Open Problems

Ian Jauslin

Gabriel Coutinho, Krystal Guo

Abstract Quantum walks on graphs are fundamental to quantum computing and have led to many interesting open problems in algebraic graph theory. This review article highlights three key classes of open problems in this domain: perfect state transfer, instantaneous uniform mixing, and average mixing matrices. In highlighting these open problems, our aim is to stimulate further research and exploration in this rapidly evolving field.

Yousef Hindy, Jessica Pointing, Meltem Tolunay +3 more

The determination of optimal open-pit profiles is a well-studied combinatorial optimization problem, with profound technical and conceptual relevance in computational mining. The ongoing evolution of quantum computing hardware and the recent advances of heuristic quantum algorithms make it worthwhile to explore the solution of the open-pit mining problem on quantum computers. In this work, we cast the open-pit mining problem as a Hamiltonian ground-state search problem, which in turn we solve with a dedicated implementation of the variational quantum eigensolver algorithm, and we propose a domain decomposition approach to extend the reach of today's small scale quantum hardware. The procedure is demonstrated on IBMQ devices using four qubits. This is the first example, to the best of our knowledge, of open-pit profile calculations being performed on quantum hardware.

Luca Salasnich, Francesco Lorenzi

Gennaro Auletta

Haoxiang Li

Quantum mechanics has a history more than 100 years, and there are still many challenging open-ended questions. The problems encountered in modern physics such as matrix mechanics and quantum information often involve both quantum theory and open-ended problems in modern mathematics such as linear algebra. Distillability problem is an important open-ended problem in the field of quantum information and linear algebra. This article recalls the development history and application of linear algebra, summarizes the background and progress of distillability conjecture, and attempts to explore this problem by summarizing and using the knowledge of linear algebra and matrix theory learned in the course.