Circuit lower bounds for low-energy states of quantum code Hamiltonians
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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 work, 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 and polynomial-distance LDPC stabilizer codes, we prove 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 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 in physically relevant systems.
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