Pseudoentanglement in constant depth: How trivial states can have non-trivial entanglement structure
Abstract
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 x ↦ 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.
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BibTeX
@misc{gheorghiu2026pseudoentanglement,
title = {{Pseudoentanglement in constant depth: How trivial states can have non-trivial entanglement structure}},
author = {Alexandru Gheorghiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.31448},
eprint = {2605.31448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.31448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pseudoentanglement in constant depth: How trivial states can have non-trivial entanglement structure
AU - Alexandru Gheorghiu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.31448
UR - https://arxiv.org/abs/2605.31448
AB - 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 x ↦ 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.
ER -