We Need Faster . . . What Physical . . . How Can We Find . . . This Leads to . . . From Quantum Computing Types of Physical . . . to Computers Quantum Processes . . . Randomness in General of Generation Omega Completely “Lawless” . . . (a brief overview of Fall 2020 class Another Possibility: . . . CS 5354/CS 4365) Home Page Title Page Vladik Kreinovich ◭◭ ◮◮ Department of Computer Science University of Texas at El Paso, USA ◭ ◮ vladik@utep.edu Page 1 of 20 Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 1. We Need Faster Computers How Can We Find . . . • Modern computers are much faster than in the past. This Leads to . . . Types of Physical . . . • However, there are still many practical problems for Quantum Processes . . . which they are too slow. Randomness in General • E.g., it is possible to predict, with high probability, Completely “Lawless” . . . where a tornado will go in the next 15 minutes. Another Possibility: . . . Home Page • However, even on modern high performance comput- ers, this computation will require several hours. Title Page • This is too late for this result to be useful. ◭◭ ◮◮ ◭ ◮ Page 2 of 20 Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 2. What Physical Processes Can We Use to Speed How Can We Find . . . up Computations This Leads to . . . • We have been unable to achieve a drastic speedup by Types of Physical . . . using the traditionally used physical processes. Quantum Processes . . . Randomness in General • So, a natural idea is to analyze whether using other Completely “Lawless” . . . physical processes can help. Another Possibility: . . . • This analysis is the main topic of this class. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 20 Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 3. How Can We Find Physical Processes that Can How Can We Find . . . Help to Speed up Computations? This Leads to . . . • A natural idea is to find processes whose future behav- Types of Physical . . . ior are computationally difficult to predict; indeed: Quantum Processes . . . Randomness in General – if this behavior was not difficult to predict, Completely “Lawless” . . . – then we would be able to replace the use of these Another Possibility: . . . processes with the corresponding computations; Home Page – thus, we would get a traditional computer that uses Title Page almost the same computation time; ◭◭ ◮◮ – however, we want a drastic increase in computa- tional speed. ◭ ◮ • We want to decide which physical processes are appro- Page 4 of 20 priate for computation speed-up. Go Back • So, we need to analyze the computational complexity Full Screen of different physical phenomena. Close Quit
We Need Faster . . . What Physical . . . 4. This Leads to Computational Complexity: the How Can We Find . . . 1st Topic of this Class This Leads to . . . • We want to perform computational complexity analysis Types of Physical . . . of different physical phenomena. Quantum Processes . . . Randomness in General • To be able to do it, we will first recall the main defini- Completely “Lawless” . . . tions of computational complexity: Another Possibility: . . . – worst-case time complexity, Home Page – average time complexity, Title Page – feasible algorithms, ◭◭ ◮◮ – P and NP, and ◭ ◮ – NP-hard problems. Page 5 of 20 • After that, we will start analyzing computational com- Go Back plexity of different physical phenomena. Full Screen Close Quit
We Need Faster . . . What Physical . . . 5. Types of Physical Processes How Can We Find . . . • Depending on what we can determine – we can divide This Leads to . . . physical processes into three main types. Types of Physical . . . Quantum Processes . . . • For some processes, we know the models that predict Randomness in General the results. Completely “Lawless” . . . • For some processes, the results are partly unpredictable. Another Possibility: . . . Home Page • For these processes, we can predict some characteristics – e.g., probabilities of different outcomes. Title Page • Some processes are completely “lawless”. ◭◭ ◮◮ • For such processes, any predicting model will eventu- ◭ ◮ ally turn out to be wrong. Page 6 of 20 • We will analyze if and how processes of each type can Go Back be used to speed up computations. Full Screen Close Quit
We Need Faster . . . What Physical . . . 6. Processes for Which We Know the Models that How Can We Find . . . Predict the Results This Leads to . . . • Most such processes are described by partial differen- Types of Physical . . . tial equations. Quantum Processes . . . Randomness in General • In these equations, the time derivative of all the quan- Completely “Lawless” . . . tities x ( t ) depends on their current values. Another Possibility: . . . • Usually, the dependence of the time derivative v ( t ) on Home Page the current values is computationally feasible. Title Page • So, to predict the value x ( t + h ) for small h > 0, we ◭◭ ◮◮ can simply compute x ( t ) + h · v ( t ). ◭ ◮ • Thus, such processes cannot lead to a drastic compu- Page 7 of 20 tational speedup. Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 7. Processes for Which the Results are Partly Un- How Can We Find . . . predictable, but for Which We Can Predict This Leads to . . . Some Characteristics – e.g., Probabilities Of Types of Physical . . . Different Outcomes: Main Example Quantum Processes . . . • There are such process – e.g., radioactive decay. Randomness in General Completely “Lawless” . . . • These processes are described by quantum mechanics. Another Possibility: . . . • In quantum mechanics: Home Page – in addition to differential equations that describe a Title Page smooth change in the system’s state, ◭◭ ◮◮ – we also have abrupt – and probabilistic – changes ◭ ◮ corresponding to measurements. Page 8 of 20 • And measurements are ubiquitous, since they are the Go Back only way by which we can gain information. Full Screen Close Quit
We Need Faster . . . What Physical . . . 8. Quantum Processes Can Indeed Speed up Com- How Can We Find . . . putations This Leads to . . . • For quantum systems, prediction indeed turns out to Types of Physical . . . be NP-hard. Quantum Processes . . . Randomness in General • Not surprisingly, several schemes have been discovered Completely “Lawless” . . . for using quantum processes to speed up computations. Another Possibility: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 20 Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 9. Quantum Computing Can Help in Solving All How Can We Find . . . Practical Problems This Leads to . . . • From the general viewpoint, these schemes cover all Types of Physical . . . possible applications of computers. Quantum Processes . . . Randomness in General • Indeed, from this general viewpoint, what do we want? Completely “Lawless” . . . • We want to understand how the world works, predict Another Possibility: . . . what will happen. Home Page • This is, crudely speaking, what science is about. Title Page • For example, we want to understand where the tornado ◭◭ ◮◮ will turn. ◭ ◮ • We also want to understand how can we improve the Page 10 of 20 situation. Go Back • This is, crudely speaking, what engineering is about. Full Screen Close Quit
We Need Faster . . . What Physical . . . 10. Quantum Computing Can Help (cont-d) How Can We Find . . . • For example, how can be make tornadoes change their This Leads to . . . course? Types of Physical . . . Quantum Processes . . . • How can we make houses less vulnerable to tornadoes? Randomness in General • Finally, we want to communicate – or not – with others. Completely “Lawless” . . . • So we need to develop techniques for communication Another Possibility: . . . Home Page only with the intended folks. Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 20 Go Back Full Screen Close Quit
We Need Faster . . . What Physical . . . 11. Quantum Computing is Useful in Solving the How Can We Find . . . Main Problems of Science And Engineering This Leads to . . . • In the general prediction problem, we need to find a Types of Physical . . . model that fits all the observations. Quantum Processes . . . Randomness in General • In a usual engineering problem, we need to find a design Completely “Lawless” . . . and/or a control that satisfies a given specification. Another Possibility: . . . • In most of these problems: Home Page – once we have a model, a design, or a control, Title Page – it is computationally feasible to check whether this ◭◭ ◮◮ model, design, etc. satisfies the given specs. ◭ ◮ • It is searching for a satisfactory model, design, etc. Page 12 of 20 which is computationally intensive. Go Back • To speed up such problem, we can use Grover’s quan- Full Screen tum search algorithm. Close Quit
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