Information recoverability of noisy quantum states

Xuanqiang Zhao1,2, Benchi Zhao1, Zihan Xia1, and Xin Wang1

1Institute for Quantum Computing, Baidu Research, Beijing 100193, China
2QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

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Extracting classical information from quantum systems is an essential step of many quantum algorithms. However, this information could be corrupted as the systems are prone to quantum noises, and its distortion under quantum dynamics has not been adequately investigated. In this work, we introduce a systematic framework to study how well we can retrieve information from noisy quantum states. Given a noisy quantum channel, we fully characterize the range of recoverable classical information. This condition allows a natural measure quantifying the information recoverability of a channel. Moreover, we resolve the minimum information retrieving cost, which, along with the corresponding optimal protocol, is efficiently computable by semidefinite programming. As applications, we establish the limits on the information retrieving cost for practical quantum noises and employ the corresponding protocols to mitigate errors in ground state energy estimation. Our work gives the first full characterization of information recoverability of noisy quantum states from the recoverable range to the recovering cost, revealing the ultimate limit of probabilistic error cancellation.

In many quantum algorithms and protocols, extracting shadow information, i.e., some observables’ expectation values, from quantum systems is one of the most crucial steps. However, undesirable noises modeled as quantum channels inevitably distort the information, yielding misleading results. Though a practically relevant and theoretically significant question, how quantum channels affect shadow information embedded in quantum systems is poorly understood.

In this work, we propose a systematic framework to answer this question from an operational perspective. Building a connection between the Schrödinger picture and the Heisenberg picture, we give a necessary and sufficient condition that fully characterizes the shadow information recoverability. Based on this condition, we define the first measure that quantifies a quantum channel’s destructivity to shadow information, which is related to the rank of the channel's matrix representation.

Our results also provide an error mitigation scheme with optimal costs, allowing the efficient recovery of shadow information from a wide range of noises, ensuring algorithms such as variational quantum eigensolver run faithfully on noisy intermediate-scale quantum devices.

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