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On estimating the entropy of shallow circuit outputs

Alexandru Gheorghiu, Matty J. Hoban

TL;DR

The paper studies the computational difficulty of estimating the entropy of outputs from shallow circuits, formalizing the entropy-difference problems ED and QED and their depth-bounded variants. It shows that under the LWE assumption, entropy estimation remains intractable for log-depth and constant-depth circuits with unbounded fan-out, and provides oracle separations suggesting an intermediate hardness regime relative to polynomial-depth circuits. The authors construct reductions from LWE to ED_log and ED_O(1) (and lift to QED_O(1) in the quantum setting) via extended trapdoor claw-free functions and randomized encodings, and they relate the quantum problem HQED to holographic AdS/CFT via history-state constructions. These results tie entropy-distinguishing tasks to cryptographic hardness and to the bulk-boundary dictionary, with implications for the quantum extended Church-Turing thesis and potential physical thought experiments.

Abstract

Estimating the entropy of probability distributions and quantum states is a fundamental task in information processing. Here, we examine the hardness of this task for the case of probability distributions or quantum states produced by shallow circuits. Specifically, we show that entropy estimation for distributions or states produced by either log-depth circuits or constant-depth circuits with gates of bounded fan-in and unbounded fan-out is at least as hard as the Learning with Errors (LWE) problem, and thus believed to be intractable even for efficient quantum computation. This illustrates that quantum circuits do not need to be complex to render the computation of entropy a difficult task. We also give complexity-theoretic evidence that this problem for log-depth circuits is not as hard as its counterpart with general polynomial-size circuits, seemingly occupying an intermediate hardness regime. Finally, we discuss potential future applications of our work for quantum gravity research by relating our results to the complexity of the bulk-to-boundary dictionary of AdS/CFT.

On estimating the entropy of shallow circuit outputs

TL;DR

The paper studies the computational difficulty of estimating the entropy of outputs from shallow circuits, formalizing the entropy-difference problems ED and QED and their depth-bounded variants. It shows that under the LWE assumption, entropy estimation remains intractable for log-depth and constant-depth circuits with unbounded fan-out, and provides oracle separations suggesting an intermediate hardness regime relative to polynomial-depth circuits. The authors construct reductions from LWE to ED_log and ED_O(1) (and lift to QED_O(1) in the quantum setting) via extended trapdoor claw-free functions and randomized encodings, and they relate the quantum problem HQED to holographic AdS/CFT via history-state constructions. These results tie entropy-distinguishing tasks to cryptographic hardness and to the bulk-boundary dictionary, with implications for the quantum extended Church-Turing thesis and potential physical thought experiments.

Abstract

Estimating the entropy of probability distributions and quantum states is a fundamental task in information processing. Here, we examine the hardness of this task for the case of probability distributions or quantum states produced by shallow circuits. Specifically, we show that entropy estimation for distributions or states produced by either log-depth circuits or constant-depth circuits with gates of bounded fan-in and unbounded fan-out is at least as hard as the Learning with Errors (LWE) problem, and thus believed to be intractable even for efficient quantum computation. This illustrates that quantum circuits do not need to be complex to render the computation of entropy a difficult task. We also give complexity-theoretic evidence that this problem for log-depth circuits is not as hard as its counterpart with general polynomial-size circuits, seemingly occupying an intermediate hardness regime. Finally, we discuss potential future applications of our work for quantum gravity research by relating our results to the complexity of the bulk-to-boundary dictionary of AdS/CFT.

Paper Structure

This paper contains 18 sections, 22 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Discussion and open questions
  4. Preliminaries
  5. Notation
  6. Complexity theory
  7. Cryptography
  8. Learning with Errors (LWE)
  9. Claw-free functions
  10. Randomized encodings
  11. Entropy difference for shallow circuits with unbounded fan-out gates
  12. Entropy difference is easier for shallow circuits
  13. Hardness based on Learning with Errors
  14. Hamiltonian quantum entropy difference
  15. Applications to holography
  16. ...and 3 more sections

Key Result

Theorem 1

There exists an oracle $\mathcal{O}$ such that $\mathsf{SZK^{\mathcal{O}}_{polylog}} \neq \mathsf{SZK}^{\mathcal{O}}$ and $\mathsf{QSZK^{\mathcal{O}}_{polylog}} \neq \mathsf{QSZK}^{\mathcal{O}}$.

Figures (3)

  • Figure 1: Circuit with oracle calls. The unitaries $U_j$ are assumed to be polylogarithmic depth quantum circuits. The unitaries $U_{\mathcal{O}}$ represent calls to the oracle $\mathcal{O}$. The circuit has polylogarithmic depth overall.
  • Figure 2: Circuit $C_j$ padded with identities.
  • Figure 3: The two situations for the bulk observer. The observer should measure the area of the surface $\gamma$ to determine which situation it is in.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1: Quantum Statistical Zero-Knowledge ($\sf QSZK$)
  • Definition 2: Quantum Entropy Difference (QED) bst
  • Definition 3: Extended Trapdoor Claw-free Functions (ETCFs) brakerski2018mahadev2018classical
  • Definition 4: Uniform randomized encoding aik
  • Lemma 1: Unique randomness aik
  • Theorem 5: aik
  • ...and 36 more