Physical Implementability of Linear Maps and Its Application in Error Mitigation

Abstract

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.

25 pages, v2 accepted by Quantum

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Cited by in corpus (32)

• Quantum Error Mitigation
• Fundamental limits of quantum error mitigation
• Quasiprobability decompositions with reduced sampling overhead
• Universal Sampling Lower Bounds for Quantum Error Mitigation
• Universal cost bound of quantum error mitigation based on quantum estimation theory
• Quantum error mitigation via matrix product operators
• Detecting and quantifying entanglement on near-term quantum devices
• Virtual quantum resource distillation
• Virtual quantum broadcasting
• Optimal quantum dataset for learning a unitary transformation
• Virtual quantum resource distillation: General framework and applications
• Noise effects on purity and quantum entanglement in terms of physical implementability
• Qubit noise deconvolution
• Uniqueness of quantum state over time function
• Single Qubit Error Mitigation by Simulating Non-Markovian Dynamics
• Reversing Unknown Quantum Processes via Virtual Combs for Channels with Limited Information
• Experimental virtual distillation of entanglement and coherence
• Information recoverability of noisy quantum states
• Mitigating Quantum Errors via Truncated Neumann Series
• Optimal unilocal virtual quantum broadcasting
• On the unraveling of open quantum dynamics
• Compressed quantum error mitigation
• Shadow Simulation of Quantum Processes
• Multiqubit noise deconvolution and characterization
• Noisy Probabilistic Error Cancellation and Generalized Physical Implementability
• Power of quantum measurement in simulating unphysical operations
• Detecting and Eliminating Quantum Noise of Quantum Measurements
• Agnostic Process Tomography
• Virtual Quantum Markov Chains
• Virtual phase-covariant quantum broadcasting for qubits
• Virtual Cloning of Quantum States
• Reduced Sampling Overhead for Probabilistic Error Cancellation by Pauli Error Propagation