Quantum Bits, Gates and Registers. Quantum Fourier Transform and Applications including integer factorization. Fault Tolerant Quantum Error Correction. Fundamentals of Quantum Theory:
Cryptography[ edit ] Integer factorizationwhich 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.
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.
AES would have the same security against an attack using Grover's algorithm that AES has against classical brute-force search see Key size. 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.
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. 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: There is no searchable structure in the collection of possible answers, The number of possible answers to check is the same as the number of inputs to the algorithm, and There exists a boolean function which evaluates each input and determines whether it is the correct answer 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 databaseas opposed to the linear scaling of classical algorithms.
A general class of problems to which Grover's algorithm can be applied  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.
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.
Solving linear equations[ edit ] 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.Ewin made the problem into his senior thesis.
He poked holes in a particularly well-known quantum algorithm, which some computer scientists from another institution had famously suggested would require a quantum computer.
Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement. A quantum computer is a device that performs quantum computing. Such a computer is different from binary digital electronic computers based on transistors.
protocols that are safe to use in a ubiquitous quantum computing world. Along the way, we resolve In this thesis, we address the questions above by giving new quantum security models for classical protocols, and new constructions and analysis techniques for these models.
Topics in quantum computing.
There are a variety of topics that "traditionally" are covered in a quantum computing course. I list some of this on this website to give a first impression of what is required of a quantum computing course.
Quantum computers oﬀer a credible challenge to the strong Church-Turing thesis, as there exist eﬃcient algorithms for which there are no classical counterparts. Schoelkopf Lab’s research focuses on the development of superconducting devices for quantum information processing, which will lead to revolutionary advances in computing.