PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA 通过无维度玻色子最大值原理
PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax
- University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
- Tel-Aviv University（特拉维夫大学）
- Penn State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明了 QMA 等于 PureSuperQMA 及其指数检查变体和 BellPureSymQMA 多项式变体，通过无维度稳定性界和 SWAP 测试实现，并得出精确 k-局部纯态一致性是 QMA-完全的。
中文摘要 AI 辅助
纯态一致性问题的自然形式是量子证明系统，其中单个纯见证必须满足许多接受约束。相应的类 $\mathsf{PureSuperQMA}$ 此前已知位于 $\mathsf{QMA}$ 和 $\mathsf{QMA}(2)$ 之间，Kamminga 和 Rudolph (ITCS'26) 猜想这两个包含关系都是严格的。在本文中，我们证明了以下令人惊讶的复杂性塌缩：$$ \mathsf{QMA} = \mathsf{PureSuperQMA} = \mathsf{PureSuperQMA}(\text{exp}) = \mathsf{BellPureSymQMA}(\text{poly}) $$ 这里 $\mathsf{PureSuperQMA}(\text{exp})$ 允许指数多个检查，这些检查被均匀索引并高效生成，同时要求 NO 情形具有逆多项式违反边际和逆多项式比例的违反检查。$\mathsf{BellPureSymQMA}(\text{poly})$ 是一个相关模型，要求证明者给验证者多项式多个纯态副本，验证者分别测量每个局部测量，输出长度对数级，然后联合处理结果。主要技术成分是对称张量态的无维度稳定性界。我们的模拟使用多项式多个见证寄存器，并结合随机对 SWAP 测试和原始验证过程的置换不变提升。关键步骤是证明在对称子空间上，极值验证值接近某个张量幂见证的值，误差与维度无关。将此论证应用于两个验证模型，得到两个模拟。作为结果，精确 $k$-局部纯态一致性对于每个固定 $k\ge2$ 是 $\mathsf{QMA}$-完全的，相应的精确玻色子和费米子纯 $N$-可表示性问题也是如此。
英文摘要
Pure-state consistency problems naturally lead to quantum proof systems in which a single pure witness must satisfy many acceptance constraints. The corresponding class $\mathsf{PureSuperQMA}$ was previously known to lie between $\mathsf{QMA}$ and $\mathsf{QMA}(2)$, and Kamminga and Rudolph (ITCS'26) conjectured that both containments are strict. In this paper, we prove the following surprising complexity collapses $$ \mathsf{QMA} = \mathsf{PureSuperQMA} = \mathsf{PureSuperQMA}(\text{exp}) = \mathsf{BellPureSymQMA}(\text{poly}) $$ Here $\mathsf{PureSuperQMA}(\text{exp})$ allows exponentially many checks which are uniformly indexed and efficiently generated, while requiring an inverse-polynomial violation margin and an inverse-polynomial fraction of violated checks for the NO cases. $\mathsf{BellPureSymQMA}(\text{poly})$ is a related model that requires the prover to give the verifier polynomially many copies of a pure state, which the verifier measures separately with logarithmic output length for each local measurement, before processing the outcomes jointly. The main technical ingredient is a dimension-free stability bound for symmetric tensor states. Our simulations use polynomially many witness registers and combine a random-pair SWAP test with a permutation-invariant lift of the original verification procedure. The key step is to show that, on the symmetric subspace, the extremal verification value is close to that of some tensor-power witness with dimension-independent error. Applying this argument to the two verification models yields both simulations. As a consequence, exact $k$-local pure-state consistency is $\mathsf{QMA}$-complete for every fixed $k\ge2$, and so are the corresponding exact bosonic and fermionic pure $N$-representability problems.
