Estimating operator norms using covering nets (1509.05065v1)

Abstract: We present several polynomial- and quasipolynomial-time approximation schemes for a large class of generalized operator norms. Special cases include the $2\rightarrow q$ norm of matrices for $q>2$, the support function of the set of separable quantum states, finding the least noisy output of entanglement-breaking quantum channels, and approximating the injective tensor norm for a map between two Banach spaces whose factorization norm through $\ell_1^n$ is bounded. These reproduce and in some cases improve upon the performance of previous algorithms by Brand~ao-Christandl-Yard and followup work, which were based on the Sum-of-Squares hierarchy and whose analysis used techniques from quantum information such as the monogamy principle of entanglement. Our algorithms, by contrast, are based on brute force enumeration over carefully chosen covering nets. These have the advantage of using less memory, having much simpler proofs and giving new geometric insights into the problem. Net-based algorithms for similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in each case with a run-time that is exponential in the rank of some matrix. We achieve polynomial or quasipolynomial runtimes by using the much smaller nets that exist in $\ell_1$ spaces. This principle has been used in learning theory, where it is known as Maurey's empirical method.

• The paper introduces a novel covering net approach that approximates operator norms in polynomial- and quasipolynomial-time, notably over separable quantum states.
• The methodology leverages statistical concentration inequalities and geometric insights to bypass SDP hierarchies for quantum channel and matrix norm problems.
• The work extends the framework to multipartite settings and Banach spaces, offering significant implications for both theoretical research and practical quantum algorithms.

Estimating Operator Norms Using Covering Nets: A New Approach

This paper presents algorithms for estimating operator norms, with a focus on generalized operator norms arising in fields such as quantum information theory, theoretical computer science, and Banach space theory. The primary contribution is a set of polynomial- and quasipolynomial-time approximation schemes that improve or match the performance of previous algorithms, especially for problems related to separable states in quantum mechanics. The authors employ a novel approach based on brute force enumeration over covering nets instead of utilizing semidefinite programming (SDP) hierarchies.

Key Contributions and Methodology

The algorithms developed in the paper provide solutions for several complex problems that involve generalized operator norms:

1. Optimization over Separable States: The authors propose a more efficient net-based algorithm that estimates optimization over separable quantum states represented by bipartite density matrices. Prior approaches used quantum information-theoretic ideas to analyze the complexity through SDP hierarchies. The current work achieves similar precision by leveraging statistical concentration inequalities, such as the matrix Hoeffding bound, offering geometric insights into the problem.
2. Maximum Output Norms of Quantum Channels: The paper extends the scope to approximating the maximum output of entanglement-breaking quantum channels in the Schatten norms. The presented methods present an improvement by demonstrating feasible polynomial-time approximations not previously available due to complexity barriers in more general quantum scenarios.
3. Hypercontractive Norms and Matrix Norms: Building from the approach for separable states, the paper provides a framework for approximating matrix norms, specifically focusing on $2 \rightarrow q$ norms for matrices. Using the type-$\gamma$ constant of Banach spaces, the authors offer polynomial-time complexity approximations for various norm computations.
4. Extension to Multipartite Settings and Banach Spaces: The algorithms are scalable to multipartite tensor product spaces, extending the framework to general norms between Banach spaces. This generalization covers operator norms through factoring, which significantly helps in understanding and optimizing across complex input-output systems in high dimensions.

Implications and Speculative Future Directions

The approach delineated in the paper holds promising practical and theoretical implications. By providing more memory efficient and conceptually simpler proofs through geometric insights and covering net methods, there could be broader applications not only within quantum information processing but also in classical settings wherein similar complexity barriers are posed by tensor norms.

Theoretically, these advancements suggest potential new avenues for examining the inherent complexity of operator norms in Banach spaces. The construct of nets and their roles in approximating complex spaces can provide additional research directions in mathematical optimization and functional analysis.

Furthermore, the advancements achieved in this paper may inspire future research efforts toward exploring the interplay between covering nets and SDP hierarchies beyond combinatorial optimization and game theory, touching upon areas like machine learning, where such computational efficiency and insights can unlock new algorithmic capabilities. The paper also raises a foundational open question: the optimization of explicit decomposition forms for operator proofs, a challenge likely to fascinate both the combinatorial optimization community and theorists focused on semidefinite methodologies.

In closing, the methods introduced here not only enhance computational efficiency but also contribute to a better understanding of geometric and operational properties inherent to advanced spaces in quantum mechanics and beyond. As such, this work marks a significant methodological shift worth investigating in further contexts where problem structure leads naturally to formulations amenable to the techniques discussed.