Quantum computing and quantum information pdf

Quantum computing and quantum information pdf how to#

ShorвЂ™s approach to integer factorisation is based on reducing the task to a special case of a mathematical problem known as the hidden subgroup problem (HSP), 15, 16 then giving an efficient quantum algorithm for this problem. This is clearly beyond current technology, but does not seem unrealistic as a long-term goal. A detailed analysis of one fault-tolerant quantum computing architecture, 7 making reasonable assumptions about the underlying hardware, suggests that a 2,000-bit number could be factorised by a quantum computer using ~3Г—10 11 quantum gates, and approximately a billion qubits, running for just over a day at a clock rate of 10вЂ‰MHz. 14 reported classical factorisation of a 768-bit number, using hundreds of modern computers over a period of 2 years, with a total computational effort of ~10 20 operations. ShorвЂ™s efficient factorisation algorithm implies that this cryptosystem is insecure against attack by a large quantum computer.ДЄs a more specific comparison than the above asymptotic runtimes, in 2010 Kleinjung et al. This result might appear only of mathematical interest, were it not for the fact that the widely used RSA public-key cryptosystem 13 relies on the hardness of integer factorisation. The best classical algorithm known (the general number field sieve) runs in time exp( O(log N) 1/3(log log N) 2/3)) 12 (in fact, this is a heuristic bound the best rigorous bound is somewhat higher), while ShorвЂ™s quantum algorithm solves this problem substantially faster, in time O(log N) 3). 11 In the factorisation problem, given an integer N= pГ— q for some prime numbers p and q, our task is to determine p and q. One of the first applications of quantum computers discovered was ShorвЂ™s algorithm for integer factorisation. If a problem is said to be complete for a complexity class, then this means that it is one of the вЂhardestвЂ™ problems within that class: it is contained within that class, and every other problem within that class reduces to it. See Table 1 for informal descriptions of some important complexity classes. We sometimes use basic ideas from computational complexity theory, 10 and in particular the notion of complexity classes, which are groupings of problems by difficulty. To compare the performance of algorithms, we use computer science style notation O( f( n)), which should be interpreted as вЂasymptotically upper-bounded by f( n)вЂ™. In the case of quantum computation, this can be measured using the quantum circuit model, where a quantum circuit is a sequence of elementary quantum operations called quantum gates, each applied to a small number of qubits (quantum bits). In both the classical and quantum settings, we measure runtime by the number of elementary operations used by an algorithm. What does it mean to say that a quantum computer solves a problem more quickly than a classical computer? As is typical in computational complexity theory, we will generally consider asymptotic scaling of complexity measures such as runtime or space usage with problem size, rather than individual problems of a fixed size.

Quantum computing and quantum information pdf how to#

We will also not cover the important topics of how to actually build a quantum computer 6 (in theory or in practice) and quantum error-correction, 7 nor quantum communication complexity 8 or quantum Shannon theory. In particular, we omit all discussion of how the quantum algorithms mentioned work. 1 There are now a number of excellent surveys about quantum algorithms, 2вЂ“ 5 and we defer to these for details of the algorithms we cover here, and many more. Indeed, at the time of writing the вЂQuantum Algorithm ZooвЂ™ website cites 262 papers on quantum algorithms. Here we aim to give a broad overview of quantum algorithmics, focusing on algorithms with clear applications and rigorous performance bounds, and including recent progress in the field.Д¬ontrary to a rather widespread popular belief that quantum computers have few applications, the field of quantum algorithms has developed into an area of study large enough that a brief survey such as this cannot hope to be remotely comprehensive. Although large-scale general-purpose quantum computers do not yet exist, the theory of quantum algorithms has been an active area of study for over 20 years. These applications are based on quantum algorithms-algorithms that run on a quantum computer and achieve a speedup, or other efficiency improvement, over any possible classical algorithm. Eventual applications of quantum computing range from breaking cryptographic systems to the design of new medicines. A quantum computer is a machine designed to use quantum mechanics to do things which cannot be done by any machine based only on the laws of classical physics.