Quantum Processes

Kazuya Yasuda, Takahiro Kubota, Yoshihiko Kakutani

In the study of quantum process algebras, researchers have introduced different notions of equivalence between quantum processes like bisimulation or barbed congruence. However, there are intuitively equivalent quantum processes that these notions do not regard as equivalent. In this paper, we introduce a notion of equivalence named observational equivalence into qCCS. Since quantum processes have both probabilistic and nondeterministic transitions, we introduce schedulers that solve nondeterministic choices and obtain probability distribution of quantum processes. By definition, the restrictions of schedulers change observational equivalence. We propose some definitions of schedulers, and investigate the relation between the restrictions of schedulers and observational equivalence.

Henry Pierce Stapp

Yuan Feng, Yuxin Deng, Mingsheng Ying

With the previous notions of bisimulation presented in the literature, to check if two quantum processes are bisimilar, we have to instantiate their free quantum variables with arbitrary quantum states, and verify the bisimilarity of the resulting configurations. This makes checking bisimilarity infeasible from an algorithmic point of view, because quantum states constitute a continuum. In this article, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.

Bob Coecke, Aleks Kissinger

Kanaga Priya Palanisamy, Thanga Revathi Shanmugakani, B. Gomathy +1 more

This chapter delves deeply into quantum process tomography (QPT) and regression: two critical approaches in the field of processing quantum information. The process of characterizing and estimating unknown quantum processes executed on quantum systems is known as quantum process tomography, while regression aims to find optimal parameters for input states to achieve desired unitary transformations. The chapter begins by introducing the concept of channel-state duality, which establishes a fundamental connection between quantum channels and quantum states. This duality enables the analysis of quantum processes by examining the input and output states, providing insights into their properties and behavior. Quantum process tomography is then discussed in detail, covering various tomographic methods and strategies used to reconstruct unknown quantum processes. Then the mathematical foundations, experimental techniques, and statistical analysis involved in quantum process tomography are explored to provide a comprehensive understanding of the field. Next, the chapter delves into the role of groups, the unitary group, and compact lie groups in the study of quantum processes. These mathematical concepts provide a framework for understanding the structure and symmetries of quantum processes, and representation theory is introduced as a powerful tool for decomposing and analyzing quantum processes into irreducible representations. The concept of storage of unitary and parallel application operations is crucial for the practical implementation of quantum processes insights about techniques such as parallelization and matrix factorization, which optimize the computational resources required for large-scale quantum systems, enabling efficient and scalable execution of quantum processes. The chapter also addresses the concept of finding optimal input states for learning quantum processes, adaptive measurement, and statistical optimization techniques, which enables researchers to identify the input states that provide the most informative data for accurate process estimation. Finally, the chapter concentrates on establishing the unitary operation and defining the parameterization for the initial state to achieve desired transformations. The effective learning criteria and quantum machine learning (QML) for leveraging quantum states provides advantages of QML over classical methods. In summary, this chapter provides a comprehensive exploration of quantum process tomography and regression. By leveraging channel-state duality, representation theory, and optimization techniques, researchers and practitioners can accurately characterize and estimate quantum processes. It also offers insights into the mathematical foundations, experimental methodologies, and practical applications of these techniques, contributing to the advancement of quantum information processing and technology.

Wolfram Schommers