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Journal of Guizhou Education Institute

Six mathematical enumerative problems in quantum information and quantum computation

Guo Zhi-fang

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

By using the constructing proof methods, some mathematics problems in quantum communication are solved and some equalities in quantum information are proved with the knowledge of combinatorial computation, sequence of number and series. These results will be used in other scientific frelds deeply.

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We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, were well-studied in the literature, but are technically not easy. As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential. The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, absolutely maximally entangled states for four subsystems with six levels each and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is 2-copy distillable. An award for solving each of them is announced.
The development of quantum technologies will open up new perspectives in the use of quantum algorithms, the creation and modeling of complex physical and biological systems, new physical methods of transmitting, receiving and processing information. In turn, this will give impetus to the development of a large number of applications in the scientific, technical, economic and social spheres of society. Quantum cryptography is a communication protection method based on certain phenomena of quantum physics. Unlike traditional cryptography, which uses mathematical methods to ensure the secrecy of information, quantum cryptography focuses on physics, where information is carried using objects of quantum mechanics. The process of sending and receiving is always carried out by physical means, for example using electrons in an electric current, or photons in fiber-optic communication lines. In this process, the current situation was determined and the problems encountered and the solution suggestions for these problems were tried to be addressed.
All-to-all connections are required in general quantum annealing machines to solve various combinatorial optimization problems. The Lechner, Hauke, and Zoller method, which is used to realize the all-to-all connections, requires many-body interactions in locally connected qubits. Because most of the qubit interactions are two-body interactions, Lechner also proposed the construction of each four-body interaction by six controlled-NOT (CNOT) gates between two qubits. However, it is difficult to construct many CNOT gates. Herein, we show more concrete sequences to produce four-body and three-body interactions based on a two-dimensional solid-state qubit system. We show that the number of operations needed to construct the many-body interactions can be reduced using appropriate pulse sequences. These findings will help reduce quantum computation costs for solving combinatorial problems.
Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.