Computing quantum discord is NP-complete (1305.5941v3)

Abstract: We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NP-complete. Therefore, quantum discord is computationally intractable: the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum discord in a quantum system of moderate size is not possible in practice. As by-products, some entanglement measures (namely entanglement cost, entanglement of formation, relative entropy of entanglement, squashed entanglement, classical squashed entanglement, conditional entanglement of mutual information, and broadcast regularization of mutual information) and constrained Holevo capacity are NP-hard/NP-complete to compute. These complexity-theoretic results are directly applicable in common randomness distillation, quantum state merging, entanglement distillation, superdense coding, and quantum teleportation; they may offer significant insights into quantum information processing. Moreover, we prove the NP-completeness of two typical problems: linear optimization over classical states and detecting classical states in a convex set, providing evidence that working with classical states is generically computationally intractable.

• The paper conclusively proves that computing quantum discord is NP-complete, indicating exponential complexity as quantum systems scale.
• It leverages complexity theory to demonstrate that quantum discord extends beyond entanglement, impacting quantum computing and information protocols.
• The findings suggest that similar quantum measures and classical state optimizations may be NP-hard, prompting new research on efficient approximation methods.

An Analysis of NP-Completeness in Quantum Discord Computation

The research presented in "Computing quantum discord is NP-complete" by Yichen Huang provides a detailed inquiry into the computational complexity associated with quantum discord, a key measure of quantum correlations that extends beyond classical entanglement. The paper conclusively establishes that computing quantum discord is NP-complete, denoting its intractability within conventional computing methodologies.

Computational Complexity and Quantum Discord

Quantum discord has gained substantial attention as it captures quantum correlations that are not accounted for by entanglement alone. While entanglement is quantified by various metrics, discord encompasses a broader scope of correlations that can exist in quantum systems. Previous to this work, the computational complexity of quantum discord remained an open question, despite its practical relevance in quantum information protocols such as quantum computation and cryptography.

This work rigorously proves that determining quantum discord is NP-complete, implying that any algorithm aiming to compute this measure will experience an exponential increase in complexity as the dimension of the quantum system grows. The findings draw on the framework of computational complexity theory, where NP-complete problems hold the characteristic of being feasibly verifiable by a deterministic Turing machine, while no known polynomial-time algorithm can solve these problems for all instances.

Intractability of Related Problems

The implications of this discovery extend beyond quantum discord itself. The paper identifies several entanglement measures, such as entanglement cost, relative entropy of entanglement, and others as being NP-hard or NP-complete to compute. The computational intractability touches various quantum protocols and transformations, providing insights that impact the feasibility of efficiently processing quantum information.

Moreover, Huang explores the NP-completeness of optimizing over classical states, specifically highlighting the intrinsic difficulties when classical states are involved in quantum systems. This complexity suggests broad intractability in activities like linear optimization involving classical states within quantum mechanics.

Implications and Future Exploration

By establishing the NP-completeness of quantum discord, the paper sets the stage for several new research directions:

• Approximation Algorithms: Is it possible to formulate an efficient approximation algorithm that can compute quantum discord within a manageable error margin? Such a development could prove crucial for practical applications.
• Special Case Efficiency: Can quantum discord be computed efficiently for specific categories of quantum states, where structure might reduce complexity?
• Complexity of Other Measures: As quantum mechanics further explores correlation measures beyond entanglement, determining their computational complexity will remain pivotal.
• Applications to Gaussian States: Given their prevalence in quantum optics and quantum information, analyzing the computational complexity of quantum discord and related measures in Gaussian states could provide valuable insights.

Conclusion

This investigation underscores the computational limits present in current quantum computing tasks. Huang’s findings on the NP-completeness of quantum discord highlight significant challenges for large-scale quantum computation and information processing, necessitating innovative solutions to mitigate these barriers. As quantum technologies evolve, understanding these constraints will be central to advancing both theoretical understanding and practical capabilities in quantum science.