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PDQMA = DQMA = NEXP: QMA With Hidden Variables and Non-collapsing Measurements


Core Concepts
The author explores the equivalence between PDQMA, DQMA, and NEXP by introducing hidden variables and non-collapsing measurements in the context of Quantum Merlin Arthur (QMA).
Abstract
The content delves into the relationship between quantum complexity classes like PDQMA, DQMA, and NEXP by incorporating hidden variables and non-collapsing measurements. It discusses various findings in quantum mechanics, computational power implications, and proofs of equivalence between these classes.
Stats
Our main result is that PDQMA = NEXP. QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP. A quantum computer augmented with quantum advice can solve any decision problem in polynomial time. The class PDQP/qpoly equals ALL. DQP/qpoly also equals ALL.
Quotes
"We define complexity classes based on 'fantasy' versions of quantum mechanics." - Scott Aaronson "Quantum mechanics over different fields leads to similar computational power." - Research Findings "A surprising new wrinkle came in 2018 when Aaronson observed that PDQP/qpoly = ALL." - Scott Aaronson "Non-collapsing measurements combined with one other resource yield extraordinary computational power." - The Paper "Our proof that MIP ⊆ PDQMA builds on previous work by Aaronson." - The Authors

Key Insights Distilled From

by Scott Aarons... at arxiv.org 03-06-2024

https://arxiv.org/pdf/2403.02543.pdf
PDQMA = DQMA = NEXP

Deeper Inquiries

Is it possible to replace the verifiers in these quantum complexity classes with variants that do not lead to NEXP

While the current findings show that modifying quantum complexity classes by giving verifiers access to non-collapsing measurements, hidden-variable histories, or non-negative witnesses leads to an increase in computational power up to NEXP, it is theoretically possible to explore variants of these classes that do not result in such a significant jump. One potential approach could involve restricting the capabilities of the verifier or introducing constraints on the types of computations allowed within the class. By carefully designing and defining new variants with limited computational abilities, it may be feasible to avoid reaching NEXP-level complexity.

What are the implications of these findings for practical applications in quantum computing

The implications of these findings for practical applications in quantum computing are profound. The ability to leverage non-collapsing measurements and hidden variables to achieve computational tasks up to NEXP demonstrates a significant expansion in quantum computing capabilities beyond traditional boundaries. This advancement opens up new possibilities for solving complex problems efficiently using quantum algorithms and protocols that incorporate these enhanced features. Practical applications could include optimization tasks, cryptography, simulation of physical systems, and other computationally intensive processes where classical methods fall short.

How does the introduction of hidden variables impact traditional understandings of quantum mechanics

The introduction of hidden variables into traditional understandings of quantum mechanics challenges some fundamental principles of the standard interpretation. In conventional quantum theory, hidden variables were often considered incompatible due to their potential impact on concepts like superposition and entanglement. However, incorporating hidden variables can lead to novel models like DQMA (Dynamical Quantum Merlin-Arthur) that demonstrate increased computational power compared to standard QMA (Quantum Merlin Arthur). This suggests a more nuanced relationship between observable phenomena and underlying mechanisms at play in quantum systems.
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