Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.
Valid transformations between quantum states are necessarily described by completely positive maps, instead of just positive maps. Positive but not completely positive maps such as the transposition map cannot be implemented due to the existence of entanglement in composite quantum systems, but there are classes of states for which the positivity is guaranteed, e.g., states not correlated to other systems. In this paper, we introduce the concept of N-copy extension of maps to quantitatively analyze the difference between positive maps and completely positive maps. We consider implementations o…
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels -- that is, not completely positive -- can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simu…
Probing the Error-Mitigation Threshold with Matrix Product States
Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学联合量子研究所与联合量子信息与计算科学中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）
Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication or thinking would be possible. Efficient implementability for creating and reading language expressions is discussed and illustrated with a quantum mechanical model. The reason for interest in language is that language expressions can have meaning, either as an informal language or as a formal language associated with a mathematical or physical theory. It i…
Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the v…
Quantum system characterization techniques represent the front line in the identification and mitigation of noise in quantum computing, but can be expensive in terms of quantum resources and time to repeatedly employ. Another challenging aspect is that parameters governing the performance of various operations tend to drift over time, and monitoring these is hence a difficult task. One of the most promising characterization techniques, gate set tomography (GST), provides a self-consistent estimate of the completely positive, trace-preserving (CPTP) maps for a complete set of gates, as well as …
QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院量子信息与计算倡议）；School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）；Hubei Center for Applied Mathematics（湖北应用数学中心）；Hubei Key Laboratory of Computational Science（湖北省计算科学重点实验室）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）；School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学科学学院当代应用数学上海市重点实验室）；Key Laboratory of Computational Physical Sciences, Ministry of Education（教育部计算物理科学研究院）；Quantum Group, Department of Computer Science, University of Oxford（牛津大学计算机系量子研究组）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
