Quantum Problems in Higher Dimensions

Guowu Meng

The Kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance. The MICZ-Kepler problems are its natural cousins and have been previously generalized from dimension 3 to dimension 5. In this paper, we construct and analyze the (quantum) MICZ-Kepler problems in all dimensions higher than 2.

Edward L. Hill

Sabri Gündüz, İhsan Yılmaz

Abstract The Grover Adaptive Search (GAS) is an innovative approach offering solutions to optimization problems thanks to quadratic speedup enabled by the Grover Search Algorithm. However, the need for more qubits, depending on the number of variables in optimization problems, creates problems in providing optimal solutions. The central premise of this study is the principle that d -dimension is more efficient than 2-dimension could lead to more advantageous results when the GAS algorithm is adapted to d -dimension. To that end, this study adapts the GAS algorithm to the d -dimension drawing on quantum computing in the d -dimension, allowing computing using fewer quantum resources, namely, qudits. The proposed algorithm is validated against the Quadratic and Higher-Order Unconstrained Binary Optimization (QUBO-HUBO). This study concludes the d -dimensional GAS algorithm achieves solutions using fewer qudits for these problems in the QUBO and the HUBO. This study underscores the progress achieved in the advancement of high-dimensional quantum computing.

Amin Faraji Astaneh, Gary William Gibbons, Sergey N. Solodukhin

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces that minimize the Willmore (bending) energy and eventually to the Willmore conjecture. We propose a generalization of the Willmore energy in higher dimensions and analyze its minimizers in a general class of topologies ${S}^{m}\ifmmode\times\else\texttimes\fi{}{S}^{n}$ and make certain observations and conjectures that may have some mathematical significance.

Thalanayar S. Santhanam, A. R. Tekumalla