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arXiv:2311.09503 · Leibniz international proceedings in informatics

NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes

Louis Golowich, Tali Kaufman

Published Nov 16, 2023• 1 citation

Abstract

Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22). In this work, we obtain improvements to both of these results using qLDPC codes of low rate: - Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes of linear dimension, we show the stronger result that qLDPC codes of arbitrarily small positive dimension yield NLTS Hamiltonians. - The SoS lower bounds of Hopkins and Lin are only weakly explicit because they require running Gaussian elimination to find a nontrivial codeword, which takes polynomial time. We resolve this shortcoming by introducing a new method of planting a strongly explicit nontrivial codeword in linear-distance qLDPC codes, which in turn yields strongly explicit SoS lower bounds. Our "planted" qLDPC codes may be of independent interest, as they provide a new way of ensuring a qLDPC code has positive dimension without resorting to parity check counting, and therefore provide more flexibility in the code construction.

DOI: 10.4230/lipics.itcs.2024.54

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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.

We provide a completely self-contained construction of a family of NLTS Hamiltonians [Freedman and Hastings, 2014] based on ideas from [Anshu, Breuckmann, and Nirkhe, 2022], [Cross, He, Natarajan, Szegedy, and Zhu, 2022] and [Eldar and Harrow, 2017]. Crucially, it does not require optimal-parameter quantum LDPC codes and can be built from simple classical LTCs such as the repetition code on an expander graph. Furthermore, it removes the constant-rate requirement from the construction of Anshu, Breuckmann, and Nirkhe.