Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms. 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. Eventual applications of quantum computing range from breaking cryptographic systems to the design of new medicines. 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.
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. 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.Contrary 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. Indeed, at the time of writing the ‘Quantum Algorithm Zoo’ website cites 262 papers on quantum algorithms. There are now a number of excellent surveys about quantum algorithms, – and we defer to these for details of the algorithms we cover here, and many more.
Shor's Algorithm
In particular, we omit all discussion of how the quantum algorithms mentioned work. We will also not cover the important topics of how to actually build a quantum computer (in theory or in practice) and quantum error-correction, nor quantum communication complexity or quantum Shannon theory. Measuring quantum speedupWhat 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.
In both the classical and quantum settings, we measure runtime by the number of elementary operations used by an algorithm. 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). To compare the performance of algorithms, we use computer science style notation O( f( n)), which should be interpreted as ‘asymptotically upper-bounded by f( n)’.We sometimes use basic ideas from computational complexity theory, and in particular the notion of complexity classes, which are groupings of problems by difficulty. See for informal descriptions of some important complexity classes. If a problem is said to be complete for a complexity class, then this means that it is one of the ‘hardest’ problems within that class: it is contained within that class, and every other problem within that class reduces to it.
One of the first applications of quantum computers discovered was Shor’s algorithm for integer factorisation. In the factorisation problem, given an integer N= p× q for some prime numbers p and q, our task is to determine p and q. The best classical algorithm known (the general number field sieve) runs in time exp( O(log N) 1/3(log log N) 2/3)) (in fact, this is a heuristic bound; the best rigorous bound is somewhat higher), while Shor’s quantum algorithm solves this problem substantially faster, in time O(log N) 3). This result might appear only of mathematical interest, were it not for the fact that the widely used RSA public-key cryptosystem relies on the hardness of integer factorisation. Shor’s efficient factorisation algorithm implies that this cryptosystem is insecure against attack by a large quantum computer.As a more specific comparison than the above asymptotic runtimes, in 2010 Kleinjung et al.
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. A detailed analysis of one fault-tolerant quantum computing architecture, making reasonable assumptions about the underlying hardware, suggests that a 2,000-bit number could be factorised by a quantum computer using 3×10 11 quantum gates, and approximately a billion qubits, running for just over a day at a clock rate of 10 MHz. This is clearly beyond current technology, but does not seem unrealistic as a long-term goal.Shor’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), then giving an efficient quantum algorithm for this problem. The HSP is parametrised by a group G, and Shor’s algorithm solves the case G= ℤ. Efficient solutions to the HSP for other groups G turn out to imply efficient algorithms to break other cryptosystems; we summarise some important cases of the HSP and some of their corresponding cryptosystems in.
Quantum Mechanics And Algorithms Book
Two particularly interesting cases of the HSP for which polynomial-time quantum algorithms are not currently known are the dihedral and symmetric groups. A polynomial-time quantum algorithm for the former case would give an efficient algorithm for finding shortest vectors in lattices; an efficient quantum algorithm for the latter case would give an efficient test for isomorphism of graphs (equivalence under relabelling of vertices). Pr A outputs w such that f ( w ) = 1 = k N,so we can find a w such that f( w)=1 with O ( N / k ) queries to f. However, we could imagine A being a more complicated algorithm or heuristic targeted at a particular problem we would like to solve.
For example, one of the most efficient classical algorithms known for the fundamental NP-complete constraint satisfaction problem 3-SAT is randomised and runs in time O((4/3) npoly( n)). Amplitude amplification can be applied to this algorithm to obtain a quantum algorithm with runtime O((4/3) n/2poly( n)), illustrating that quantum computers can speedup non-trivial classical algorithms for NP-complete problems.An interesting future direction for quantum algorithms is finding accurate approximate solutions to optimisation problems.
Recent work of Farhi, Goldstone and Gutmann gave the first quantum algorithm for a combinatorial task (simultaneously satisfying many linear equations of a certain form) which outperformed the best efficient classical algorithm known in terms of accuracy; in this case, measured by the fraction of equations satisfied. This inspired a more efficient classical algorithm for the same problem, leaving the question open of whether quantum algorithms for optimisation problems can substantially outperform the accuracy of their classical counterparts.
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