Unentangled stoquastic Merlin-Arthur proof systems: the power of unentanglement without destructive interference

Abstract: Stoquasticity, originating in sign-problem-free physical systems, gives rise to $\sf StoqMA$, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between $\sf MA$ and $\sf AM$. Unentanglement similarly gives rise to ${\sf QMA}(2)$, introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes $\sf QMA$ to two unentangled proofs and still has only the trivial $\sf NEXP$ upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via ${\sf StoqMA}(2)$, the class of unentangled stoquastic Merlin-Arthur proof systems. Although $\sf StoqMA$ is semi-quantum and may collapse to $\sf MA$, ${\sf StoqMA}(2)$ turns out to be surprisingly powerful. We establish the following results: - ${\sf NP} \subseteq {\sf StoqMA}(2)$ with $\widetilde{O}(\sqrt{n})$-qubit proofs and completeness error $2^{-{\rm polylog}(n)}$. Conversely, ${\sf StoqMA}(2) \subseteq {\sf EXP}$ via the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014); with our lower bound, our refined analysis yields the optimality of this algorithm under ETH. - ${\sf StoqMA}(2)_1 \subseteq {\sf PSPACE}$, and the containment holds with completeness error $2^{{-2^{{\rm} poly}(n)}}$. - ${\sf PreciseStoqMA}(2)$, a variant of ${\sf StoqMA}(2)$ with exponentially small promise gap, cannot achieve perfect completeness unless ${\sf EXP}={\sf NEXP}$. In contrast, ${\sf PreciseStoqMA}$ achieves perfect completeness, since ${\sf PSPACE} \subseteq {\sf PreciseStoqMA}_1$. - When the completeness error is negligible, ${\sf StoqMA}(k) = {\sf StoqMA}(2)$ for $k\geq 2$. Our lower bounds are obtained by stoquastizing the short-proof ${\sf QMA}(2)$ protocols via distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our new rectangular closure testing framework.

• The paper demonstrates that NP is contained in StoqMA(2) using O(√n)-qubit proofs with a 2-polylog(n) completeness error.
• It establishes that PreciseStoqMA(2) equals NEXP under exponentially small promise gaps, paralleling QMA(2) lower bounds.
• The work introduces a rectangular closure testing framework that enables efficient verification and tighter EXP and PSPACE upper bounds.

Unentangled Stoquastic Merlin-Arthur Proof Systems: Computational Power Without Destructive Interference

Introduction and Context

The paper "Unentangled stoquastic Merlin-Arthur proof systems: the power of unentanglement without destructive interference" (2604.27886) undertakes a systematic complexity-theoretic analysis of unentangled stoquastic Merlin-Arthur proof systems (StoqMA(2)). The classical and quantum counterparts, MA and QMA(2), are well-studied: MA leverages randomness and a classical witness, while QMA(2) generalizes quantum verification to two unentangled proofs. Stoquasticity—where off-diagonal matrix elements are real and non-positive—enables simulation without a sign problem, leading to the intermediate complexity class StoqMA. The paper merges this with unentanglement, forming StoqMA(2), and explores how computational power persists in absence of destructive interference.

Main Results

Lower Bounds (Expressive Power)

One central result is that NP is contained in StoqMA(2) employing $O(\sqrt{n})$-qubit proofs with completeness error $2$-polylog$(n)$, paralleling best-known QMA(2) lower bounds with sublinear quantum proofs—apart from the loss of perfect completeness. Furthermore, StoqMA(2) achieves NEXP when promise gaps are exponentially small, with the result PreciseStoqMA(2) = NEXP = PreciseQMA(2), thus unentanglement robustly elevates stoquastic proof power to match highly expressive quantum classes.

Short proofs (e.g., logarithmic length) with inverse-polynomial gap suffice for NP certification in StoqMA(2), tightly paralleling quantum proofs' known phenomena. These protocols leverage the distribution-testing perspective: the acceptance probability is tied to Hellinger affinity between distributions induced by the verification circuit and the witness; this enables certifying NP with short unentangled quantum proofs using stoquastic verification, a feat unattainable in classical MA without violating ETH.

Upper Bounds (Algorithmic and Structural Limitations)

The paper establishes StoqMA(2) $\subseteq$ EXP via an improved Sum-of-Squares (SoS) algorithm, tightening previous bounds and demonstrating essential optimality under ETH. For nearly perfect completeness, StoqMA(2) proofs with doubly-exponentially small completeness error are shown to be in PSPACE; and PreciseStoqMA(2) with triple-exponentially small gaps is in EXP. This strongly contrasts with QMA(2), where the best upper bound remains NEXP, indicating that stoquasticity imposes additional structure enabling feasible upper bounds.

Furthermore, a new combinatorial technique—the rectangular closure testing framework—permits the efficient verification of nearly perfect completeness, capturing non-negative product witness structure and supporting the PSPACE and EXP inclusions.

Structural Robustness and Collapses

The work proves that for negligible completeness error, StoqMA($k$) collapses to StoqMA(2) for any $k \geq 2$, preserving robustness akin to QMA(2)—via prover compression and witness symmetrization techniques with quadratic gap loss. Strikingly, PreciseStoqMA(2) fails to be closed under perfect completeness unless EXP = NEXP, differing from PreciseQMA(2) and PreciseStoqMA, which are closed and equal to PSPACE. This reveals polyadic subtleties: the presence of unentanglement can eliminate perfect completeness closure, diverging from quantum analogs.

Proof Techniques

Lower bounds are achieved via stoquastization of QMA(2) short-proof protocols through distribution testing, exploiting non-negativity to avoid sign problem and leveraging probabilistic method (generalized birthday paradox) for soundness analysis. Upper bounds employ the rectangular closure testing framework for nearly perfect completeness, and a refined SoS analysis for general proofs. The SoS algorithm exploits the symmetric structure and non-negativity to deliver optimal parameter dependence.

Implications and Significance

Complexity Landscape and Quantum-Class Separation

StoqMA(2) fills a complexity-theoretic gap between classical and fully quantum multi-prover systems, demonstrating that unentanglement bolstered by stoquasticity can reach the expressive heights of QMA(2), while remaining algorithmically tractable. The existence of efficient upper bounds and lower bounds matching quantum models elucidates a new intermediate class whose power is dictated by the absence of destructive interference but whose structure enforces tractability.

The essential optimality under ETH of both lower and upper bounds shows that further improvements would yield subexponential SAT algorithms—indicating deep interplay between quantum-inspired models and classical hardness.

Practical and Theoretical Directions

Practically, the results suggest that stoquastic verification plus unentangled proofs creates powerful yet simulable quantum-inspired protocols, possibly relevant for quantum simulation scenarios devoid of sign interference. The rectangular closure framework could inform algorithmic strategies for separability testing and zero-knowledge proofs in physically accessible quantum systems.

Theoretically, the failure of perfect completeness closure in PreciseStoqMA(2) but not in PreciseQMA(2), combined with the collapse behavior for negligible completeness error, motivates further investigation into how promise gap scaling and unentanglement interrelate in complexity class stability.

Specifically, this work indicates the potential for stoquastic, unentangled models to serve as testbeds for quantum-class separation or hardness results, given their proximity to both classical and quantum boundaries.

Future Research Directions

• Refining error reduction for general StoqMA: The search for error reduction protocols in StoqMA could yield or close gaps with classical MA.
• Exploring collapses under derandomization assumptions: If MA = NP, StoqMA may collapse to MA, impacting the intermediate class' existence.
• Extension to physical Hamiltonian models: Leveraging stoquastic protocols for real-world quantum computational tasks without sign problems.
• Applications in quantum separability testing and convex optimization: The rectangular closure and SoS strategies may augment hierarchical relaxations and tensor optimization.
• Analogues in multi-prover quantum interactive proofs: Investigating whether stoquasticity admits similar collapses or hardness as other quantum proof systems.

Conclusion

This paper articulates a comprehensive complexity-theoretic characterization of StoqMA(2), explicating the computational power imparted by unentangled proofs in stoquastic systems absent destructive interference. The results bridge quantum and classical verification, demonstrating expressivity parity with quantum multi-prover systems while facilitating algorithmic upper bounds via structural properties. The revelations regarding promise gap-induced class collapses and closure under completeness accentuate new directions in the study of quantum-inspired proof systems, both as vehicles for understanding computational hardness and as practical tools for quantum protocols within tractable domains.