum Monte Carlo]] (or other stochastic technique), and thus obtain a heuristic algorithm for finding the ground state of the classical glass.
 
In the case of annealing a purely mathematical ''objective function'', one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that has non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.
 
It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal  annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima. Since thermal transition probabilities (~<math>\exp{(-\Delta/k_{B}T)}</math>; <math>T</math> => Temperature, <math>k_{B}</math> => Boltzmann constant) depend only on the height <math>\Delta</math> of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier,<ref>P. Ray, B. K. Chakrabarti and A. Chakrabarti, "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations",  [http://prb.aps.org/abstract/PRB/v39/i16/p11828_1 Phys. Rev. B '''39''' 11828 (1989)]</ref> the quantum tunneling probability through the same barrier depends not only the height <math>\Delta</math> of the barrier, but also on its width <math>w</math> and is approximately given by <math>\exp{(- \Delta^{1/2}w/ \Gamma) }</math>; <math>\Gamma</math>=> Tunneling field.<ref>A. Das, B.K. Chakrabarti, and R.B. Stinchcombe, "Quantum annealing in a kinetically constrained system", [http://pre.aps.org/abstract/PRE/v72/i2/e026701 Phys. Rev. E '''72''' art. 026701 (2005)]</ref> If the barriers are thin enough <math>( w \ll \Delta^{1/2})</math>, quantum fluctuations can surely bring the system out of the shallow local minima. This <math>O(N^{1/2})</math> advantage in quantum search (compared to the classical effort growing linearly with <math> \Delta </math> or <math> N </math>, the problem size) is well established.<ref>J. Roland and N.J. Cerf, "Quantum search by local adiabatic evolution",[http://pra.aps.org/abstract/PRA/v65/i4/e042308 Phys. Rev. A '''65''', 042308 (2002)]</ref>
 
It is speculated that in a [[quantum computer]], such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the [[quantum entanglement]] used in more traditional quantum algorithms.
 
==Implementations==
[[File:DWave 128chip.jpg|thumb|Photograph of a chip constructed by D-Wave Systems Inc., mounted and wire-bonded in a sample holder.  The D-Wave processor is designed to use 128 [[superconductivity|superconducting]] logic elements that exhibit controllable and tunable coupling to perform operations.]]
 
In 2011, [[D-Wave Systems]] announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature <ref>M. W. Johnson et al., "Quantum annealing with manufactured spins",  [http://www.nature.com/nature/journal/v473/n7346/abs/nature10012.html Nature '''473''' 194 (2011)]</ref> on its performance. The company claims this system uses a 128 [[qubit]] processor chipset.<ref>{{cite web |title=Learning to program the D-Wave One |url=http://dwave.wordpress.com/2011/05/11/learning-to-program-the-d-wave-one/ |accessdate=11 May 2011}}</ref>  On May 25, 2011 D-Wave announced that [[Lockheed Martin]] Corporation entered into an agreement to purchase a D-Wave One system.<ref>{{cite web | url=http://www.dwavesys.com/en/pressreleases.html#lm_2011 |title= D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation |accessdate=2011-05-30 |date=2011-05-25}}</ref> On October 28, 2011 [[University of Southern California|USC]]'s [[Information Sciences Institute]] took delivery of Lockheed's D-Wave One, where it has become the first operational commercial "quantum computer".<ref>{{cite web | url=http://www.dwavesys.com/en/pressreleases.html#usc_2011 |title= USC To Establish First Operational Quantum Computing System at an Academic Institution|accessdate=2011-10-30|date=2011-10-28}}</ref>
 
In May 2013 it was announced that a consortium of Google, NASA AMES and the non-profit Universities Space Research Association purchased an adiabatic quantum computer from D-Wave Systems with 512 qubits.<ref>N. Jones, Google and NASA snap up quantum computer, Nature (2013), doi: 10.1038/nature.2013.12999, http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999.</ref><ref>V. N. Smelyanskiy, E. G. Rieffel, S. I. Knysh, C. P. Williams, M. W. Johnson, M. C. Thom, W. G. Macready, K. L. Pudenz, "A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration", [http://arxiv.org/abs/1204.2821 arXiv:1204.2821]</ref>
 
D-Wave's architecture differs from traditional quantum computers (none of which exist in practice as of today) in that it has noisy, high error-rate qubits, since it is designed specifically for quantum annealing.
 
== References ==
{{Reflist}}
 
== General review articles and books ==
* G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" [http://iopscience.iop.org/0305-4470/39/36/R01 J. Phys. A '''39''', R393 (2006)].
* A. Das and B. K. Chakrabarti, "Colloquium: Quantum annealing and analog quantum computation" [http://rmp.aps.org/abstract/RMP/v80/i3/p1061_1  Rev. Mod. Phys. '''80''', 1061 (2008)].
* S. Suzuki, J.-i. Inoue & B. K. Chakrabarti,[http://www.amazon.com/Quantum-Transitions-Transverse-Lecture-Physics/dp/364233038X/ref=la_B00BJ8LIAS_1_1?s=books&ie=UTF8&qid=1389576675&sr=1-1 "Quantum Ising Phases & Transitions in Transverse Ising Models", Springer, Heidelberg (2013)], Chapter 8 on Quantum Annealing.
* V. Bapst, L. Foini, F. Krzakala, G. Semerjian and F. Zamponi, "The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective", [http://www.sciencedirect.com/science/article/pii/S037015731200347X Physics Reports '''523''' 127 (2013)].
* Arnab Das and [[Bikas K Chakrabarti]] (Eds.), [http://www.amazon.com/Quantum-Annealing-Related-Optimization-Methods/dp/3540279873/ref=sr_1_1?s=books&ie=UTF8&qid=1389577046&sr=1-1 "Quantum Annealing and Related Optimization Methods", Lecture Note in Physics, Vol. '''679''', Springer, Heidelberg (2005)].
* Anjan K. Chandra, Arnab Das and [[Bikas K Chakrabarti]] (Eds.),[http://www.amazon.com/Quantum-Quenching-Annealing-Computation-Lecture/dp/3642114695/ref=sr_1_1?s=books&ie=UTF8&qid=1389576962&sr=1-1 "Quantum Quenching, Annealing and Computation", Lecture Note in Physics, Vol. '''802''', Springer, Heidelberg (2010)].
* A. Ghosh and S. Mukherjee,  "Quantum Annealing and Computation: A Brief Documentary Note",  [http://arxiv.org/abs/1310.1339  arXiv:1310.1339].
 
{{Optimization algorithms}}
 
[[Category:Stochastic optimization]]
[[Category:Optimization algorithms and methods]]
[[Category:Quantum algorithms]]

Revision as of 23:52, 13 November 2012

I'm Fernando (21) from Seltjarnarnes, Iceland.
I'm learning Norwegian literature at a local college and I'm just about to graduate.
I have a part time job in a the office.

my site; wellness [continue reading this..]

Quantum annealing (QA) is a general method for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations.[1][2] It is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding the ground state of a spin glass.

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse-field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian, i.e., adiabatic quantum computation.[3] The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal.[4]

Comparison to simulated annealing

Quantum annealing can be compared to simulated annealing, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical [5] and numerical [6] evidence suggests that quantum annealing outperforms simulated annealing under certain conditions.

Quantum mechanics: Analogy & advantage

The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using quantum Monte Carlo (or other stochastic technique), and thus obtain a heuristic algorithm for finding the ground state of the classical glass.

In the case of annealing a purely mathematical objective function, one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that has non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.

It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima. Since thermal transition probabilities (~; => Temperature, => Boltzmann constant) depend only on the height of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier,[7] the quantum tunneling probability through the same barrier depends not only the height of the barrier, but also on its width and is approximately given by ; => Tunneling field.[8] If the barriers are thin enough , quantum fluctuations can surely bring the system out of the shallow local minima. This advantage in quantum search (compared to the classical effort growing linearly with or , the problem size) is well established.[9]

It is speculated that in a quantum computer, such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the quantum entanglement used in more traditional quantum algorithms.

Implementations

Photograph of a chip constructed by D-Wave Systems Inc., mounted and wire-bonded in a sample holder. The D-Wave processor is designed to use 128 superconducting logic elements that exhibit controllable and tunable coupling to perform operations.

In 2011, D-Wave Systems announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature [10] on its performance. The company claims this system uses a 128 qubit processor chipset.[11] On May 25, 2011 D-Wave announced that Lockheed Martin Corporation entered into an agreement to purchase a D-Wave One system.[12] On October 28, 2011 USC's Information Sciences Institute took delivery of Lockheed's D-Wave One, where it has become the first operational commercial "quantum computer".[13]

In May 2013 it was announced that a consortium of Google, NASA AMES and the non-profit Universities Space Research Association purchased an adiabatic quantum computer from D-Wave Systems with 512 qubits.[14][15]

D-Wave's architecture differs from traditional quantum computers (none of which exist in practice as of today) in that it has noisy, high error-rate qubits, since it is designed specifically for quantum annealing.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

General review articles and books

Template:Optimization algorithms

  1. ↑ T. Kadowaki and H. Nishimori, "Quantum annealing in the transverse Ising model" Phys. Rev. E 58, 5355 (1998)
  2. ↑ A. B. Finilla, M. A. Gomez, C. Sebenik and D. J. Doll, "Quantum annealing: A new method for minimizing multidimensional functions" Chem. Phys. Lett. 219, 343 (1994)
  3. ↑ E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Ludgren and D. Preda, "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem" Science 292, 472 (2001)
  4. ↑ J. Brooke, D. Bitko, T. F. Rosenbaum and G. Aeppli, "Quantum annealing of a disordered magnet", Science 284 779 (1999)
  5. ↑ S. Morita and H. Nishimori, "Mathematical foundation of quantum annealing", J.Math. Phys. 49, 125210 (2008)
  6. ↑ G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" J. Phys. A 39, R393 (2006)
  7. ↑ P. Ray, B. K. Chakrabarti and A. Chakrabarti, "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations", Phys. Rev. B 39 11828 (1989)
  8. ↑ A. Das, B.K. Chakrabarti, and R.B. Stinchcombe, "Quantum annealing in a kinetically constrained system", Phys. Rev. E 72 art. 026701 (2005)
  9. ↑ J. Roland and N.J. Cerf, "Quantum search by local adiabatic evolution",Phys. Rev. A 65, 042308 (2002)
  10. ↑ M. W. Johnson et al., "Quantum annealing with manufactured spins", Nature 473 194 (2011)
  11. ↑ Template:Cite web
  12. ↑ Template:Cite web
  13. ↑ Template:Cite web
  14. ↑ N. Jones, Google and NASA snap up quantum computer, Nature (2013), doi: 10.1038/nature.2013.12999, http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999.
  15. ↑ V. N. Smelyanskiy, E. G. Rieffel, S. I. Knysh, C. P. Williams, M. W. Johnson, M. C. Thom, W. G. Macready, K. L. Pudenz, "A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration", arXiv:1204.2821
Retrieved from "https://en.formulasearchengine.com/index.php?title=Craps_principle&oldid=13914"