Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal – in some reasonable sense – or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries). Specifically, we show that the following algorithms are optimal: Grover’s algorithm for fast search in an unsorted array, teleportation algorithm – which is important for parallel quantum computations, and quantum annealing optimization algorithm. This covers many algorithms related to quantum computing. We also mention that algorithms for quantum communication and Deutsch-Josza algorithm – for fast checking whether a bit affect computation results – are optimal. In all these cases, optimality is shown not just for one specific optimality criterion, but for all possible optimality criteria that satisfy the natural invariance requirement.
Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal – in some reasonable sense – or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries). Specifically, we show that the following algorithms are optimal: Grover’s algorithm for fast search in an unsorted array, teleportation algorithm – which is important for parallel quantum computations, and quantum annealing optimization algorithm. This covers many algorithms related to quantum computing. We also mention that algorithms for quantum communication and Deutsch-Josza algorithm – for fast checking whether a bit affect computation results – are optimal. In all these cases, optimality is shown not just for one specific optimality criterion, but for all possible optimality criteria that satisfy the natural invariance requirement.
We are now ready to assemble quantum subroutines into quantum algorithms. A reasonable first question is: Are there problems which can be solved efficiently on a quantum computer? In his analysis of a universal quantum computer [25], Deutsch illustrated quantum parallelism by constructing a procedure to compute the parity f (0) ⊕ f (1) of a function on one bit. In a generalization of this context, Deutsch and Jozsa [27] defined a quantum algorithm that could solve a problem more efficiently than is possible on a classical computer. This problem helped motivate the search for algorithms that could solve “real” problems, and Simon [67] provided an example that inspired Shor’s subsequent work on the factoring problem and on the discrete logarithm problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Deutsch's algorithm and Grover's quantum search are theoretically interesting, but it is not obvious that these two algorithms are actually useful for anything important. We next consider a selection of more advanced algorithms, several of which may have real-life applications. Some of these will be too complicated to explain fully, and their properties will only be sketched briefly. The Deutsch–Jozsa algorithm Deutsch's algorithm is simple, but important, as it shows that a quantum algorithm can find a property of an unknown function (its parity) with a smaller number of queries than any possible classical algorithm (one rather than two). For this reason we can say that quantum computing is more efficient than classical computing within the oracle model of function evaluation. (It is widely believed that quantum computing is more efficient than classical computing in general, but this is a surprisingly hard thing to prove.) The simplicity of the algorithm is also an advantage, as it can be implemented on very primitive quantum computers. Beyond this, however, Deutsch's algorithm is also important as the simplest member of a large family of quantum algorithms, including most notably Shor's quantum factoring algorithm. The second simplest algorithm in the family is the Deutsch–Jozsa algorithm, which solves a very closely related problem. Consider an unknown binary function with n input bits, giving N = 2 n possible inputs, and a single output bit.
Search is one of the most commonly used primitives in quantum algorithm design. It is known that quadratic speedups provided by Grover’s algorithm are optimal, and no faster quantum algorithms for Search exist. While it is known that at least some quantum computation is required to achieve these speedups, the existing bounds do not rule out the possibility of an equally fast hybrid quantum-classical algorithm where most of the computation is classical. In this work, we study such hybrid algorithms, and we show that classical computation, unless it by itself can solve the Search problem, cannot assist quantum computation. In addition, we generalize this result to algorithms with subconstant success probabilities.
One of the main reasons for the current interest in quantum computing is that, in principle, quantum algorithms can break the RSA encoding, the encoding that is used for the majority secure communications - in particular, the majority of e-commerce transactions are based on this encoding. This does not mean, of course, that with the emergence of quantum computers, there will no more ways to secretly communicate: while the existing non-quantum schemes will be compromised, there exist a quantum cryptographic scheme that will enables us to secretly exchange information. In this scheme, however, there is a certain probability that an eavesdropper will not be detected. A natural question is: can we decrease this probability by an appropriate modification of the current quantum cryptography algorithm? In this paper, we show that such a decrease is not possible: the current quantum cryptography algorithm is, in some reasonable sense, optimal.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.
This article analyzes mathematical models of quantum parallelism based on the transfer of a quantum system into a superposition state. The principles of quantum parallelism and its mathematical foundations are substantiated. To demonstrate the quantum advantage, mathematical models of fundamental quantum algorithms Deutsch-Joža and Grover are analyzed, which illustrate the efficiency of quantum computing compared to classical methods. The Deutsch-Joshy algorithm is considered, which generalizes the problem of determining the type of a binary function by extending it to a set of arguments written in a quantum register. This algorithm allows determining whether a function is constant or balanced in a single quantum call, while in classical calculations this requires exponential time. It is noted that although the Deutsch-Joshy algorithm demonstrates the possibilities of parallel quantum computing, it is of mainly theoretical importance. Grover’s algorithm implements quantum parallel computing for the practical problem of searching for an element in an unordered database. It is based on iterative amplification of the amplitude of the state corresponding to the searched element and provides quadratic speedup compared to classical algorithms. In addition to database search, Grover’s algorithm can be adapted to solve other optimization problems. The work of the considered algorithms is illustrated by numerical examples, which simplifies the understanding of their principles and contributes to the further methodological development of parallel quantum computing. The article also outlines the advantages of quantum parallelism over classical computing and identifies the prospects for the application of parallel quantum algorithms.