We construct a family of 2D-local constant-depth quantum circuits that output states whose entanglement entropy across a specified cut cannot be estimated in quantum polynomial time. As constant-depth quantum circuits can be learned from polynomially many quantum samples, our resulting pseudoentangled states are implicitly public-key and not pseudorandom. This separates pseudoentanglement from pseudorandomness in the shallow-circuit regime: the former is possible, while the latter is not. The construction is based on the quantum intractability of the Dense-Sparse Learning Parity with Noise problem introduced in [DJ25] and uses a bounded-fan-in, bounded-fan-out classical randomized encoding for linear maps $\mathbf{x} \mapsto \mathbf{Mx},$ which could be of independent interest. As applications, we obtain quantum hardness for the problem of learning the entanglement structure (across a fixed cut) of the ground-state of 1D and 2D local Hamiltonians. The 1D Hamiltonian has an inverse polynomial gap, whereas the 2D one has a constant gap. This complements the result of [BZZ24] that showed only factoring-based hardness for the 1D case, though achieving a volume versus area entanglement difference.

翻译：我们构造了一族二维局部常深度量子线路，其输出态跨越特定划分的纠缠熵无法在量子多项式时间内估计。由于常深度量子线路可从多项式数量的量子样本中学习，所得伪纠缠态本质上是公钥型且非伪随机的。这揭示了浅层线路体系中伪纠缠与伪随机性的分离：前者可实现而后者不可行。该构造基于[DJ25]提出的稠密-稀疏带噪声学习奇偶性问题的量子难解性，并采用了线性映射$\mathbf{x} \mapsto \mathbf{Mx}$的有界扇入有界扇出经典随机化编码方案——这一方法本身可能具有独立研究价值。作为应用，我们证明了学习一维与二维局域哈密顿量基态（沿固定划分）纠缠结构问题具备量子困难性。其中一维哈密顿量具有逆多项式能隙，而二维情形具有恒定能隙。这补充了[BZZ24]的结果：该工作虽实现了一维情形下体积律与面积律纠缠的差异，但仅证明了基于因子分解的困难性。