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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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This content will become publicly available on December 1, 2026
Approximation algorithms for network design in non-uniform fault models
Abstract The Survivable Network Design problem (SNDP) is a well-studied problem, (partly) motivated by the design of networks that are robust to faults under the assumption that any subset of edges up to a specific number can fail. We considernon-uniformfault models where the subset of edges that fail can be specified in different ways. We consider three models: the flexible graph connectivity model (Adjiashvili, D.: Fault-tolerant shortest paths - beyond the uniform failure model (unpublished).) (Adjiashvili, D. et al. in Math. Program.192(1–2), 409–441 (2022)) (Boyd, S. et al. in Math. Program. pp. 1–24 (2023)) which was our initial motivation, the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) (Adjiashvili, D. et al,: in Math. Program.149(1–2), 361–390 (2015)) and the relative survivable network design model (Dinitz, M. et al.: in Algorithms and Techniques (APPROX/RANDOM 2022), Leibniz International Proceedings in Informatics (LIPIcs),245, pp. 41:1–41:19. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022)) (Dinitz, M. et al.: in Algorithms, pp. 190–204. Springer Nature Switzerland, Cham (2023)). While SNDP admits a 2-approximation (Jain, K.: in Combinatorica21(1), 39–60 (2001)), the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, we consider two important special cases, namely the single pair case and the global connectivity case. For both these, we obtain constant factor approximations in several parameter ranges of interest. These are based on an augmentation framework and via decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second set of results are poly-logarithmic approximations for the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) when the “width” of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. The results are obtained via two algorithmic approaches and they have different tradeoffs in terms of the approximation ratio and generality.
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- PAR ID:
- 10682517
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Mathematical Programming
- ISSN:
- 0025-5610
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model [Adjiashvili, 2013; Adjiashvili et al., 2020; Adjiashvili et al., 2022; Boyd et al., 2023], in which the edge set is partitioned into safe and unsafe edges. Given parameters p,q ≥ 1, the goal is to find a cheap subgraph that remains p-connected even after the failure of q unsafe edges. We also discuss the bulk-robust model [Adjiashvili et al., 2015; Adjiashvili, 2015] and the relative survivable network design model [Dinitz et al., 2022]. While SNDP admits a 2-approximation [K. Jain, 2001], the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when p and q are fixed, we consider two special cases. For the s-t case we obtain an approximation ratio that depends only on p,q whenever p+q > pq/2 which includes (p,2) and (2,q) for all p,q ≥ 1. For the global connectivity case we obtain an O(q) approximation for (2,q), and an O(p) approximation for (p,2) and (p,3) for any p ≥ 1, and for (p,4) when p is even. These are based on an augmentation framework and decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second result is a poly-logarithmic approximation for a generalization of the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. We utilize a recent framework due to Chen et al. [Chen et al., 2022] that was designed for handling group connectivity.more » « less
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