Virtual Quantum Markov Chains

The framework of virtual quantum recovery.

Abstract

Quantum Markov chains generalize classical Markov chains for random variables to the quantum realm and exhibit unique inherent properties, making them an important feature in quantum information theory. In this work, we propose the concept of virtual quantum Markov chains (VQMCs), focusing on scenarios where subsystems retain classical information about global systems from measurement statistics. As a generalization of quantum Markov chains, VQMCs characterize states where arbitrary global shadow information can be recovered from subsystems through local quantum operations and measurements. We present an algebraic characterization for virtual quantum Markov chains and show that the virtual quantum recovery is fully determined by the block matrices of a quantum state on its subsystems. Notably, we find a distinction between two classes of tripartite entanglement by showing that the W state is a VQMC while the GHZ state is not. Furthermore, we establish semidefinite programs to determine the optimal sampling overhead and the robustness of virtual quantum Markov chains. We demonstrate the optimal sampling overhead is additive, indicating no free lunch to further reduce the sampling cost of recovery from parallel calls of the VQMC states. Our findings elucidate distinctions between quantum Markov chains and virtual quantum Markov chains, extending our understanding of quantum recovery to scenarios prioritizing classical information from measurement statistics.

Publication
arXiv:2312.02031
Yu-Ao Chen
Yu-Ao Chen
Research Associate

I obtained my BS in Mathematics and Applied Mathematics from University of Science and Technology of China. I obtained my PhD degree in Applied Mathematics from University of Chinese Academy of Sciences under the supervision of Prof. Xiao-Shan Gao. My research interests include quantum computing, symbolic computation and cryptanalysis.

Chengkai Zhu
Chengkai Zhu
PhD Student

I obtained my BS in Applied Mathematics from China Agricultural University under the supervision of Prof. Zhencai Shen. I obtained my MS degree in Cyberspace Security from University of Chinese Academy of Sciences under the supervision of Prof. Zhenyu Huang. My research interests include quantum information theory and quantum computation.

Keming He
Keming He
Research Assistant

I obtained my BS in Electronic Information Science and Technology from Chongqing University. I obtained my MS degree in Electrical Engineering from University of Southern California. My research interests include quantum information theory and quantum computation.

Mingrui Jing
Mingrui Jing
PhD Student

I obtained my BS and MS degrees in physics from the University of Melbourne. My research interests include distributed quantum computing, quantum entanglement and quantum machine learning.

Xin Wang
Xin Wang
Associate Professor

The main focus of my research is to better understand the limits of information processing with quantum systems and the power of quantum artificial intelligence.