Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians

Authors Anurag Anshu , Chinmay Nirkhe

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Anurag Anshu
  • Simons Institute for the Theory of Computing, Berkeley, California, USA
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
Chinmay Nirkhe
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA


We thank Lior Eldar, Zeph Landau, Umesh Vazirani, and anonymous conference reviewers for detailed comments on the manuscript that greatly improved the presentation. Additional thanks to Dorit Aharanov, Matt Hastings, Aleksander Kubica, Rishabh Pipada, Elizabeth Yang, and Henry Yuen for helpful discussions.

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Anurag Anshu and Chinmay Nirkhe. Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 6:1-6:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings [Freedman and Hastings, 2014] - which posits the existence of a local Hamiltonian with a super-constant quantum 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 or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). 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 even in physically relevant systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • quantum pcps
  • local hamiltonians
  • error-correcting codes


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