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.
Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to establish code families with constant encoding rate and linear distance, simultaneously achieving optimal asymptotic scaling in both parameters. In particular, we characterize explicit logical representatives of the full gauged code space, and provide a fundamental treatment of non-Abelian code distance in the sense of Knill--Laflamme based on combining expansion with cleaning to establish protection against arbitrary errors. We further construct an almost-good family with entire code spaces exhibiting long-range magic. Moreover, we use linearly addressable cup product to directly generate linearly many encoded magic states from gauging measurements of Clifford operators. These results extend good qLDPC codes beyond the Pauli stabilizer setting, lay the foundation for pursuing the no low-energy trivial magic conjecture and shed light on non-Abelian quantum phases and orders beyond geometric locality.
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.
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.
We establish a systematic mathematical framework for constructing and computing (co)homological invariants that induce logical gates on quantum low-density parity-check (qLDPC) codes. By synthesizing tools from graph theory, algebraic topology, and number theory, we study constructions ranging from hypergraph product (HGP) codes to sheaf codes. Using commutative algebra, we generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a longstanding challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, {we establish a nearly linear family of independent cup product cochains on almost-good qLDPC codes. Under a separate nonvanishing pairing condition, they are expected to yield parallel logical multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes, we construct interlaced families through odd-degree lifts that preserve compatible invariant pairings, allowing nontrivial, constant-depth logical gates throughout an infinite family to be certified on a constant-size HGP seed. Treating local code constructions and expansion bounds as external inputs, our framework identifies the compatibility conditions for combining large dimension and distance, local testability, and constant-depth non-Clifford gates, thereby provides a mathematical foundation for separate constructions that realize these properties simultaneously.