QMA vs. QCMA and Pseudorandomness (2411.14416v4)

Abstract: We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $\mathsf{QMA}$ from $\mathsf{QCMA}$. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions. Our result can be viewed as establishing a "win-win" scenario: either there is a classical oracle separation of $\mathsf{QMA}$ from $\mathsf{QCMA}$, or there is quantum advantage in distinguishing pseudorandom distributions on permutations.

• The paper presents a conditional result demonstrating that a classical oracle can separate QMA from QCMA when quantum pseudorandomness holds.
• It employs permutation-based analyses to argue that dense sources appear pseudorandom to polynomial-query quantum algorithms.
• The findings suggest significant implications for cryptography and theoretical advances in distinguishing quantum from classical proofs.

$QMA$ vs. $QCMA$ and Pseudorandomness: An Analysis

The paper addresses a prominent question in quantum complexity theory regarding the relationship between the complexity classes $\mathsf{QMA}$ (Quantum Merlin-Arthur) and $\mathsf{QCMA}$ (Quantum-Classical Merlin-Arthur). Specifically, it explores whether there exists a classical oracle capable of separating these two classes. Achieving a separation would deepen our understanding of the difference in computational power conferred by quantum versus classical proofs. The authors present a significant conditional result: such an oracle exists if a conjectured quantum pseudorandomness property holds true. This conjecture posits that quantum algorithms making relatively few queries cannot distinguish between a uniform distribution over permutations and distributions drawn from dense sources.

The exploratory framework is built on demonstrating a win-win scenario. Either a classical oracle can be found that separates $\mathsf{QMA}$ from $\mathsf{QCMA}$, or quantum algorithms can distinguish pseudorandom distributions on permutations from uniform ones, thereby showcasing a quantum advantage. The implications of this work extend to cryptographic applications, particularly concerning the post-quantum security of protocols reliant on pseudorandom permutations.

Introduction and Context

The problem elaborates on the complexity dichotomy proposed by Aaronson and Kuperberg, who initially demonstrated a quantum oracle separation of $\mathsf{QMA}$ and $\mathsf{QCMA}$. Despite advancements in considering alternative oracle models and restricted verifier versions, a definitive classical oracle remained elusive. By building on these foundations, the present work advances the discourse by introducing the conjecture that underlies their main theorem: if quantum pseudorandomness holds—a belief inspired by the complexity-theoretic guarantee that high min-entropy distributions appear uniform to polynomial time algorithms—then a classical oracle can separate $\mathsf{QMA}$ and $\mathsf{QCMA}$.

Main Theorem and Conjecture

The critical conjecture suggests that random permutations, both direct and inverse, form $\delta$-dense sources, appear pseudorandom to polynomial-query quantum algorithms. Confirming this would infer a $QMA$-QCMA separation given a special classical oracle. Focusing on permutations sidesteps issues of non-uniformity and reduces the problem to understanding the properties of well-structured quantum pseudorandomness conjectures over non-product distributions.

Implications and Further Directions

Practically, the paper could influence areas requiring rigid computational assumptions such as cryptography, offering insights into quantum advantages in settings where permutation-based security protocols like Feistel networks or hash functions are used. Theoretically, the work aligns closely with longstanding questions in quantum complexity theory, similar to the unresolved Aaronson-Ambainis conjecture relating to polynomial influences on symmetric distributions.

Unconditional Interactive Game and Bounded-Adaptive Models

The paper also provides an unconditional result for $\mathsf{QMA}$ vs. $\mathsf{QCMA}$ separation in a sequential setting or interactive game setup, demonstrating non-adaptive $QMA$ and $QCMA$ proof distinctions. By embedding this separation within logical constructs and leveraging multi-instance security reductions, crucial insights are drawn about proof protocols which become constrained with partial witness guessing.

Conclusions and Future Research

This paper builds a bridge between different conjectures in quantum complexity theory and lays out several paths for future exploration. Further research could explore establishing direct links between conjectures or extending the applicability range of known tools to confirm pseudorandomness assumptions. Moreover, exploration into bounded-adaptivity paradigms and permutations under scrutiny for various security protocols remains a fertile area for demonstrating potential limitations of classical proofs against quantum operations. Overall, this work exemplifies a critical juncture where notions of pseudorandomness could decisively influence our understanding of quantum capabilities vis-à-vis classical computations.