The decision problem version of estimating the Shannon entropy is the Entropy Difference problem (ED): given descriptions of two circuits, determine which circuit produces more entropy in its output when acting on a uniformly random input. The analogous problem with quantum circuits (QED) is to determine which circuit produces the state with greater von Neumann entropy, when acting on a fixed input state and after tracing out part of the output. Based on plausible complexity-theoretic assumptions, both of these problems are believed to be intractable for polynomial-time quantum computation. In this paper, we investigate the hardness of these problems in the case where the input circuits have logarithmic and constant depth, respectively. We show that, relative to an oracle, these problems cannot be as hard as their counterparts with polynomial-size circuits. Furthermore, we show that if a certain type of reduction from QED to the log-depth version exists, it implies that any polynomial-time quantum computation can be performed in log depth. While this suggests that having shallow circuits makes entropy estimation easier, we give indication that the problem remains intractable for polynomial-time quantum computation by proving a reduction from Learning-With-Errors (LWE) to constant-depth ED. We then consider a potential application of our results to quantum gravity research. First, we introduce a Hamiltonian version of QED where one is given two local Hamiltonians and asked to estimate the entanglement entropy difference in their ground states. We show that this problem is at least as hard as the circuit version and then discuss a potential experiment that would make use of the AdS/CFT correspondence to solve LWE efficiently. We conjecture that unless the AdS/CFT bulk to boundary map is exponentially complex, this experiment would violate the intractability assumption of LWE.

中文翻译：

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估计浅电路输出的熵很难

估计香农熵的决策问题版本是熵差问题 (ED)：给定两个电路的描述，确定哪个电路在作用于均匀随机输入时在其输出中产生更多的熵。量子电路 (QED) 的类似问题是确定哪个电路产生具有更大冯诺依曼熵的状态，当作用于固定的输入状态并追踪输出的一部分之后。基于看似合理的复杂性理论假设，这两个问题都被认为是多项式时间量子计算难以处理的。在本文中，我们分别研究了在输入电路具有对数深度和恒定深度的情况下这些问题的难度。我们证明，相对于预言机，这些问题不像多项式大小的电路那样难。此外，我们表明，如果存在从 QED 到对数深度版本的某种类型的减少，这意味着任何多项式时间量子计算都可以在对数深度中执行。虽然这表明具有浅层电路使熵估计更容易，但我们通过证明从误差学习 (LWE) 到恒定深度 ED 的减少，表明该问题对于多项式时间量子计算仍然难以解决。然后我们考虑我们的结果在量子引力研究中的潜在应用。首先，我们介绍了 QED 的哈密顿版本，其中给定两个局部哈密顿量，并要求估计它们基态的纠缠熵差。我们证明这个问题至少与电路版本一样难，然后讨论一个潜在的实验，该实验将利用 AdS/CFT 对应关系有效地解决 LWE。我们推测，除非 AdS/CFT 体块到边界图呈指数级复杂，否则该实验将违反 LWE 的难以处理的假设。