insight - Quantum Computing Complexity Theory - # Perfect Zero Knowledge Protocols for Multiprover Interactive Proof Systems with Entangled Provers

Efficient Two-Prover Perfect Zero Knowledge Protocols for Recursively Enumerable Languages

Q: How can the techniques developed in this paper be extended to obtain perfect zero knowledge protocols with polylog question length, constant answer length, and constant soundness gap?

To achieve perfect zero knowledge protocols with polylog question length, constant answer length, and a constant soundness gap, we can explore the following extensions of the techniques developed in the paper: Improved Analysis of Subdivision: Enhance the analysis of context subdivision transformations to ensure that the resulting protocols maintain a constant soundness gap while reducing question length to polylogarithmic levels. This may involve refining the subdivision process to optimize the trade-off between question length reduction and soundness preservation. Exploring New Classical Transformations: Investigate additional classical transformations of constraint systems that can be shown to preserve quantum soundness. By identifying and utilizing transformations that efficiently maintain soundness properties in the quantum setting, it may be possible to construct perfect zero knowledge protocols with the desired characteristics. Utilizing Parallel Repetition: Leverage parallel repetition techniques to reduce an inverse-polynomial soundness gap to a constant soundness gap. By applying parallel repetition judiciously in the protocol construction, it is possible to achieve perfect zero knowledge with polylog question length and constant answer length while ensuring a constant soundness gap. By combining these strategies and potentially exploring new avenues for protocol design and analysis, it is feasible to extend the techniques developed in the paper to create perfect zero knowledge protocols meeting the specified criteria.

Q: What is the exact characterization of the complexity class MIPco, and can the results be extended to show that all languages in MIPco have a perfect zero knowledge commuting operator protocol?

The complexity class MIPco consists of languages that can be decided by multi-prover interactive proof systems with commuting operator strategies. The exact characterization of MIPco is not definitively known, but there is a conjecture that MIPco is equivalent to the co-RE (complement of the recursively enumerable) class. This conjecture suggests that MIPco captures languages whose complements can be recognized by a multi-prover interactive proof system with commuting operator strategies. The results obtained in the paper can potentially be extended to demonstrate that all languages in MIPco have a perfect zero knowledge commuting operator protocol. By leveraging the techniques developed for BCS-MIPco protocols and adapting them to the setting of commuting operator strategies, it may be feasible to construct efficient and secure perfect zero knowledge protocols for languages in MIPco. This extension would involve adapting the protocol construction and analysis to accommodate the unique characteristics of commuting operator strategies while ensuring perfect zero knowledge properties.

Q: Are there other classical transformations of constraint systems that can be shown to preserve quantum soundness, and how can these be used to construct efficient perfect zero knowledge protocols?

There are various classical transformations of constraint systems that can be demonstrated to preserve quantum soundness when transitioning to the quantum setting. One approach is to explore transformations that maintain the structure and constraints of the classical system while ensuring compatibility with quantum principles. These transformations can be used to construct efficient perfect zero knowledge protocols by leveraging the following strategies: Constraint-by-Constraint Transformations: Implement transformations that operate on individual constraints of the system, ensuring that the quantum version retains the soundness properties of the classical constraints. By systematically applying these transformations, it is possible to construct quantum protocols with preserved soundness. Algebraic Mapping Techniques: Utilize algebraic mappings between classical and quantum constraint systems to ensure that the quantum version maintains the integrity and soundness of the original constraints. By establishing clear mappings and preserving the logical structure of the constraints, efficient perfect zero knowledge protocols can be developed. Optimizing Reductions: Optimize reductions between classical and quantum constraint systems to enhance the efficiency and soundness of the resulting protocols. By refining the reduction techniques and ensuring soundness preservation, it is feasible to construct robust perfect zero knowledge protocols based on classical transformations. By employing these classical transformation techniques and adapting them to the quantum context while focusing on soundness preservation, it is possible to construct efficient perfect zero knowledge protocols for a wide range of applications in quantum computing and cryptography.

Core Concepts

Every language in the complexity class MIP*, which contains all recursively enumerable languages, has a two-prover one-round perfect zero knowledge interactive proof system.

Abstract

The content discusses the construction of efficient two-prover perfect zero knowledge (PZK) protocols for the complexity class MIP*, which contains all recursively enumerable languages.

Key highlights:

The recent MIP* = RE theorem shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages.
Prior work showed that every language in MIP* has a PZK-MIP* protocol, but the construction required six provers.
This paper shows that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering a natural question.
The proof uses a new method based on the key consequence of the MIP* = RE theorem that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games.
The authors develop a toolkit for analyzing quantum soundness of reductions between BCS games, which applies to both quantum and commuting operator strategies.
The results also have applications for the membership problem of quantum and commuting operator correlations.

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Quotes

"Every language in MIP* (and hence in RE) admits a two-prover one-round perfect zero knowledge MIP* protocol with completeness probability c = 1 and soundness probability s = 1/2, in which the verifier chooses questions uniformly at random."

Key Insights Distilled From

Two prover perfect zero knowledge for MIP*

by Kieran Maste... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00926.pdf

Two prover perfect zero knowledge for MIP*

Deeper Inquiries

How can the techniques developed in this paper be extended to obtain perfect zero knowledge protocols with polylog question length, constant answer length, and constant soundness gap?

To achieve perfect zero knowledge protocols with polylog question length, constant answer length, and a constant soundness gap, we can explore the following extensions of the techniques developed in the paper:

Improved Analysis of Subdivision: Enhance the analysis of context subdivision transformations to ensure that the resulting protocols maintain a constant soundness gap while reducing question length to polylogarithmic levels. This may involve refining the subdivision process to optimize the trade-off between question length reduction and soundness preservation.

Exploring New Classical Transformations: Investigate additional classical transformations of constraint systems that can be shown to preserve quantum soundness. By identifying and utilizing transformations that efficiently maintain soundness properties in the quantum setting, it may be possible to construct perfect zero knowledge protocols with the desired characteristics.

Utilizing Parallel Repetition: Leverage parallel repetition techniques to reduce an inverse-polynomial soundness gap to a constant soundness gap. By applying parallel repetition judiciously in the protocol construction, it is possible to achieve perfect zero knowledge with polylog question length and constant answer length while ensuring a constant soundness gap.

By combining these strategies and potentially exploring new avenues for protocol design and analysis, it is feasible to extend the techniques developed in the paper to create perfect zero knowledge protocols meeting the specified criteria.

What is the exact characterization of the complexity class MIPco, and can the results be extended to show that all languages in MIPco have a perfect zero knowledge commuting operator protocol?

The complexity class MIPco consists of languages that can be decided by multi-prover interactive proof systems with commuting operator strategies. The exact characterization of MIPco is not definitively known, but there is a conjecture that MIPco is equivalent to the co-RE (complement of the recursively enumerable) class. This conjecture suggests that MIPco captures languages whose complements can be recognized by a multi-prover interactive proof system with commuting operator strategies.
The results obtained in the paper can potentially be extended to demonstrate that all languages in MIPco have a perfect zero knowledge commuting operator protocol. By leveraging the techniques developed for BCS-MIPco protocols and adapting them to the setting of commuting operator strategies, it may be feasible to construct efficient and secure perfect zero knowledge protocols for languages in MIPco. This extension would involve adapting the protocol construction and analysis to accommodate the unique characteristics of commuting operator strategies while ensuring perfect zero knowledge properties.

Are there other classical transformations of constraint systems that can be shown to preserve quantum soundness, and how can these be used to construct efficient perfect zero knowledge protocols?

There are various classical transformations of constraint systems that can be demonstrated to preserve quantum soundness when transitioning to the quantum setting. One approach is to explore transformations that maintain the structure and constraints of the classical system while ensuring compatibility with quantum principles. These transformations can be used to construct efficient perfect zero knowledge protocols by leveraging the following strategies:

Constraint-by-Constraint Transformations: Implement transformations that operate on individual constraints of the system, ensuring that the quantum version retains the soundness properties of the classical constraints. By systematically applying these transformations, it is possible to construct quantum protocols with preserved soundness.

Algebraic Mapping Techniques: Utilize algebraic mappings between classical and quantum constraint systems to ensure that the quantum version maintains the integrity and soundness of the original constraints. By establishing clear mappings and preserving the logical structure of the constraints, efficient perfect zero knowledge protocols can be developed.

Optimizing Reductions: Optimize reductions between classical and quantum constraint systems to enhance the efficiency and soundness of the resulting protocols. By refining the reduction techniques and ensuring soundness preservation, it is feasible to construct robust perfect zero knowledge protocols based on classical transformations.

By employing these classical transformation techniques and adapting them to the quantum context while focusing on soundness preservation, it is possible to construct efficient perfect zero knowledge protocols for a wide range of applications in quantum computing and cryptography.