Inapproximability Results for Computational Problems on Lattices

Miklós Ajtai

Louay Bazzi, Georges Khater

Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the $X$- and $Z$-syndromes. For color codes under independent $X$- and $Z$-error models, we consider Separate Minimum-Weight decoding. Assuming $P\neq NP$, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the $4.8.8$ color code on the torus, there exists a constant $c>0$ such that no polynomial-time algorithm can always produce a solution whose weight is within $cN^{1/14}$ of the optimum, where $N$ is the number of qubits, unless $P=NP$. For the planar surface code, we obtain an $Ω(N^{1/18})$ gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.

Subhash Khot

In this article, we present a survey of known inapproximability results for computational problems on lattices, viz. the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), the Closest Vector Problem with Preprocessing (CVPP), the Covering Radius Problem (CRP), the Shortest Independent Vectors Problem (SIVP), and the Shortest Basis Problem (SBP).

Daniele Micciancio

Abstract: We prove that the Shortest Vector Problem (SVP) on point lattices is NPhard to approximate for any constant factor under polynomial-time randomized reductions with one-sided error that produce no false positives. We also prove inapproximability for quasi-polynomial factors under the same kind of reductions running in subexponential time. Previous hardness results for SVP either incurred two-sided error, or only proved hardness for small constant approximation factors. Close similarities between our reduction and recent results on the complexity of the analogous problem for linear codes make our new proof an attractive target for derandomization, paving the road to a possible NP-hardness proof for SVP under deterministic polynomial-time reductions.

Sanjeev Arora, László Babai, Jacques Stern +1 more

Marek Hyčko, Mirko Navara