A construction of Combinatorial NLTS

TL;DR

This paper proves a weaker version of the NLTS conjecture, called combinatorial NLTS, by constructing Hamiltonians with high complexity states that violate only a small fraction of local terms, using tensor networks and expander codes.

Contribution

The paper introduces a new combinatorial NLTS construction combining tensor networks with expander codes, deviating from traditional CSS code Hamiltonians.

Findings

Established a lower bound on quantum circuit complexity for states violating few local terms.

Constructed Hamiltonians with high complexity states using tensor networks and expander codes.

Generalized previous NLETS results to a broader combinatorial setting.

Abstract

The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity measured by the quantum circuit depth preparing the state). Here, we prove a weaker version called the combinatorial NLTS, where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms. This generalizes the prior NLETS results (Eldar and Harrow [2017]; Nirkhe, Vazirani and Yuen [2018]). Our construction is obtained by combining tensor networks with expander codes (Sipser and Spielman [1996]). The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the `uncle Hamiltonian' of Fernandez-Gonzalez et. al. [2015]. Thus, we deviate from the quantum CSS code Hamiltonians considered in most prior works.