Why Quantum . . . Quantum . . . Remaining Problems . . . Quantum Physics: . . . Current Quantum Measurements in . . . Cryptography Algorithm Main Idea of Quantum . . . A General Family of . . . Is Optimal: A Proof What Do We Want to . . . Analyzing the . . . Oscar Galindo, Vladik Kreinovich, and Home Page Olga Kosheleva Title Page University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ ogalindomo@miners.utep.edu, vladik@utep.edu, olgak@utep.edu Page 1 of 40 Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 1. Why Quantum Computing Remaining Problems . . . • In many practical problems, we need to process large Quantum Physics: . . . amounts of data in a limited time. Measurements in . . . Main Idea of Quantum . . . • To be able to do it, we need computations to be as fast A General Family of . . . as possible. What Do We Want to . . . • Computations are already fast. Analyzing the . . . Home Page • However, there are many important problems for which we still cannot get the results on time. Title Page • For example, we can predict with a reasonable accuracy ◭◭ ◮◮ where the tornado will go in the next 15 minutes. ◭ ◮ • However, these computations take days on the fastest Page 2 of 40 existing high performance computer. Go Back • One of the main limitations: the speed of all the pro- Full Screen cesses is limited by the speed of light c ≈ 3 · 10 5 km/sec. Close Quit
Why Quantum . . . Quantum . . . 2. Why Quantum Computing (cont-d) Remaining Problems . . . • For a laptop of size ≈ 30 cm, the fastest we can send a Quantum Physics: . . . 30 cm 3 · 10 5 km / sec ≈ 10 − 9 sec. Measurements in . . . signal across the laptop is Main Idea of Quantum . . . • During this time, a usual few-Gigaflop laptop performs A General Family of . . . quite a few operations. What Do We Want to . . . Analyzing the . . . • To further speed up computations, we thus need to Home Page further decrease the size of the processors. Title Page • We need to fit Gigabytes of data – i.e., billions of cells ◭◭ ◮◮ – within a small area. ◭ ◮ • So, we need to attain a very small cell size. Page 3 of 40 • At present, a typical cell consists of several dozen molecules. Go Back • As we decrease the size further, we get to a few-molecule Full Screen size. Close Quit
Why Quantum . . . Quantum . . . 3. Why Quantum Computing (cont-d) Remaining Problems . . . • At this size, physics is different: quantum effects be- Quantum Physics: . . . come dominant. Measurements in . . . Main Idea of Quantum . . . • At first, quantum effects were mainly viewed as a nui- A General Family of . . . sance. What Do We Want to . . . • For example, one of the features of quantum world is Analyzing the . . . that its results are usually probabilistic. Home Page • So, if we simply decrease the cell size but use the same Title Page computer engineering techniques, then: ◭◭ ◮◮ – instead of getting the desired results all the time, ◭ ◮ – we will start getting other results with some prob- Page 4 of 40 ability. Go Back • This probability of undesired results increases as we Full Screen decrease the size of the computing cells. Close Quit
Why Quantum . . . Quantum . . . 4. Why Quantum Computing (cont-d) Remaining Problems . . . • However, researchers found out that: Quantum Physics: . . . Measurements in . . . – by appropriately modifying the corresponding al- Main Idea of Quantum . . . gorithms, A General Family of . . . – we can avoid the probability-related problem and, What Do We Want to . . . even better, make computations faster. Analyzing the . . . • The resulting algorithms are known as algorithms of Home Page quantum computing . Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 40 Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 5. Quantum Computing Will Enable Us to De- Remaining Problems . . . code All Traditionally Encoded Messages Quantum Physics: . . . • One of the spectacular algorithms of quantum comput- Measurements in . . . ing is Shor’s algorithm for fast factorization. Main Idea of Quantum . . . A General Family of . . . • Most encryption schemes – the backbone of online com- What Do We Want to . . . merce – are based on the RSA algorithm. Analyzing the . . . • This algorithm is based on the difficulty of factorizing Home Page large integers. Title Page • To form an at-present-unbreakable code, the user se- ◭◭ ◮◮ lects two large prime numbers P 1 and P 2 . ◭ ◮ • These numbers form his private code. Page 6 of 40 • He then transmits to everyone their product n = P 1 · P 2 Go Back that everyone can use to encrypt their messages. Full Screen • At present, the only way to decode this message is to know the values P i . Close Quit
Why Quantum . . . Quantum . . . 6. Quantum Computing Can Decode All Tradi- Remaining Problems . . . tionally Encoded Messages (cont-d) Quantum Physics: . . . • Shor’s algorithm allows quantum computers to effec- Measurements in . . . tively find P i based on n . Main Idea of Quantum . . . A General Family of . . . • Thus, it can read practically all the secret messages What Do We Want to . . . that have been sent so far. Analyzing the . . . • This is one governments invest in the design of quan- Home Page tum computers. Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 40 Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 7. Quantum Cryptography: an Unbreakable Al- Remaining Problems . . . ternative to the Current Cryptographic Schemes Quantum Physics: . . . • That RSA-based cryptographic schemes can be broken Measurements in . . . by quantum computing. Main Idea of Quantum . . . A General Family of . . . • However, this does not mean that there will be no se- What Do We Want to . . . crets. Analyzing the . . . • Researchers have invented a quantum-based encryp- Home Page tion scheme that cannot be thus broken. Title Page • This scheme, by the way, is already used for secret ◭◭ ◮◮ communications. ◭ ◮ Page 8 of 40 Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 8. Remaining Problems And What We Do in This Remaining Problems . . . Talk Quantum Physics: . . . • In addition to the current cryptographic scheme, one Measurements in . . . can propose its modifications. Main Idea of Quantum . . . A General Family of . . . • This possibility raises a natural question: which of What Do We Want to . . . these scheme is the best? Analyzing the . . . • In this talk, we show that the current cryptographic Home Page scheme is, in some reasonable sense, optimal. Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 40 Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 9. Quantum Physics: Possible States Remaining Problems . . . • One of the main ideas behind quantum physics is that Quantum Physics: . . . in the quantum world, Measurements in . . . Main Idea of Quantum . . . – in addition to the regular states, A General Family of . . . – we can also have linear combinations of these states, What Do We Want to . . . with complex coefficients. Analyzing the . . . • Such combinations are known as superpositions . Home Page • A single 1-bit memory cell in the classical physics can Title Page only have states 0 and 1. ◭◭ ◮◮ • In quantum physics, these states are denoted by | 0 � ◭ ◮ and | 1 � . Page 10 of 40 • We can also have superpositions c 0 ·| 0 � + c 1 ·| 1 � , where Go Back c 0 and c 1 are complex numbers. Full Screen Close Quit
Why Quantum . . . Quantum . . . 10. Measurements in Quantum Physics Remaining Problems . . . • What will happen if we try to measure the bit in the Quantum Physics: . . . superposition state c 0 · | 0 � + c 1 · | 1 � ? Measurements in . . . Main Idea of Quantum . . . • According to quantum physics, as a result of this mea- A General Family of . . . surement, we get: What Do We Want to . . . – 0 with probability | c 0 | 2 and Analyzing the . . . – 1 with probability | c 1 | 2 . Home Page • After the measurement, the state also changes: Title Page ◭◭ ◮◮ – if the measurement result is 0, the state will turn into | 0 � , and ◭ ◮ – if the measurement result is 1, the state will turn Page 11 of 40 into | 1 � . Go Back Full Screen Close Quit
Why Quantum . . . Quantum . . . 11. Measurements in Quantum Physics (cont-d) Remaining Problems . . . • Since we can get either 0 or 1, the corresponding prob- Quantum Physics: . . . abilities should add up to 1; so: Measurements in . . . Main Idea of Quantum . . . – for the expression c 0 · | 0 � + c 1 · | 1 � to represent a A General Family of . . . physically meaningful state, What Do We Want to . . . – the coefficients c 0 and c 1 must satisfy the condition Analyzing the . . . | c 0 | 2 + | c 1 | 2 = 1 . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 40 Go Back Full Screen Close Quit
Recommend
More recommend
