Classically Replaceable Operations

Guoding Liu, Xingjian Zhang, and Xiongfeng Ma

Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084 China

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Quantum information science provides powerful technologies beyond the scope of classical physics. In practice, accurate control of quantum operations is a challenging task with current quantum devices. The implementation of high fidelity and multi-qubit quantum operations consumes massive resources and requires complicated hardware design to fight against noise. An approach to alleviating this problem is to replace quantum operations with classical processing. Despite the common practice of this approach, rigorous criteria to determine whether a given quantum operation is replaceable classically are still missing. In this work, we define the classically replaceable operations in four general scenarios. In each scenario, we provide their necessary and sufficient criteria and point out the corresponding classical processing. For a practically favorable case of unitary classically replaceable operations, we show that the replaced classical processing is deterministic. Beyond that, we regard the irreplaceability of quantum operations by classical processing as a quantum resource and relate it to the performance of a channel in a non-local game, as manifested in a robustness measure.

Implementation of high-fidelity multi-qubit quantum operations is still challenging with current quantum devices. On the contrary, classical processing is generally easier to implement and has higher accuracy. The tractability of classical processing motivates us to replace some quantum operations in a task with classical ones. Such an idea has helped quantum key distribution become the earliest practical application in quantum information science. It is also implicitly employed in recent-developed quantum-classical-hybrid algorithms, such as variational quantum algorithms.

In our work, we consolidate the concept of “classical replacement” with a mathematical definition and provide necessary and sufficient criteria for the replaceability of a quantum operation. Based on this, we apply this classical replacement to variational quantum algorithms. From the quantum foundation point of view, the clarification of classical replaceability and classical irreplaceability manifests a new kind of quantumness. Using the characterization of classically replaceable operations, we reveal the links between classical replaceability, coherence, and entanglement. Regarding irreplaceability as a quantum resource, we establish a channel resource theory. Additionally, we use a robustness measure to quantify the irreplaceability and link this measure to the advantage that a quantum channel can provide over classically replaceable operations in a nonlocal game.

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