arXiv subjects

Louay Bazzi

Publications and source records attributed to Louay Bazzi.

8 recordsLinked to original sources

arXiv2026

Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes

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 $\Omega(N^{1/18})$ gap. Our inapproximability results use H{\aa}stad'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.

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arXiv2026

Improved decoding algorithms for surface codes under independent bit-flip and phase-flip errors

We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent $X/Z$ noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an $O(n^{3/2}\log n)$-time decoder is achievable for surface and toric codes, improving over the $O(n^{3}\log n)$ worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~$1$ for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in $\mathrm{NC}$. For SMLC decoding, we show that the planar surface code admits an exact decoder with $O(n^{3/2})$ algebraic complexity and that the problem lies in $\mathrm{NC}$, improving over the $O(n^{2})$ algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with $O(n^{3})$ algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.

cs.IT↗

arXiv2018

On the covering radius of small codes versus dual distance

Tietäväinen's upper and lower bounds assert that for block-length-$n$ linear codes with dual distance $d$, the covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$ and typically at least $\frac{n}{2}-Θ(\sqrt{dn\log{\frac{n}{d}}})$. The gap between those bounds on $R -\frac{n}{2}$ is an $Θ(\sqrt{\log{\frac{n}{d}}})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$, i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap can be eliminated by relaxing the covering requirement to allow for missing $o(1)$ fraction of points. Namely, if the dual distance $d = o(n)$, then for sufficiently large $d$, almost all points can be covered with radius $R\leq\frac{n}{2}-Θ(\sqrt{dn\log{\frac{n}{d}}})$. Compared to random linear codes, our bound on $R-\frac{n}{2}$ is asymptotically tight up to a factor less than $3$. We give applications to dual BCH codes. The proof builds on the author's previous work on the weight distribution of cosets of linear codes, which we simplify in this paper and extend from codes to probability distributions on $\{0,1\}^n$, thus enabling the extension of the above result to $(d-1)$-wise independent distributions.

cs.IT↗

arXiv2017

Weight distribution of cosets of small codes with good dual properties

The bilateral minimum distance of a binary linear code is the maximum $d$ such that all nonzero codewords have weights between $d$ and $n-d$. Let $Q\subset \{0,1\}^n$ be a binary linear code whose dual has bilateral minimum distance at least $d$, where $d$ is odd. Roughly speaking, we show that the average $L_\infty$-distance -- and consequently the $L_1$-distance -- between the weight distribution of a random cosets of $Q$ and the binomial distribution decays quickly as the bilateral minimum distance $d$ of the dual of $Q$ increases. For $d = Θ(1)$, it decays like $n^{-Θ(d)}$. On the other $d=Θ(n)$ extreme, it decays like and $e^{-Θ(d)}$. It follows that, almost all cosets of $Q$ have weight distributions very close to the to the binomial distribution. In particular, we establish the following bounds. If the dual of $Q$ has bilateral minimum distance at least $d=2t+1$, where $t\geq 1$ is an integer, then the average $L_\infty$-distance is at most $\min\{\left(e\ln{\frac{n}{2t}}\right)^{t}\left(\frac{2t}{n}\right)^{\frac{t}{2} }, \sqrt{2} e^{-\frac{t}{10}}\}$. For the average $L_1$-distance, we conclude the bound $\min\{(2t+1)\left(e\ln{\frac{n}{2t}}\right)^{t} \left(\frac{2t}{n}\right)^{\frac{t}{2}-1},\sqrt{2}(n+1)e^{-\frac{t}{10}}\}$, which gives nontrivial results for $t\geq 3$. We given applications to the weight distribution of cosets of extended Hadamard codes and extended dual BCH codes. Our argument is based on Fourier analysis, linear programming, and polynomial approximation techniques.

cs.IT↗

arXiv2017

On the tightness of Tietäväinen's bound for distributions with limited independence

In 1990, Tietäväinen showed that if the only information we know about a linear code is its dual distance $d$, then its covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$. While Tietäväinen's bound was later improved for large values of $d$, it is still the best known upper bound for small values including the $d = o(n)$ regime. Tietäväinen's bound holds also for $(d-1)$-wise independent probability distributions on $\{0,1\}^n$, of which linear codes with dual distance $d$ are special cases. We show that Tietäväinen's bound on $R-\frac{n}{2}$ is asymptotically tight up to a factor of $2$ for $k$-wise independent distributions if $k\leq\frac{n^{1/3}}{\log^2{n}}$. Namely, we show that there exists a $k$-wise independent probability distribution $μ$ on $\{0,1\}^n$ whose covering radius is at least $\frac{n}{2}-\sqrt{kn}$. Our key technical contribution is the following lemma on low degree polynomials, which implies the existence of $μ$ by linear programming duality. We show that, for sufficiently large $k\leq\frac{n^{1/3}}{\log^2{n}}$ and for each polynomial $f(v)\in {\mathbb R}[v]$ of degree at most $k$, the expected value of $f$ with respect to the binomial distribution cannot be positive if $f(w)\leq 0$ for each integer $w$ such that $|w-n/2|\leq\sqrt{kn}$. The proof uses tools from approximation theory.

cs.IT↗

arXiv2015

LP decoding excess over symmetric channels

We consider the problem of Linear Programming (LP) decoding of binary linear codes. The LP excess lemma was introduced by the first author, B. Ghazi, and R. Urbanke (IEEE Trans. Inf. Th., 2014) as a technique to trade crossover probability for "LP excess" over the Binary Symmetric Channel. We generalize the LP excess lemma to discrete, binary-input, Memoryless, Symmetric and LLR-Bounded (MSB) channels. As an application, we extend a result by the first author and H. Audah (IEEE Trans. Inf. Th., 2015) on the impact of redundant checks on LP decoding to discrete MSB channels.

cs.IT↗

arXiv2015

Impact of redundant checks on the LP decoding thresholds of LDPC codes

Feldman et al.(2005) asked whether the performance of the LP decoder can be improved by adding redundant parity checks to tighten the LP relaxation. We prove that for LDPC codes, even if we include all redundant checks, asymptotically there is no gain in the LP decoder threshold on the BSC under certain conditions on the base Tanner graph. First, we show that if the graph has bounded check-degree and satisfies a condition which we call asymptotic strength, then including high degree redundant checks in the LP does not significantly improve the threshold in the following sense: for each constant delta>0, there is a constant k>0 such that the threshold of the LP decoder containing all redundant checks of degree at most k improves by at most delta upon adding to the LP all redundant checks of degree larger than k. We conclude that if the graph satisfies a rigidity condition, then including all redundant checks does not improve the threshold of the base LP. We call the graph asymptotically strong if the LP decoder corrects a constant fraction of errors even if the LLRs of the correct variables are arbitrarily small. By building on the work of Feldman et al.(2007) and Viderman(2013), we show that asymptotic strength follows from sufficiently large expansion. We also give a geometric interpretation of asymptotic strength in terms pseudocodewords. We call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity. Under the assumptions that the graph girth is logarithmic and the minimum check degree is at least 3, rigidity is equivalent to the nondegeneracy property that adding at least logarithmically many checks does not give a constant weight check. We argue that nondegeneracy is a typical property of random check-regular graphs.

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arXiv2013

Linear Programming Decoding of Spatially Coupled Codes

For a given family of spatially coupled codes, we prove that the LP threshold on the BSC of the graph cover ensemble is the same as the LP threshold on the BSC of the derived spatially coupled ensemble. This result is in contrast with the fact that the BP threshold of the derived spatially coupled ensemble is believed to be larger than the BP threshold of the graph cover ensemble as noted by the work of Kudekar et al. (2011, 2012). To prove this, we establish some properties related to the dual witness for LP decoding which was introduced by Feldman et al. (2007) and simplified by Daskalakis et al. (2008). More precisely, we prove that the existence of a dual witness which was previously known to be sufficient for LP decoding success is also necessary and is equivalent to the existence of certain acyclic hyperflows. We also derive a sublinear (in the block length) upper bound on the weight of any edge in such hyperflows, both for regular LPDC codes and for spatially coupled codes and we prove that the bound is asymptotically tight for regular LDPC codes. Moreover, we show how to trade crossover probability for "LP excess" on all the variable nodes, for any binary linear code.

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