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Add to Wish List. Close Preview. Toggle navigation Additional Book Information. Description Table of Contents Editor s Bio. Summary Among the most exciting developments in science today is the design and construction of the quantum computer. Its realization will be the result of multidisciplinary efforts, but ultimately, it is mathematics that lies at the heart of theoretical quantum computer science.

Mathematics of Quantum Computation brings together leading computer scientists, mathematicians, and physicists to provide the first interdisciplinary but mathematically focused exploration of the field's foundations and state of the art.

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Each section of the book addresses an area of major research, and does so with introductory material that brings newcomers quickly up to speed. The preceding discussion is of course a very brief introduction to the concept of a quantum logic gate.

Please see the article on quantum logic gates for further information. To put the story together, we can describe a quantum computation as a network of quantum logic gates and measurements. One can always 'defer' a measurement to the end of a quantum computation, though this can come at a computational cost according to some cost models. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. For more details on the sequences of operations used for various quantum algorithms , see universal quantum computer , Shor's algorithm , Grover's algorithm , Deutsch—Jozsa algorithm , amplitude amplification , quantum Fourier transform , quantum gate , quantum adiabatic algorithm and quantum error correction.

One can represent any quantum computation as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem. Integer factorization , which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers e.

This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time in the number of digits of the integer algorithm for solving the problem.

## The Mathematics of Quantum Information

In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

However, other cryptographic algorithms do not appear to be broken by those algorithms. Quantum cryptography could potentially fulfill some of the functions of public key cryptography. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking. Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, [19] including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials , and solving Pell's equation.

No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. The most well-known example of this is quantum database search , which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by classical algorithms. In this case, the advantage is not only provable but also optimal, it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups.

## Mathematics of Quantum Computing

Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees. Problems that can be addressed with Grover's algorithm have the following properties:. For problems with all these properties, the running time of Grover's algorithm on a quantum computer will scale as the square root of the number of inputs or elements in the database , as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied [21] is Boolean satisfiability problem.

In this instance, the database through which the algorithm is iterating is that of all possible answers. An example and possible application of this is a password cracker that attempts to guess the password or secret key for an encrypted file or system. Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.

Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question.

The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. The Quantum algorithm for linear systems of equations or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts. John Preskill has introduced the term quantum supremacy to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field.

### Computing + Mathematical Sciences

IBM said in that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark. There are a number of technical challenges in building a large-scale quantum computer, [34] and thus far quantum computers have yet to solve a problem faster than a classical computer.

One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits.

Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T 2 for NMR and MRI technology, also called the dephasing time , typically range between nanoseconds and seconds at low temperature.

As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.

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These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time. As described in the Quantum threshold theorem , if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence.

This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them.

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Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L 2 , where L is the number of qubits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L.

For a bit number, this implies a need for about 10 4 bits without error correction. A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons , quasi-particles used as threads and relying on braid theory to form stable logic gates.

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are:. The quantum Turing machine is theoretically important but the direct implementation of this model is not pursued. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.

For physically implementing a quantum computer, many different candidates are being pursued, among them distinguished by the physical system used to realize the qubits :. A large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. There is also a vast amount of flexibility. In , Paul Benioff describes the first quantum mechanical model of a computer. In this work, Benioff showed that a computer could operate under the laws of quantum mechanics by describing a Schrodinger equation description of Turing machines , laying a foundation for further work in quantum computing.

The paper [2] was submitted in June and published in April of Russian mathematician Yuri Manin then motivates the development of quantum computers. Benioff built on his earlier work showing that a computer can operate under the laws of quantum mechanics. In , Paul Benioff further develops his original model of a quantum mechanical Turing machine.

## reference request - On Mathematical Arguments Against Quantum Computing - MathOverflow

In , David Deutsch describes the first universal quantum computer. Just as a Universal Turing machine can simulate any other Turing machine efficiently Church-Turing thesis , so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown. In , Bikas K. In , David Deutsch and Richard Jozsa propose a computational problem that can be solved efficiently with the determinist Deutsch—Jozsa algorithm on a quantum computer, but for which no deterministic classical algorithm is possible. This was perhaps the earliest result in the computational complexity of quantum computers, proving that they were capable of performing some well-defined computational task more efficiently than any classical computer.

In , an international group of six scientists, including Charles Bennett, showed that perfect quantum teleportation is possible [65] in principle, but only if the original is destroyed. Shor's algorithm can theoretically break many of the Public-key cryptography systems in use today, [66] sparking a tremendous interest in quantum computers. In , the DiVincenzo's criteria are published, which are a list of conditions that are necessary for constructing a quantum computer, proposed by the theoretical physicist David P.

In , researchers demonstrated Shor's algorithm to factor 15 using a 7- qubit NMR computer. In , researchers at the University of Michigan built a semiconductor chip ion trap. Such devices from standard lithography may point the way to scalable quantum computing.