We discuss possible applications of the 1-D direct and inverse scattering problem to design of universal quantum gates for quantum computation. The potentials generating some universal gates are described.
There has been a fast-growing interest in quantum computation in the context of high energy and nuclear physics. One important motivation is calculating real-time evolution in large Hilbert spaces which can be prohibitive with classical computers. It is possible to deal with the rapid oscillations of real-time evolution using quantum computers with a so far limited number of qubits. We have been able to measure phase shifts during the scattering process with real-time evolution for the collision by calculating the time delay between models with and without an external potential. To explore scattering problems with more intricate potentials, we use the phase shift to calculate the transmission probability, to find resonances, and to compare with results obtained by finite volume spectroscopy. Quantum computing shows possible advantages in simulating large quantum systems, which means that we can construct a suitable model in IBM qiskit with quantum gates and circuits to explore more resonances.
One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. One of the tools used in it is the Yang-Baxter equation whose solutions are interpreted as universal quantum gates. Lately, more general Yang-Baxter equations have been investigated, making progress as two-spectral equations and Yang-Baxter systems. This paper intends to apply these new findings to the field of topological quantum computation, more specifically, the proposition of the two-spectral Yang-Baxter operators as universal quantum gates for 2 qubits and 2 qutrits systems, obtaining 4x4 and 9x9 matrices respectively, and further elaboration of the corresponding Hamiltonian by the use of computer algebra software Mathematica® and its Qucalc package. In addition, possible physical systems to which the Yang-Baxter operators obtained can be applied are considered. In the present work it is demonstrated the utility of the Yang-Baxter equation to generate universal quantum gates and the power of computer algebra to design them; it is expected that these mathematical studies contribute to the further development of quantum computers
In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S-matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.