Efficient Computation of the Quantum Rate-Distortion Function Using Symmetry Reduction and Inexact Mirror Descent

Core Concepts

The authors present an efficient algorithm to compute the quantum rate-distortion function by exploiting symmetry properties of common problem instances and using an inexact mirror descent optimization approach.

Abstract

The paper focuses on efficiently computing the quantum rate-distortion function, which is an important tool in quantum information theory. The authors make the following key contributions:

Symmetry reduction: They show that for certain common distortion measures, the quantum rate-distortion problem possesses inherent symmetries that can be exploited to significantly reduce the problem dimension and computational complexity. For the case of the entanglement fidelity distortion measure and maximally mixed input state, they are able to obtain an explicit solution to the rate-distortion function.

Inexact mirror descent algorithm: The authors propose an inexact variant of the mirror descent algorithm to solve the quantum rate-distortion problem. They show how this algorithm is related to the Blahut-Arimoto and expectation-maximization methods previously used for similar problems. The inexact approach allows them to retain convergence guarantees while improving computational efficiency.

Numerical experiments: The authors present the first numerical experiments computing the quantum rate-distortion function for multi-qubit quantum channels, demonstrating the scalability of their approach compared to existing methods.

Overall, the paper provides a comprehensive framework for efficiently computing the quantum rate-distortion function, combining theoretical insights about problem structure with practical algorithmic developments.

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Efficient Computation of the Quantum Rate-Distortion Function

by Kerry He,Jam... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2309.15919.pdf

Deeper Inquiries

How can the symmetry reduction techniques presented in this paper be extended to other common distortion measures beyond the entanglement fidelity

The symmetry reduction techniques presented in the paper for the entanglement fidelity distortion measure can be extended to other common distortion measures by identifying suitable subgroups of the unitary group that preserve the structure of the input state. By finding subgroups that leave the input state invariant, similar to how the unitary subgroup G was chosen for the entanglement fidelity case, we can reduce the problem dimension and simplify the optimization process. This involves selecting subgroups that satisfy the condition gρAg† = ρA for all g in the subgroup, ensuring that the distortion measure remains unchanged under the group action. By leveraging the properties of these subgroups, we can effectively apply symmetry reduction techniques to a broader range of distortion measures in the quantum rate-distortion problem.

What are the limitations of the mirror descent approach, and are there other optimization frameworks that could be applied to the quantum rate-distortion problem

The mirror descent approach, while effective in certain cases, has limitations that may impact its applicability to the quantum rate-distortion problem. One limitation is the requirement for the function to be L-smooth relative to the Legendre function, which may not always hold for complex optimization problems. Additionally, the linear convergence of mirror descent is not guaranteed in all scenarios, leading to potential slow convergence rates for certain instances of the quantum rate-distortion function.
To address these limitations, alternative optimization frameworks such as primal-dual methods, interior-point methods, or stochastic optimization techniques could be applied to the quantum rate-distortion problem. Primal-dual methods offer the advantage of handling both primal and dual variables simultaneously, potentially improving convergence rates and efficiency. Interior-point methods are known for their effectiveness in solving convex optimization problems with large-scale constraints, which could be beneficial for the quantum rate-distortion problem with complex constraints. Stochastic optimization techniques could also be explored to handle the high-dimensional and non-convex nature of the problem, providing a different approach to finding optimal solutions efficiently.

What are the potential applications of efficiently computing the quantum rate-distortion function, and how could the insights from this work impact other areas of quantum information theory

Efficiently computing the quantum rate-distortion function has several potential applications in quantum information theory and related fields. One key application is in quantum communication, where understanding the optimal rate-distortion trade-off for quantum channels can lead to improved data transmission and storage protocols. By efficiently computing the quantum rate-distortion function, researchers and practitioners can design more robust quantum communication systems with enhanced error correction and data compression capabilities.
The insights gained from this work could also have implications for quantum cryptography, quantum error correction, and quantum machine learning. Understanding the properties of quantum channels that achieve the optimal rate-distortion trade-off can inform the development of secure communication protocols, error-resilient quantum computing schemes, and efficient quantum data processing algorithms. Overall, the advancements in computing the quantum rate-distortion function could pave the way for advancements in various areas of quantum information theory and quantum technologies.