Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them, even approximately, must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be “robust” - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any marginal of a ground state to a fraction at least 1-10-8of the qubits must be globally entangled. This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of “good” quantum LDPC codes, and the NLTS conjecture [1] positing the existence of local Hamiltonians in which any low-energy state is highly entangled. Our Hamiltonian is based on applying the hypergraph product by Tillich-Zemor [2] to the repetition code with checks from an expander graph. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings, 2014 -- which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states -- identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings
(Quantum Information and Computation, 2014) -- which posits the existence of a
local Hamiltonian with a super-constant circuit lower bound on the complexity
of all low-energy states -- identifies a fundamental obstacle to the resolution
of the quantum PCP conjecture. In this paper, we provide new techniques, based
on entropic and local indistinguishability arguments, for proving circuit lower
bounds for the low-energy states of local Hamiltonians arising from quantum
error-correcting codes. For local Hamiltonians arising from nearly linear-rate and
polynomial-distance LDPC stabilizer codes, we prove a super-constant circuit
lower bounds for the complexity of all states of energy o(n) (which can be
viewed as an almost linear NLTS theorem). Such codes are known to exist and, in
general, are not locally-testable, a property previously suspected to be
essential for the NLTS conjecture. Curiously, such codes can also be
constructed on a constant-dimensional lattice, showing that complex
entanglement at high energies can persist in physically relevant systems.
Quantum entanglement is considered, by and large, to be a very delicate and non-robust phenomenon that is very hard to maintain in the presence of noise, or non-zero temperatures. In recent years however, and motivated, in part, by a quest for a quantum analog of the PCP theorem researches have tried to establish whether or not we can preserve quantum entanglement at "constant" temperatures that are independent of system size. This would imply that any quantum state with energy at most, say 0.05 of the total available energy of the Hamiltonian, would be highly-entangled. To this date, no such systems were found, and moreover, it became evident that even embedding local Hamiltonians on robust, albeit "non-physical" topologies, namely expanders, does not guarantee entanglement robustness. In this study, we indicate that such robustness may be possible after all: We construct an infinite family of O(1)-local Hamiltonians, corresponding to check terms of a quantum error-correcting code with the following property of inapproximability: any quantum state with energy at most 0.05 w.r.t. the total available energy cannot be even approximately simulated by classical circuits of bounded (sub-logarithmic) depth. In a sense, this implies that even providing a "witness" to the fact that the local Hamiltonian can be "almost" satisfied, already requires some measure of long-range entanglement. Our construction is but a first step in what, we believe, is a whole range of possible entanglement - robust local Hamiltonians. A natural next step, we believe, is to devise such local Hamiltonians that resist approximation in terms of bounded-depth quantum circuits (e.g. NLTS), and even find such robust forms of entanglement that are useful for some computation.
Understanding the complexity of low-energy quantum states is a central problem in quantum complexity theory and many-body physics. Here we prove the no low-energy trivial magic (NLTM) conjecture, constructing a family of local Hamiltonians for which every state within a positive energy density window requires circuit depth
Ω(logn) to prepare from arbitrary stabilizer state that may themselves be highly entangled. Notably, our result is intrinsic in the sense that the same Hamiltonian family and energy density threshold apply to arbitrary qudit stabilizer inputs with any bounded local dimensions. Our construction builds on non-Abelian gauged sheaf codes obtained using the cup product structure on good quantum locally testable codes. We establish constant operator soundness for the associated Hamiltonians, enabling their protected logical structure to constrain all states at sufficiently low energy density. By excluding shallow stabilizer-based classical witnesses, Our result substantially strengthens NLTS and advances our understanding of low-energy complexity relevant to quantum PCP.
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum
p-spin glasses, random Hamiltonians whose interactions act on
p-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement.
We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with
n, we prove that, for all sufficiently large fixed
p, any circuit preparing an
n-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth
Ωp(logn). In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed
D, a sufficiently large degree prefactor rules out depth-
D preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits.
Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.