NLTS conjecture

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The no low-energy trivial states (NLTS) conjecture is a idea in quantum information that says there should exist families of quantum systems whose lowest-energy states cannot be created by shallow, easy-to-run quantum circuits. In other words, even when a system is near its ground (lowest) energy, its most important states are still inherently complex to prepare. The conjecture was put forward by Michael Freedman and Matthew Hastings in 2013 as a stepping stone toward the bigger quantum PCP idea.

To understand NLTS, it helps to know a bit about local Hamiltonians. A local Hamiltonian describes a quantum system built from many small, interacting parts, with each interaction involving only a few parts at a time. The ground-state energy is the lowest possible energy the system can have. If a family of such Hamiltonians has the NLTS property, then any state whose energy is within a small fixed range above the ground energy cannot be produced by a circuit of limited depth; it must require a more complex quantum circuit to prepare.

NLTS sits alongside the broader question known as the quantum PCP (qPCP) conjecture. The classical PCP theorem shows that certain problems remain hard to approximate even with powerful algorithms. A quantum analogue would say that approximating ground energies of quantum systems is hard even for quantum computers. NLTS provides a more approachable target: showing that some quantum systems have low-energy states that are not “trivial” or easy to generate helps explain why such quantum problems may be hard to approximate, supporting the idea behind qPCP.

Early progress included an announced solution by Eldar and Harrow in 2015, which was later revised to a weaker result called NLETS (no low-energy trivial states) after a mistake was found. A full, accepted proof of NLTS arrived in 2023 from Anurag Anshu, Nikolas Breuckmann, and Chinmay Nirkhe, presented at STOC 2023. Their work built on constructing local Hamiltonians whose low-energy states cannot be produced by simple circuits, strengthening the connection between entanglement, complexity, and physical properties of quantum matter.

Over time, the definitions have been broadened. Early NLTS constructions used stabilizer codes, which are classically simulable thanks to the Gottesman-Knill theorem. Later work extended these ideas so that the low-energy space can exclude not only simple circuit-preparable states but also a wider range of easy-to-describe or stabilizer-like states. A stronger version proposed in 2021 by Ghariabian and Le Gall asks for Hamiltonians whose low-energy states cannot even be efficiently simulated classically when measured in the computational basis.

In short, NLTS captures a fundamental link between physics and computation: there are quantum systems where the most relevant low-energy states remain computationally complex to produce, reinforcing the intuition that some quantum problems resist efficient approximation and shedding light on the quest for a full quantum PCP theorem.