A construction of Combinatorial NLTS

• URL: http://arxiv.org/abs/2206.02741v2
• Date: Wed, 20 Jul 2022 17:16:03 GMT
• Title: A construction of Combinatorial NLTS
• Authors: Anurag Anshu and Nikolas P. Breuckmann
• Abstract summary: 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. Here, we prove a weaker version called the NLTS, where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms.
• Score: 22.539300644593936
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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