Why Majority Rule Does Transitions Between . . . Not Work in - PowerPoint PPT Presentation

Quantum Computing: . . . Still, Reliability Is a . . . Duplication: A . . . Quantum States Why Majority Rule Does Transitions Between . . . Not Work in Quantum States of Several . . . What Would a . . . Computing: A Pedagogical Let Us Show Why All . . . Discussion Explanation Home Page Title Page Oscar Galindo 1 , Olga Kosheleva 2 , and Vladik Kreinovich 1 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science 2 Department of Teacher Education Page 1 of 24 University of Texas at El Paso 500 W. University Go Back El Paso, TX 79968, USA Full Screen ogilndomo@miners.utep.edu, olgak@utep.edu, vladik@utep.edu Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 1. Quantum Computing: A Brief Introduction Duplication: A . . . • Modern computers are very fast. Quantum States Transitions Between . . . • However, for many important practical problems, it is States of Several . . . still not possible to solve them in reasonable time. What Would a . . . • E.g., in principle, we can use computer simulations to Let Us Show Why All . . . find which biochemical compound can block a virus. Discussion • However, even on the existing high-performance com- Home Page puters, this would take thousands of years. Title Page • It is therefore desirable to design faster computers. ◭◭ ◮◮ • One of the main obstacles to this design is the speed ◭ ◮ of light. Page 2 of 24 • According to relativity theory, no physical process can Go Back be faster than a speed of light. Full Screen • On a usual 30-cm-size laptop, light takes 1 nanosecond to go from one side to another. Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 2. Quantum Computing (cont-d) Duplication: A . . . • During this time even the cheapest laptop can perform Quantum States four operations. Transitions Between . . . States of Several . . . • Thus, the only way to speed up computations is to What Would a . . . further shrink computers. Let Us Show Why All . . . • Thus, to shrink their elements. Discussion Home Page • Already an element of the computer consists of a few hundred or thousand molecules. Title Page • So if we shrink it even more, we will get to the level of ◭◭ ◮◮ individual molecules. ◭ ◮ • At this level, we need to take into account quantum Page 3 of 24 physics – the physics of the micro-world. Go Back • Computations on this level are known as quantum com- Full Screen puting . Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 3. Quantum Computing: Challenges and Successes Duplication: A . . . • In Newton’s mechanics, we can, e.g., predict the mo- Quantum States tions of celestial bodies hundreds of years ahead. Transitions Between . . . States of Several . . . • In contrast, in quantum physics, only probabilistic pre- What Would a . . . dictions are possible. Let Us Show Why All . . . • This is a major challenge for quantum computing. Discussion Home Page • However, several algorithms were invented that pro- duce the results with probability close to 1. Title Page • Some even produce them much faster than all known ◭◭ ◮◮ non-quantum algorithms ◭ ◮ • Grover’s quantum algorithm can find an element in an Page 4 of 24 unsorted n -element array in time proportional to √ n . Go Back • The fastest possible non-quantum algorithm needs to Full Screen look, in the worst case, at all n elements. Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 4. Quantum Computing: Successes (cont-d) Duplication: A . . . • Thus, it requires, in the worst case, n computational Quantum States steps. Transitions Between . . . States of Several . . . • An even more impressive speed-up occurs with Shor’s What Would a . . . algorithm for factoring large numbers. Let Us Show Why All . . . • This algorithm requires time bounded by a polynomial Discussion of the number’s length. Home Page • However, all known non-quantum algorithms requires Title Page exponential time. ◭◭ ◮◮ • This is very important since: ◭ ◮ – most existing computer security techniques Page 5 of 24 – are based on the difficulty of factoring large num- Go Back bers. Full Screen Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 5. Still, Reliability Is a Problem for Quantum Com- Duplication: A . . . puting Quantum States • In the ideal case, all quantum operations are performed Transitions Between . . . exactly. States of Several . . . What Would a . . . • Then, we get correct results with probability practi- Let Us Show Why All . . . cally indistinguishable from 1. Discussion • In reality, however, operations can only be implemented Home Page with some accuracy. Title Page • As a result, the probability of an incorrect answer be- ◭◭ ◮◮ comes non-negligible. ◭ ◮ • How can we increase the reliability of quantum com- Page 6 of 24 putations? Go Back Full Screen Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 6. Duplication: A Natural Idea Duplication: A . . . • There is a probability that a pen will not work when Quantum States needed, so a natural idea is to carry two pens. Transitions Between . . . States of Several . . . • There is a probability that a computer on board of a What Would a . . . spacecraft will malfunction. Let Us Show Why All . . . • So, a natural idea is to have two computers. Discussion Home Page • If there is a probability that a hardware problem will cause data to be lost, a natural idea is to have a backup. Title Page • Better yet, have two (or more) backups, to make the ◭◭ ◮◮ probability of losing the data truly negligible. ◭ ◮ • Similarly, for usual (non-quantum) algorithms: Page 7 of 24 – a natural way to increase their reliability Go Back – is to have several computers performing the same Full Screen computations. Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 7. Duplication (cont-d) Duplication: A . . . • Then, if the results are different, we select the result of Quantum States the majority. Transitions Between . . . States of Several . . . • This way, we increase the probability of having a cor- What Would a . . . rect result. Let Us Show Why All . . . • Indeed, suppose, e.g., that we use three computers in- Discussion dependently working in parallel. Home Page • For each of then, the probability of malfunctioning is Title Page some small (but not negligible) value p . ◭◭ ◮◮ • Since the computers are independent, the probability ◭ ◮ that all three of them malfunction is equal to p 3 . Page 8 of 24 • For each pair, the probability that these two malfunc- Go Back tion and the remaining one perform correctly is: Full Screen p 2 · (1 − p ) . Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 8. Duplication (cont-d) Duplication: A . . . • There are three possible pairs. Quantum States Transitions Between . . . • So the overall probability that this majority scheme States of Several . . . will produce a wrong result is equal to 3 p 2 · (1 − p )+ p 3 . What Would a . . . • For small p , this is much much smaller than the prob- Let Us Show Why All . . . ability p that a single computer will malfunction. Discussion Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 24 Go Back Full Screen Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 9. What About Quantum Computing? Duplication: A . . . • Nothing prevents us from having three independent Quantum States quantum computers working in parallel. Transitions Between . . . States of Several . . . • This will similarly decrease the probability of malfunc- What Would a . . . tioning. Let Us Show Why All . . . • Sometimes, however, the desired result is itself quan- Discussion tum – e.g., in quantum cryptography algorithms. Home Page • It is known that for computations with purely quantum Title Page results, the majority rule does not work. ◭◭ ◮◮ • The usual arguments why it does not work refer to ◭ ◮ rather complex results. Page 10 of 24 • In this paper, we provide a simple pedagogical expla- Go Back nation for this fact. Full Screen • OK, only as simple as it is possible when we talk about quantum computing. Close Quit

Quantum Computing: . . . Still, Reliability Is a . . . 10. Quantum States Duplication: A . . . • Let us recall the main specifics of quantum physics and Quantum States quantum computing. Transitions Between . . . States of Several . . . • One of the specifics of quantum physics is that: What Would a . . . – in addition to non-quantum states s 1 , . . . , s n , Let Us Show Why All . . . – we can also have superpositions of these states, i.e., Discussion states of the type a 1 · s 1 + . . . + a n · s n . Home Page • Here, a i are complex numbers s.t. | a 1 | 2 + . . . + | a n | 2 = 1 . Title Page ◭◭ ◮◮ • If some physical quantity has value v i on each state s i : ◭ ◮ – then, when we measure this quantity in the super- position state, Page 11 of 24 – we get each value v i with probability | a i | 2 . Go Back • These probabilities have to add to 1; this explains the Full Screen constraint on a i . Close Quit