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Shor's Algorithm
Shor's algorithm is a quantum algorithm, quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. On a quantum computer, to factor an integer N , Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in \log N , the size of the integer given as input. Specifically, it takes quantum logic gate, quantum gates of order O \! \left((\log N)^ (\log \log N) (\log \log \log N) \right) using fast multiplication, or even O \! \left((\log N)^ (\log \log N) \right) utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey and Van Der Hoven, thus demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is almost exponentially faster than the most efficient known classical factoring algorithm, the ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. Problems which are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA (cryptosystem)
RSA (Rivest–Shamir–Adleman) is a public-key cryptography, public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ) (the British signals intelligence agency) by the English mathematician Clifford Cocks. That system was classified information, declassified in 1997. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers. The security of RSA relies on the pract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Sieve
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by Carl Pomerance in 1981 as an improvement to Schroeppel's linear sieve. Basic aim The algorithm attempts to set up a congruence of squares modulo ''n'' (the integer to be factorized), which often leads to a factorization of ''n''. The algorithm works in two phases: the ''data collection'' phase, where it collects information that may lead to a congruence of squares; and the ''data processing'' phase, where it puts all the data it has collected into a matrix and solves it to obtain a congruence of square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 has th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Testing
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, ''n'', check whether it is evenly divisible by any prime number between 2 and (i.e. that the division leaves no remainder). If so, then ''n'' is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM Q System One
IBM Quantum System One is the first circuit-based commercial quantum computer, introduced by IBM in January 2019. This integrated quantum computing system is housed in a 2.7x2.7x2.7 m airtight glass cube that maintains a controlled physical environment. The cylinder protruding from the ceiling in the center is a dilution refrigerator, containing a 20-qubit transmon quantum processor. It was tested for the first time in the summer of 2018, for two weeks, in Milan, Italy. IBM Quantum System One was developed by IBM Research, with assistance from the Map Project Office and Universal Design Studio. CERN, ExxonMobil, Fermilab, Argonne National Laboratory and Lawrence Berkeley National Laboratory are among the clients signed up to access the system remotely. From April 6 to May 31, 2019, the Boston Museum of Science hosted an exhibit featuring a replica of the IBM Quantum System One. On June 15, 2021, IBM deployed the first unit of Quantum System One in Germany at its headquarters i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journal Citation Reports'' impact factor and the journal ''h''-index proposed by Google Scholar, many physicists and other scientists consider ''Physical Review Letters'' to be one of the most prestigious journals in the field of physics. ''According to Google Scholar, PRL is the journal with the 9th journal h-index among all scientific journals'' ''PRL'' is published as a print journal, and is in electronic format, online and CD-ROM. Its focus is rapid dissemination of significant, or notable, results of fundamental research on all topics related to all fields of physics. This is accomplished by rapid publication of short reports, called "Letters". Papers are published and available electronically one article at a time. When published in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Photonics
Photonics is a branch of optics that involves the application of generation, detection, and manipulation of light in form of photons through emission, transmission, modulation, signal processing, switching, amplification, and sensing. Though covering all light's technical applications over the whole spectrum, most photonic applications are in the range of visible and near-infrared light. The term photonics developed as an outgrowth of the first practical semiconductor light emitters invented in the early 1960s and optical fibers developed in the 1970s. History The word 'Photonics' is derived from the Greek word "phos" meaning light (which has genitive case "photos" and in compound words the root "photo-" is used); it appeared in the late 1960s to describe a research field whose goal was to use light to perform functions that traditionally fell within the typical domain of electronics, such as telecommunications, information processing, etc. Photonics as a field began with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nature (journal)
''Nature'' is a British weekly scientific journal founded and based in London, England. As a multidisciplinary publication, ''Nature'' features peer-reviewed research from a variety of academic disciplines, mainly in science and technology. It has core editorial offices across the United States, continental Europe, and Asia under the international scientific publishing company Springer Nature. ''Nature'' was one of the world's most cited scientific journals by the Science Edition of the 2019 ''Journal Citation Reports'' (with an ascribed impact factor of 42.778), making it one of the world's most-read and most prestigious academic journals. , it claimed an online readership of about three million unique readers per month. Founded in autumn 1869, ''Nature'' was first circulated by Norman Lockyer and Alexander Macmillan as a public forum for scientific innovations. The mid-20th century facilitated an editorial expansion for the journal; ''Nature'' redoubled its efforts in exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
