Revised: October 12, 2022
Published: November 27, 2025
Abstract: [Plain Text Version]
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Info. Comput. 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 article 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. Here, by `low-depth' states, we mean $n$-qubit quantum states that can be prepared using quantum circuits of constant depth, starting with the state $|0\rangle^{\otimes m}$, where $m \geq n$.
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A conference version of this paper appeared in the Proceedings of the 13th Innovations in Theoretical Computer Science Conference (ITCS 2022).
Licensed under a Creative Commons Attribution License (CC-BY)