PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax

• URL: http://arxiv.org/abs/2609.11854v1
• Date: Thu, 10 Sep 2026 17:37:22 GMT
• Title: PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax
• Abstract summary: Pure-state consistency problems naturally lead to quantum proof systems in which a single witness must satisfy many acceptance constraints.<n>$mathsfBellPureSymQMA(textpoly)$ is a related model that requires the prover to give the verifierly many copies of a pure state.<n>We show that exact $k$-local pure-state consistency is $mathsfQMA$-complete for every fixed $kge2$, and so are the corresponding exact bosonic and fermionic pure $N$-representability problems.
• Score: 36.23068106699028
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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