Another way to look at quantum computing: the amazing correlation with Pinball collisions

It’s interesting to find something in common in seemingly unrelated topics, and it may lead to new ways of understanding things. This paper discusses the relationship between Grover search algorithm and complete elastic collision. < / P > < p > in science and mathematics, many seemingly unrelated topics share certain characteristics. Such similarities can sometimes lead to significant advances in both areas, but often they are simply interesting. In December last year, Adam brown, a Google physicist, discovered that there is an extremely precise relationship between a basic quantum computing algorithm and a wonderful method for calculating the irrational number π. “For now, this finding is simple and interesting, but we hope to find new ways of thinking about things that people may be able to use in the future to find connections that they could not see before. “For a phenomenon, multiple perspectives are very useful,” Brown said. In a preprint paper published online, brown showed that there was a mathematical correlation between two seemingly unrelated problems. One of the problems is the well-known Grover search algorithm for quantum computers, which is theoretically faster than any classical search algorithm. The other problem is an unexpected process: by counting the number of collisions of ideal elastic spheres, we can get the π value of arbitrary precision. < p > < p > quantum computation uses qubits, each qubit can represent two states at the same time, and they are usually constructed by ion or superconducting circuits. In principle, a certain number of qubits can represent and operate on combinations of exponentially more than classical bits. Previously, people thought it was a daydream to use this probability property for computing, but researchers have successfully designed an algorithm to extract useful information from qubits. < p > < p > the first quantum algorithm was proposed by Peter Sauer in 1994, when he was working at Bell Laboratories in New Jersey. Sauer algorithm can decompose integer into prime factor efficiently, which brings potential threat to many encryption schemes. The key of the algorithm is to reconstruct the integer decomposition into a new problem: determining the repetition period of a sequence. This is essentially a Fourier transform, which can be found by using global operations on the complete set of qubits. The second basic algorithm is Grover algorithm proposed by lov Grover independently in Bell Laboratories in 1996. It has a very different working mode. “Sauer algorithm and Grover algorithm are the two most typical quantum algorithms. “Even today, most of the quantum algorithms we know are inspired by either the Schell algorithm, the Grover algorithm, or both,” said Scott Aaronson of the University of Texas at Austin. Grover algorithm is usually described as a database search process, that is, checking a list containing n items to find one item that satisfies the required properties. If the list has been sorted by a certain tag, any tag can be found by continuously halving the list; the number of queries required for this process is logn. However, for unordered lists, it takes an average of N / 2 steps to complete each item. < / P > < p > like other quantum algorithms, Grover algorithm also operates the whole set of qubits at the same time, while preserving the relationship between them. However, Grover’s research shows that it usually takes only sub global operations to find the desired item. < / P > < p > this kind of promotion is not as much as that brought by Sauer algorithm, because what Sauer algorithm brings is exponential improvement. But it can be applied to the general problem of Grover. The first step of Grover algorithm is to mix all n qubits equally. Then, the algorithm repeatedly makes all qubits perform two alternate operations. The first operation is to embed the target: it reverses the state of a specific but unknown bit. The goal of this task is to determine which bits have been modified, but the method is not to observe all the bits. The second operation does not require any information about the target. Grover found that every time the sequence was repeated, the weight of the target in the hybrid structure would increase. After the appropriate number of repetitions, when an observation is performed, there is a very high possibility of getting the correct result. < / P > < p > these complex quantum operations seem to have nothing to do with elastic spheres. However, when studying the problems related to Grover algorithm, brown saw an animation made by grant Sanderson, a mathematician, which made him notice the similarity between the two. Brown showed in his paper that there is a precise mapping between the two problems. < p > < p > Sanderson’s animation explains an unexpected observation described by mathematician Gregory Galperin of the University of Eastern Illinois in 2003. In his paper, he imagines two ideal elastic balls that can move on a horizontal surface without friction, and they can completely elastic collide with each other and with the wall on the left. < / P > < p > if the ball on the right hits the lighter still ball on the left, the small ball on the left will move to the left, and the speed of the big ball on the right will not slow down much. The small ball will bounce against the wall and then hit the big ball again, which will be repeated many times. In the end, such a collision will cause the big ball to turn around until it eventually moves to the right faster than the small one. < / P > < p > before that, the number of collisions will increase with the increase of the mass ratio of the big ball to the small ball. If the two balls have the same mass, the collision occurs three times: the first time, the right side ball transfers all the movement to the left side ball, the left side ball rebounds after hitting the wall, and then returns the momentum to the right side ball completely through the collision. If the mass of the big ball is 100 times that of the small ball, there will be 31 collisions in this process. If this mass ratio is 10000, there will be 314 collisions. According to the calculation, for every 99 times increase in the mass ratio, the number of collisions divided by the square root of the mass ratio makes the number of π more than one digit: 3.141592654. < / P > < p > when brown happened to see Sanderson’s animation, he was thinking about Grover algorithm, and then he found that there was a significant similarity between the two. For example, the two quantum operations of Grover’s algorithm can correspond to ball and wall collisions respectively. The quality ratio corresponds to the size of the database. In addition, the final result is that the operands are proportional to π and the square root of the database size.. < / P > < p > apart from the amazing connection between these two systems, what role does π play in these two situations? Of course, the most famous part of π is that it is the ratio of the circumference of a circle to its diameter, but it also appears in the corresponding ratio of higher dimensional objects such as ellipses and balls. One of the methods to define the sphere is to give the qualification condition in the abscissa and ordinate of X and y by Algebra: the point on the circle with radius r satisfies the limited condition: x + y = R. It has been proved that both the above collision problem and Grover algorithm have this form of constraints. The colliding or manipulating quantum system of the ball corresponds to the rotation on the circle defined by these constraints. < / P > < p > for example, for two elastic spheres with mass m and m, the total kinetic energy of both is retained in elastic collision. To completely retain the kinetic energy of the big ball, we need to be at coordinate v_ M and V_ A 180 ° turn is made in the plane of M, and 180 ° is equal to π radian. Similarly, in quantum systems, the probability of observing a particular result is proportional to the square of the “wave function” corresponding to the result. The sum of the probabilities of the target and all other results must be 1. < / P > < p > one might ask, “does this provide important insights into the nature of the world? Or just a little curiosity? “Perhaps Grover’s algorithm can provide us with important knowledge about the nature of the world, perhaps the study of elastic spheres is to satisfy curiosity, and perhaps the reason why they are connected is more the second than the first. “< / P > < p > however, sometimes such connections can lead to some significant progress, and there are many cases in the history of physics and mathematics. For example, physicists have spent more than 20 years exploring the amazing correspondence between a strong interacting multiparticle quantum system and a gravitational model integrating a curved space-time of one dimension. Even wormholes in space-time are expected to solve the paradox related to the entanglement of distant particles in quantum mechanics. < / P > < p > mathematics is often developed through connections with different fields. However, it was not until a few centuries later that we used the simple method of proving the elliptic equation from the large one. In January, computer scientists proved a theorem related to Alan Turing’s concept of determinable computing, which further impacted other seemingly unrelated fields. < / P > < p > in Aronson’s opinion, the “correspondence” between Grover algorithm and elastic ball is accurate, but it may also be an interesting analogy. But that’s good enough. 」Continue ReadingVideo Number assistant internal test online! Four functions let you send 1g video on the computer

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