arXiv subjects

Venkatesan Guruswami

Publications and source records attributed to Venkatesan Guruswami.

At least 55 records · Page 3Linked to original sources

arXiv2024

Decoding Quasi-Cyclic Quantum LDPC Codes

Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev & Kalachev, 2020) achieved a near-linear distance (of $Ω(N/\log N)$ where $N$ is the number of codeword qubits), and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects $Θ(N/\log N)$ adversarial errors. En route, we give such an algorithm for the hypergraph product version these codes, which have weaker $Θ(\sqrt{N})$ distance (but are simpler). Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.

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arXiv2024

Quantum LDPC Codes of Almost Linear Distance via Homological Products

We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,Θ(N),Θ(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $ε>0$, our second main result constructs close-to-linear distance $[[N,N^{1-ε},N^{1-ε}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$.

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arXiv2024

Geno-Weaving: Low-Complexity Capacity-Achieving DNA Storage

As a possible implementation of data storage using DNA, multiple strands of DNA are stored in a liquid container so that, in the future, they can be read by an array of DNA readers in parallel. These readers will sample the strands with replacement to produce a random number of noisy reads for each strand. An essential component of such a data storage system is how to reconstruct data out of these unsorted, repetitive, and noisy reads. It is known that if a single read can be modeled by a substitution channel $W$, then the overall capacity can be expressed by the "Poisson-ization" of $W$. In this paper, we lay down a rateless code along each strand to encode its index; we then lay down a capacity-achieving block code at the same position across all strands to protect data. That weaves a low-complexity coding scheme that achieves DNA's capacity.

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arXiv2024

Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$ (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$ (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent $γ=\log(n/k)/\log(d)\rightarrow 0$ as the block length $n\rightarrow\infty$, where $k$ and $d$ denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size $q$. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters ($k,d=n/2^{O(\log^*n)}$) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).

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arXiv2024

Tight bounds for stream decodable error-correcting codes

In order to communicate a message over a noisy channel, a sender (Alice) uses an error-correcting code to encode her message $x$ into a codeword. The receiver (Bob) decodes it correctly whenever there is at most a small constant fraction of adversarial error in the transmitted codeword. This work investigates the setting where Bob is computationally bounded. Specifically, Bob receives the message as a stream and must process it and write $x$ in order to a write-only tape while using low (say polylogarithmic) space. We show three basic results about this setting, which are informally as follows: (1) There is a stream decodable code of near-quadratic length. (2) There is no stream decodable code of sub-quadratic length. (3) If Bob need only compute a private linear function of the input bits, instead of writing them all to the output tape, there is a stream decodable code of near-linear length.

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arXiv2024

Capacity-Achieving Gray Codes

To ensure differential privacy, one can reveal an integer fuzzily in two ways: (a) add some Laplace noise to the integer, or (b) encode the integer as a binary string and add iid BSC noise. The former is simple and natural while the latter is flexible and affordable, especially when one wants to reveal a sparse vector of integers. In this paper, we propose an implementation of (b) that achieves the capacity of the BSC with positive error exponents. Our implementation adds error-correcting functionality to Gray codes by mimicking how software updates back up the files that are getting updated ("coded Gray code"). In contrast, the old implementation of (b) interpolates between codewords of a black-box error-correcting code ("Grayed code").

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arXiv2024

Exponential Time Approximation for Coloring 3-Colorable Graphs

The problem of efficiently coloring $3$-colorable graphs with few colors has received much attention on both the algorithmic and inapproximability fronts. We consider exponential time approximations, in which given a parameter $r$, we aim to develop an $r$-approximation algorithm with the best possible runtime, providing a tradeoff between runtime and approximation ratio. In this vein, an algorithm to $O(n^\varepsilon)$-color a 3-colorable graphs in time $2^{Θ(n^{1-2\varepsilon}\log(n))}$ is given in (Atserias and Dalmau, SODA 2022.) We build on tools developed in (Bansal et al., Algorithmic, 2019) to obtain an algorithm to color $3$-colorable graphs with $O(r)$ colors in $\exp\left(\tilde{O}\left(\frac {n\log^{11/2}r} {r^3}\right)\right)$ time, asymptotically improving upon the bound given by Atserias and Dalmau.

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arXiv2024

Almost Optimal Time Lower Bound for Approximating Parameterized Clique, CSP, and More, under ETH

The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant $\varepsilon> 0$ such that for any computable function $f:\mathbb{N}\to\mathbb{N}$, no $f(k)\cdot n^{O(1)}$-time algorithm can, on input a $k$-variable CSP instance with domain size $n$, find an assignment satisfying $1-\varepsilon$ fraction of the constraints. A recent work by Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under ETH) that approximating sparse parameterized CSPs within a constant factor requires $n^{k^{1-o(1)}}$ time. This immediately implies that, assuming ETH, finding a $(k/2)$-clique in an $n$-vertex graph with a $k$-clique requires $n^{k^{1-o(1)}}$ time. We also prove almost optimal time lower bounds for approximating $k$-ExactCover and Max $k$-Coverage. Our proof follows the blueprint of the previous work to identify a "vector-structured" ETH-hard CSP whose satisfiability can be checked via an appropriate form of "parallel" PCP. Using further ideas in the reduction, we guarantee additional structures for constraints in the CSP. We then leverage this to design a parallel PCP of almost linear size based on Reed-Muller codes and derandomized low degree testing.

cs.CC↗

arXiv2024

Threshold rates for properties of random codes

Suppose that $P$ is a property that may be satisfied by a random code $C \subset Σ^n$. For example, for some $p \in (0,1)$, ${P}$ might be the property that there exist three elements of $C$ that lie in some Hamming ball of radius $pn$. We say that $R^*$ is the threshold rate for ${P}$ if a random code of rate $R^* + ε$ is very likely to satisfy ${P}$, while a random code of rate $R^* - ε$ is very unlikely to satisfy ${P}$. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably "symmetric". For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property ${P}$ above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general.

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arXiv2024

Punctured Low-Bias Codes Behave Like Random Linear Codes

Random linear codes are a workhorse in coding theory, and are used to show the existence of codes with the best known or even near-optimal trade-offs in many noise models. However, they have little structure besides linearity, and are not amenable to tractable error-correction algorithms. In this work, we prove a general derandomization result applicable to random linear codes. Namely, in settings where the coding-theoretic property of interest is "local" (in the sense of forbidding certain bad configurations involving few vectors -- code distance and list-decodability being notable examples), one can replace random linear codes (RLCs) with a significantly derandomized variant with essentially no loss in parameters. Specifically, instead of randomly sampling coordinates of the (long) Hadamard code (which is an equivalent way to describe RLCs), one can randomly sample coordinates of any code with low bias. Over large alphabets, the low bias requirement can be weakened to just large distance. Furthermore, large distance suffices even with a small alphabet in order to match the current best known bounds for RLC list-decodability. In particular, by virtue of our result, all current (and future) achievability bounds for list-decodability of random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) "mother" code. We also show that our punctured codes emulate the behavior of RLCs on stochastic channels, thus giving a derandomization of RLCs in the context of achieving Shannon capacity as well. Thus, we have a randomness-efficient way to sample codes achieving capacity in both worst-case and stochastic settings that can further inherit algebraic or other algorithmically useful structural properties of the mother code.

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arXiv2024

How Many Matrices Should I Prepare To Polarize Channels Optimally Fast?

Polar codes that approach capacity at a near-optimal speed, namely with scaling exponents close to $2$, have been shown possible for $q$-ary erasure channels (Pfister and Urbanke), the BEC (Fazeli, Hassani, Mondelli, and Vardy), all BMS channels (Guruswami, Riazanov, and Ye), and all DMCs (Wang and Duursma). There is, nevertheless, a subtlety separating the last two papers from the first two, namely the usage of multiple dynamic kernels in the polarization process, which leads to increased complexity and fewer opportunities to hardware-accelerate. This paper clarifies this subtlety, providing a trade-off between the number of kernels in the construction and the scaling exponent. We show that the number of kernels can be bounded by $O(\ell^{3/μ-1})$ where $μ$ is the targeted scaling exponent and $\ell$ is the kernel size. In particular, if one settles for scaling exponent approaching $3$, a single kernel suffices, and to approach the optimal scaling exponent of $2$, about $O(\sqrt{\ell})$ kernels suffice.

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arXiv2024

Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars

We modify Cheraghchi-Nakos [CN20] and Price-Scarlett's [PS20] fast binary splitting approach to nonadaptive group testing. We show that, to identify a uniformly random subset of $k$ infected persons among a population of $n$, it takes only $\ln(2 - 4\varepsilon) ^{-2} k \ln n$ tests and decoding complexity $O(\varepsilon^{-2} k \ln n)$, for any small $\varepsilon > 0$, with vanishing error probability. In works prior to ours, only two types of group testing schemes exist. Those that use $\ln(2)^{-2} k \ln n$ or fewer tests require linear-in-$n$ complexity, sometimes even polynomial in $n$; those that enjoy sub-$n$ complexity employ $O(k \ln n)$ tests, where the big-$O$ scalar is implicit, presumably greater than $\ln(2)^{-2}$. We almost achieve the best of both worlds, namely, the almost-$\ln(2)^{-2}$ scalar and the sub-$n$ decoding complexity. How much further one can reduce the scalar $\ln(2)^{-2}$ remains an open problem.

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arXiv2024

Successive Cancellation Sampling Decoder: An Attempt to Analyze List Decoding Theoretically

Successive cancellation list (SCL) decoders of polar codes excel in practical performance but pose challenges for theoretical analysis. Existing works either limit their scope to erasure channels or address general channels without taking advantage of soft information. In this paper, we propose the "successive cancellation sampling" (SCS) decoder. SCS hires iid "agents" to sample codewords using posterior probabilities. This makes it fully parallel and amenable for some theoretical analysis. As an example, when comparing SCS with $a$ agents to any list decoder with list size $\ell$, we can prove that the error probability of the former is at most $\ell/ae$ more than that of the latter. In this paper, we also describe how to adjust the "temperature" of agents. Warmer agents are less likely to sample the same codewords and hence can further reduce error probability.

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arXiv2024

Isolate and then Identify: Rethinking Adaptive Group Testing

Group testing (GT) is the art of identifying binary signals and the marketplace for exchanging new ideas for related fields such as unique-element counting, compressed sensing, traitor tracing, and geno-typing. A GT scheme can be nonadaptive or adaptive; the latter is preferred when latency is ess of an issue. To construct adaptive GT schemes, a popular strategy is to spend the majority of tests in the first few rounds to gain as much information as possible, and uses later rounds to refine details. In this paper, we propose a transparent strategy called "isolate and then identify" (I@I). In the first few rounds, I@I divides the population into teams until every team contains at most one sick person. Then, in the last round, I@I identifies the sick person in each team. Performance-wise, I@I is the first GT scheme that achieves the optimal coefficient $1/$capacity$(Z)$ for the $k \log_2 (n/k)$ term in the number of tests when $Z$ is a generic channel corrupting the test outcomes. I@I follows a modular methodology whereby the isolating part and the identification part can be optimized separately.

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arXiv2024

Baby PIH: Parameterized Inapproximability of Min CSP

The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $(1-\varepsilon)$-satisfiable (where the parameter is the number of variables) for some constant $0<\varepsilon<1$. We consider a minimization version of CSPs (Min-CSP), where one may assign $r$ values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the $r \times r$ pairs of values assigned to its variables (call such a CSP instance $r$-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every $r \ge 1$, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even $r$-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well.

cs.CC↗

arXiv2024

Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold

We consider the task of certifying that a random $d$-dimensional subspace $X$ in $\mathbb{R}^n$ is well-spread - every vector $x \in X$ satisfies $c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when $d \leq O(\sqrt{n})$. On the other hand, when $d \gg \sqrt{n}$, the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the $d \gg \sqrt{n}$ regime. Our algorithm runs in time $\exp(\widetilde{O}(n^{\varepsilon}))$ when $d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$, establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.

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arXiv2024

Near-Tight Bounds for 3-Query Locally Correctable Binary Linear Codes via Rainbow Cycles

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $δ> 0$ of adversarial errors must have dimension at most $O_δ(\log^2 n \cdot \log \log n)$. This is almost tight in view of quadratic Reed-Muller codes being a $3$-query locally correctable code (LCC) with dimension $Θ(\log^2 n)$. Our result improves, for the binary field case, the $O_δ(\log^8 n)$ bound obtained in the recent breakthrough of (Kothari and Manohar, 2023) (arXiv:2311.00558) (and the more recent improvement to $O_δ(\log^4 n)$ for binary linear codes announced in (Yankovitz, 2024)). Previous bounds for $3$-query linear LCCs proceed by constructing a $2$-query locally decodable code (LDC) from the $3$-query linear LCC/LDC and applying the strong bounds known for the former. Our approach is more direct and proceeds by bounding the covering radius of the dual code, borrowing inspiration from (Iceland and Samorodnitsky, 2018) (arXiv:1802.01184). That is, we show that if $x \mapsto (v_1 \cdot x, v_2 \cdot x, \ldots, v_n \cdot x)$ is an arbitrary encoding map $\mathbb{F}_2^k \to \mathbb{F}_2^n$ for the $3$-query LCC, then all vectors in $\mathbb{F}_2^k$ can be written as a $\widetilde{O}_δ(\log n)$-sparse linear combination of the $v_i$'s, which immediately implies $k \le \widetilde{O}_δ((\log n)^2)$. The proof of this fact proceeds by iteratively reducing the size of any arbitrary linear combination of at least $\widetildeΩ_δ(\log n)$ of the $v_i$'s. We achieve this using the recent breakthrough result of (Alon, Bucić, Sauermann, Zakharov, and Zamir, 2023) (arXiv:2309.04460) on the existence of rainbow cycles in properly edge-colored graphs, applied to graphs capturing the linear dependencies underlying the local correction property.

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arXiv2024

Nonadaptive Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding

Group testing (GT) is the Boolean version of spare signal recovery and, due to its simplicity, a marketplace for ideas that can be brought to bear upon related problems, such as heavy hitters, compressed sensing, and multiple access channels. The definition of a "good" GT varies from one buyer to another, but it generally includes (i) usage of nonadaptive tests, (ii) limiting to $O(k \log n)$ tests, (iii) resiliency to test noise, (iv) $O(k \mathrm{poly}(\log n))$ decoding time, and (v) lack of mistakes. In this paper, we propose $Gacha~GT$. Gacha is an elementary and self-contained, versatile and unified scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs. The new idea Gacha brings to the market is a redesigned Reed--Solomon code for probabilistic list-decoding at diminishing code rates over reasonably-large alphabets. Normally, list-decoding a vanilla Reed--Solomon code is equivalent to the nontrivial task of identifying the subsets of points that fit low-degree polynomials. In this paper, we explicitly tell the decoder which points belong to the same polynomial, thus reducing the complexity and enabling the improvement on GT.

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