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The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that a particular family of constant-rate and linear-distance qLDPC codes correspond to NLTS local Hamiltonians, although we believe this to be true for all current constructions of good qLDPC codes.
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A construction of combinatorial NLTS
The NLTS (No Low-Energy Trivial State) conjecture [M. H. Freedman and M. B. Hastings, Quantum Inf. Comput. 14, 144 (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 no low error trivial states (NLETS), 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 [L. Eldar and A. W. Harrow, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2017), pp. 427–438] and [Nirkhe et al., in 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Leibniz International Proceedings in Informatics (LIPIcs), edited by Chatzigiannakis et al. (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018), Vol. 107, pp. 1–11]. Our construction is obtained by combining tensor networks with expander codes [M. Sipser and D. Spielman, IEEE Trans. Inf. Theory 42, 1710 (1996)]. The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the “uncle Hamiltonian” of Fernández-González et al. [Commun. Math. Phys. 333, 299 (2015)]. Thus, we deviate from the quantum Calderbank-Shor-Steane (CSS) code Hamiltonians considered in most prior works.
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- Award ID(s):
- 2013303
- PAR ID:
- 10434665
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 63
- Issue:
- 12
- ISSN:
- 0022-2488
- Page Range / eLocation ID:
- 122201
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0113731
