Estimating the entropy of shallow circuit outputs is hard
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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.
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