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arXiv:2605.31448

Pseudoentanglement in constant depth: How trivial states can have non-trivial entanglement structure

Alexandru Gheorghiu

quant-phcs.CRSubmitted May 29, 2026• Announced Jun 1, 2026• 0 citations• Revised Oct 1, 2026 (v2)
v1 · May 29, 2026v2 · Oct 1, 2026
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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 $\mathbf{x} \mapsto \mathbf{Mx},$ 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.

Comments
Corrected cryptographic parameters, revised the 1D Hamiltonian proof, and clarified entropy thresholds and hash assumptions; main results unchanged. 34 pages, 3 figures

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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,\mathbf{x} \mapsto \mathbf{Mx},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.
Characterizing the entanglement structure of ground states of local Hamiltonians is a fundamental problem in quantum information. In this work we study the computational complexity of this problem, given the Hamiltonian as input. Our main result is that to show it is cryptographically hard to determine if the ground state of a geometrically local, polynomially gapped Hamiltonian on qudits (d=O(1)d=O(1)d=O(1)) has near-area law vs near-volume law entanglement. This improves prior work of Bouland et al. (arXiv:2311.12017) showing this for non-geometrically local Hamiltonians. In particular we show this problem is roughly factoring-hard in 1D, and LWE-hard in 2D. Our proof works by constructing a novel form of public-key pseudo-entanglement which is highly space-efficient, and combining this with a modification of Gottesman and Irani's quantum Turing machine to Hamiltonian construction. Our work suggests that the problem of learning so-called "gapless" quantum phases of matter might be intractable.
Pseudoentangled states are defined by their ability to hide their entanglement structure: they are indistinguishable from random states to any observer with polynomial resources, yet can have much less entanglement than random states. Existing constructions of pseudoentanglement based on phase and/or subset states are limited in the entanglement structures they can hide, e.g., the states may have low entanglement on a single cut, on all cuts at once, or on local cuts in one dimension. Here, we introduce new constructions of pseudoentangled states based on (pseudo)random tensor networks that afford much more flexibility in the achievable entanglement structures. We illustrate our construction with the simplest example of a matrix product state, realizable as a staircase circuit of pseudorandom unitary gates, which exhibits pseudo-area-law scaling of entanglement in one dimension. We then generalize our construction to arbitrary tensor network structures that admit an isometric realization. A notable application of this result is the construction of pseudoentangled "holographic" states whose entanglement entropy obeys a Ryu-Takayanagi "minimum-cut" formula, answering a question posed in Aaronson et al. [arXiv:2211.00747].
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning AC0{AC}^0AC0 and TC0{TC}^0TC0 under the uniform distribution. Our first result concerns the concept class TC0{TC}^0TC0 (resp., AC0{AC}^0AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT{AND}, {OR}, {NOT}AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE{RLWE}RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0{TC}^0TC0; and if there exists no 2O(d1/η)2^{O(d^{1/\eta})}2O(d1/η)-time quantum algorithm to solve RLWE{RLWE}RLWE with dimension d=O(polylogn)d = O(polylog n)d=O(polylogn) (for every constant η>2\eta > 2η>2), then there exists no O(nlog⁡νn)O(n^{ \log^{\nu} n} )O(nlogνn)-time quantum learning algorithm for poly(n)poly(n)poly(n)-sized AC0{AC}^0AC0 circuits (for a constant ν>0\nu>0ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20{TC}^0_2TC20​ in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE{LWE}LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20{TC}^0_2TC20​, i.e., depth-2 TC0{TC}^0TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.
We give an efficient algorithm for learning kkk-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying UUU to the all-zero input. For a depth-ddd circuit on nnn sites with random 2ℓ2\ell2ℓ-qubit gates, the algorithm learns the original circuit UUU with high probability in poly(n,2ℓd)\text{poly}(n,2^{\ell d})poly(n,2ℓd) time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as ℓd=O(log⁡n)\ell d = O(\log n)ℓd=O(logn). In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn), and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn) scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states -- first defined by Ji, Liu and Song -- which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to log⁡n\log nlogn across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.
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