Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs

• URL: http://arxiv.org/abs/2608.07986v1
• Date: Sat, 08 Aug 2026 07:45:56 GMT
• Title: Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs
• Abstract summary: We study gap amplification of the class $mathsfQMA+(2)$ characterized by unentangled quantum proofs.<n>Our proof combines symmetric-subspace projections with the relation $mathsfQMA+(1)=mathsfNEXP$ of Bassirian, Fefferman, and Marwaha.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study gap amplification of the class $\mathsf{QMA}^{+}(2)$ characterized by unentangled quantum proofs whose amplitudes are nonnegative in the computational basis. This class was recently introduced by Jeronimo and Wu (STOC 2023), and its behavior depends sharply on the completeness-soundness gap: although it captures the power of $\mathsf{NEXP}$ for some small constant gap, it is equal to $\mathsf{QMA}(2)$ for larger constant gap. This is in stark contrast to $\mathsf{QMA}(2)$ where strong gap amplification is known due to the product test by Harrow and Montanaro (FOCS 2010, JACM 2013). In this paper, we prove for every completeness $c$ and soundness $s$ with $c-s=1/\mathrm{poly}(n)$, \[ \mathsf{NEXP} = \mathsf{QMA}^{+}(2,c,s) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14+\frac1{\mathrm{poly}(n)}\right). \] Our result gives a clean complexity phase transition for $\mathsf{QMA}^{+}(2)$ since we have \[ \mathsf{QMA}^{\mathbb R}(2) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14-\frac1{\mathrm{poly}(n)}\right), \] where $\mathsf{QMA}^{\mathbb R}(2)$ denotes $\mathsf{QMA}(2)$ with witnesses restricted to real amplitudes. Our amplification is thus optimal in the sense that a slight improvement of our soundness would have the collapse \[ \mathsf{QMA}^{\mathbb R}(2)=\mathsf{NEXP}. \] Our proof combines symmetric-subspace projections with the relation $\mathsf{QMA}^{+}(1)=\mathsf{NEXP}$ of Bassirian, Fefferman, and Marwaha (ITCS 2024). The main technical ingredient is a dimension-independent de Finetti theorem in Hilbert-Schmidt norm that applies when the number of registers under consideration grows logarithmically.

Related papers

• Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples [7.864201093845001]
  We develop a theory of discrete residual least-squares approximation for elliptic spectral equations $mathfrak L_u=f$ using linearized ReLU$k$ neural networks.<n>We establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss.<n>The key analytical ingredient is a Bernstein inequality for linearized ReLU$k$ network spaces.
  arXiv  Detail & Related papers  (2026-08-07T01:22:27Z)
• Hardness Amplification for (Sparse) LPN [5.1105538244022135]
  We prove new hardness amplification results for Learning Parity with Noise ($mathsfLPN$) and its sparse variants.<n>An algorithm that solves $mathsfLPN$ on a'small fraction of instances' can be converted into an algorithm that solves $mathsfLPN$ on 'almost all instances'<n>We extend the same amplification approach to $mathsfLPN$ over $mathbbF_q$ and to Sparse-$mathsfLPN$, where each query vector $vec a$ has
  arXiv  Detail & Related papers  (2026-05-11T06:34:37Z)
• The Complexity of Stoquastic Sparse Hamiltonians [0.30079490585515334]
  We show that the Stoquastic Sparse Hamiltonians problem ($mathsfStoqSH$) is in $mathsfStoqMA$.<n>Since Stoquastic Local Hamiltonians are $mathsfStoqMA$-hard, this implies that $mathsfStoqSH$ is $mathsfStoqMA$-complete.<n>We also show that the separable version of $mathsfStoqMA(2)$ is $mathsfStoqMA
  arXiv  Detail & Related papers  (2026-05-04T17:22:02Z)
• On the Complexity of Pure-State Consistency of Local Density Matrices [0.0]
  We define a pure state analogue of the complexity class QMA+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA.<n>We prove hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete.<n>We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete.
  arXiv  Detail & Related papers  (2024-11-05T13:43:21Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Dimension Independent Disentanglers from Unentanglement and Applications [55.86191108738564]
  We construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show that to capture NEXP, it suffices to have unentangled proofs of the form $| psi rangle = sqrta | sqrt1-a | psi_+ rangle where $| psi_+ rangle has non-negative amplitudes.
  arXiv  Detail & Related papers  (2024-02-23T12:22:03Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorization [49.090785356633695]
  We study the asymmetric low-rank factorization problem: [mathbfU in mathbbRm min d, mathbfU$ and mathV$.
  arXiv  Detail & Related papers  (2021-06-27T17:25:24Z)
• StoqMA meets distribution testing [0.0]
  We provide a novel connection between $mathsfStoqMA$ and distribution testing via reversible circuits. We show that both variants of $mathsfStoqMA$ that without any ancillary random bit and with perfect soundness are contained in $mathsfNP$. Our results make a step towards collapsing the hierarchy $mathsfMA subseteq mathsfStoqMA subseteq mathsfSBP$ [BBT06], in which all classes are contained in $
  arXiv  Detail & Related papers  (2020-11-11T12:30:42Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.