Optimization Is Important How to Jump out of a . . . Proper Selection of . . . Quantum Annealing What Is the Optimal Which Annealing . . . Annealing Schedule in Physical Meaning of . . . Need to Select a Family Quantum Annealing Definitions Results Oscar Galindo and Vladik Krenovich Home Page Title Page Department of Computer Science University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ ogalindomo@miners.utep.edu, vladik@utep.edu Page 1 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 1. Optimization Is Important Proper Selection of . . . • In practical applications, we want to find: Quantum Annealing Which Annealing . . . – the best possible decision, i.e., Physical Meaning of . . . – the decision for which the appropriate objective Need to Select a Family function attains its largest possible value. Definitions • There exist many numerical algorithms for optimiza- Results tion. Home Page • How do we know that we have reached the maximum? Title Page • We compare the quality of current alternative with the ◭◭ ◮◮ qualities of several similar alternatives. ◭ ◮ • Often, we have a situation of local maximum, when: Page 2 of 30 – an alternative is better than all nearby alternatives, Go Back but Full Screen – is still worse than some other – distant – alterna- tive. Close Quit
Optimization Is Important How to Jump out of a . . . 2. How to Jump out of a Local Maximum? Proper Selection of . . . • It is not clear in which direction we should go. Quantum Annealing Which Annealing . . . • All directions are, in this sense, equally good (or equally Physical Meaning of . . . bad). Need to Select a Family • So a reasonable idea is to pick each direction at ran- Definitions dom. Results Home Page • The corresponding change should be significant – oth- erwise, we will not get out of the local maximum. Title Page • This is the main idea behind simulated annealing . ◭◭ ◮◮ • We want this process to eventually stop. ◭ ◮ Page 3 of 30 • Thus, the probability of jumping decreases with time. Go Back • How exactly this probability decreases with time is known as the annealing schedule . Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 3. Proper Selection of Annealing Schedule Is Very Proper Selection of . . . Important Quantum Annealing • The success – or not – of simulated annealing depends Which Annealing . . . on annealing schedule. Physical Meaning of . . . Need to Select a Family • If we decrease the probability too fast, we will not give Definitions the algorithm enough time to jump out of a local max. Results • On the other hand, if we decrease it too slowly, the Home Page algorithm will work forever and never stops. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 4. Quantum Annealing Proper Selection of . . . • Traditional computations are deterministic. Quantum Annealing Which Annealing . . . • So, to perform simulated annealing, we need to artifi- Physical Meaning of . . . cially introduce randomness. Need to Select a Family • In contrast, quantum processes are, in general, proba- Definitions bilistic. Results • Thus, in quantum computing, to implement a random Home Page deviation: Title Page – there is no need to artificially introduce random- ◭◭ ◮◮ ness, ◭ ◮ – it is sufficient to somewhat modify the system’s dy- Page 5 of 30 namics. Go Back • In general, the state of a quantum system is described by a complex-valued function ψ ( t ). Full Screen • It is known as the wave function . Close Quit
Optimization Is Important How to Jump out of a . . . 5. Quantum Annealing (cont-d) Proper Selection of . . . • The dynamics of a quantum system is described by Quantum Annealing Schroedinger’s equations i · � · ∂ψ Which Annealing . . . ∂t = Hψ. Physical Meaning of . . . = √− 1 and H is a linear operator. def • Here i Need to Select a Family Definitions • In these terms, annealing-type modification means adding Results additional terms – decreasing with time – to H : Home Page i · � · ∂ψ ∂t = Hψ + γ ( t ) · H 0 ψ. Title Page ◭◭ ◮◮ • Quantum annealing is the main idea behind D-Wave ◭ ◮ Systems quantum computing devices. Page 6 of 30 • These are the only commercially available and practi- Go Back cally useful computers that use quantum computing. Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 6. Which Annealing Schedule Works Best Proper Selection of . . . • Empirically, depending on the specific optimization prob- Quantum Annealing lem, two schedules work the best: Which Annealing . . . Physical Meaning of . . . – the power law annealing schedule γ ( t ) = A · t a , for Need to Select a Family some A and a < 0; and Definitions – the exponential annealing schedule Results γ ( t ) = A · exp( a · t ) for some A and a < 0. Home Page • In this talk, we present a possible theoretical explana- Title Page tion for these empirical facts. ◭◭ ◮◮ ◭ ◮ Page 7 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 7. Physical Meaning of Annealing Proper Selection of . . . • A physical system tends to end up in a state with the Quantum Annealing smallest possible energy. Which Annealing . . . Physical Meaning of . . . • For example, in a gravitational field, this means getting Need to Select a Family to as low a position as possible. Definitions • In principle, we can place a ball on top of the mountain Results – which will constitute a local minimum of energy. Home Page • However, if a strong wind blows and disturbs the ball, Title Page it will start falling down. ◭◭ ◮◮ • So, a natural way to use simulated physical phenomena ◭ ◮ for optimization is to simulate a system for which: Page 8 of 30 – for all the values of the parameters, Go Back – its energy is equal to the value of minimized objec- Full Screen tive function. Close Quit
Optimization Is Important How to Jump out of a . . . 8. Physical Meaning (cont-d) Proper Selection of . . . • In general, in Schroedinger equations, energy is repre- Quantum Annealing sented by the operator H . Which Annealing . . . Physical Meaning of . . . • So for quantum annealing, this operator must represent Need to Select a Family the desired objective function. Definitions Results Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 9. Need to Select a Family Proper Selection of . . . • The original optimization problem is usually not for- Quantum Annealing mulated in terms of energy. Which Annealing . . . Physical Meaning of . . . • So, how we transform it into energy depends on our Need to Select a Family choice of units. Definitions • If we select a different unit, this means that in original Results unit, instead of H , we will have C · H ; Home Page – if for H , the best schedule was γ ( t ), Title Page – then for C · H , the best schedule is C · γ ( t ). ◭◭ ◮◮ • Indeed, i · � · ∂ψ ◭ ◮ ∂t = C · Hψ + C · γ ( t ) · H 0 ψ is equivalent to the original equation if we re-scale the time: t → t/C . Page 10 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 10. Need to Select a Family (cont-d) Proper Selection of . . . • The choice of the energy unit is rather arbitrary, so: Quantum Annealing Which Annealing . . . – with each such optimal schedule γ ( t ), Physical Meaning of . . . – in different energy units, a schedule C · γ ( t ) is op- Need to Select a Family timal. Definitions • Thus, we can only select a family { C · γ ( t ) } C> 0 of an- Results nealing functions. Home Page • Here, a function γ ( t ) is fixed, and the parameter C can Title Page take any positive value. ◭◭ ◮◮ ◭ ◮ Page 11 of 30 Go Back Full Screen Close Quit
Optimization Is Important How to Jump out of a . . . 11. What Does Optimal Mean? Proper Selection of . . . • What is the best depends on the situation. Quantum Annealing Which Annealing . . . • We may want to get the results as fast as possible. Physical Meaning of . . . • We may want to get the maximum as close to the global Need to Select a Family maximum as possible even if it takes more time. Definitions • There are many other possible options. Results Home Page • In all these cases, we have a criterion for comparing families, i.e., saying when: Title Page – one family F 1 is better than family F 2 (we will de- ◭◭ ◮◮ note it by F 2 < F 1 ), ◭ ◮ – or F 2 is better than F 1 ( F 1 < F 2 ), Page 12 of 30 – or F 1 and F 2 are of the same quality with respect to Go Back the given criterion; we will denote this by F 1 ∼ F 2 . Full Screen • Of course, if F 1 is better than F 2 and F 2 is better than F 2 , then F 1 should be better than F 3 . Close Quit
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