arXiv subjects

Navin Kashyap

Publications and source records attributed to Navin Kashyap.

At least 19 recordsLinked to original sources

arXiv2026

Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes

Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled-$Z$ rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. We illustrate our methodology by providing explicit constructions of CSS code families that transversally realize addressable logical single-qubit $Z$-rotations.

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arXiv2026

Error-Correcting Weakly Constrained Codes: Constructions and Achievable Rates

We investigate weakly constrained codes, in which specific patterns occur with prescribed frequencies rather than being strictly forbidden as in conventional constrained coding. We propose a capacity-achieving construction of a weakly constrained codebook based on Eulerian cycles. We then obtain, via expurgation, weakly constrained codes with linear minimum distance and positive rate, and analyze the rates achievable. Finally, we propose a practical concatenated code construction that supports polynomial-time encoding and decoding.

cs.IT↗

arXiv2026

Reed--Muller Codes Achieve the Symmetric Capacity on Finite-State Channels

We study reliable communication over finite-state channels (FSCs) using Reed--Muller (RM) codes. Building on recent symmetry-based analyses for memoryless channels, we show that a sequence of binary RM codes (with some random scrambling) can achieve the symmetric capacity (or uniform-input information rate) of a binary-input indecomposable FSC. Our approach has three components. First, we establish a capacity-via-symmetry theorem for doubly-transitive group codes on discrete memoryless channels (DMCs) with non-binary inputs, under some symmetry and puncturing conditions. Then, we reduce a binary-input FSC to an almost memoryless non-binary channel by grouping adjacent input bits into blocks and interleaving non-binary codes onto the channel. Finally, we show that the interleaved non-binary codes can be constructed from a single binary RM code.

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arXiv2026

Recoverable systems and the maximal hard-core model on the triangular lattice

In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity.

math.CO↗

arXiv2026

Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

quant-ph↗

arXiv2026

Directed Information: Estimation, Optimization and Applications in Communications and Causality

Directed information (DI) is an information measure that attempts to capture directionality in the flow of information from one random process to another. It is closely related to other causal influence measures, such as transfer entropy, Granger causality, and Pearl's causal framework. This monograph provides an overview of DI and its main application in information theory, namely, characterizing the capacity of channels with feedback and memory. We begin by reviewing the definitions of DI, its basic properties, and its relation to Shannon's mutual information. Next, we provide a survey of DI estimation techniques, ranging from classic plug-in estimators to modern neural-network-based estimators. Considering the application of channel capacity estimation, we describe how such estimators numerically optimize DI rate over a class of joint distributions on input and output processes. A significant part of the monograph is devoted to techniques to compute the feedback capacity of finite-state channels (FSCs). The feedback capacity of a strongly connected FSC involves the maximization of the DI rate from the channel input process to the output process. This maximization is performed over the class of causal conditioned probability input distributions. When the FSC is also unifilar, i.e., the next state is given by a time-invariant function of the current state and the new input-output symbol pair, the feedback capacity is the optimal average reward of an appropriately formulated Markov decision process (MDP). This MDP formulation has been exploited to develop several methods to compute exactly, or at least estimate closely, the feedback capacity of a unifilar FSC. This monograph describes these methods, starting from the value iteration algorithm, to Q-graph methods, and reinforcement learning algorithms that can handle large input and output alphabets.

cs.IT↗

arXiv2026

Estimators for Substitution Rates in Genomes from Read Data

We study the problem of estimating the mutation rate between two sequences from noisy sequencing reads. Existing alignment-free methods typically assume direct access to the full sequences. We extend these methods to the sequencing framework, where only noisy reads from the sequences are observed. We use a simple model in which both mutations and sequencing errors are substitutions. We propose multiple estimators, provide theoretical guarantees for one of them, and evaluate the others through simulations.

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arXiv2025

Trace Reconstruction of First-Order Reed-Muller Codewords Using Run Statistics

In this paper, we derive an expression for the expected number of runs in a trace of a binary sequence $x \in \{0,1\}^n$ obtained by passing $x$ through a deletion channel that independently deletes each bit with probability $q$. We use this expression to show that if $x$ is a codeword of a first-order Reed-Muller code, and the deletion probability $q$ is 1/2, then $x$ can be reconstructed, with high probability, from $\tilde{O}(n^2)$ many of its traces.

cs.IT↗

arXiv2025

The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence

Recoverable systems provide coarse models of data storage on the two-dimensional square lattice, where each site reconstructs its value from neighboring sites according to a specified local rule. To study the typical behavior of recoverable patterns, this work introduces an interaction potential on the local recovery regions of the lattice, which defines a corresponding interaction model. We establish uniqueness of the Gibbs measure at high temperature and derive bounds on the entropy in the zero- and low-temperature regimes. For the recovery rule under consideration, exactly recoverable configurations coincide with maximal independent sets of the grid. Relying on methods developed for the standard hard-core model, we show phase coexistence at high activity in the maximal case. Unlike the standard hard-core model, however, the maximal version admits nontrivial ground states even at low activity, and we manage to classify them explicitly. We further verify the Peierls condition for the associated contour model. Combined with the Pirogov-Sinai theory, this shows that each ground state gives rise to an extremal Gibbs measure, proving phase coexistence at low activity.

math.CO↗

arXiv2025

On Quantum Computation Using Bias-Preserving Gates

Certain types of quantum computing platforms, such as those realized using Rydberg atoms or Kerr-cat qubits, are natively more susceptible to Pauli-Z noise than Pauli-X noise, or vice versa. On such hardware, it is useful to ensure that computations use only gates that maintain the Z-bias (or X-bias) in the noise. This is so that quantum error-correcting codes tailored for biased-noise models can be used to provide fault-tolerance on these platforms. In this paper, we follow up on the recent work of Fellous-Asiani et al. (npj Quantum Inf., 2025) in studying the structure and properties of bias-preserving gates. Our main contributions are threefold: (1) We give a novel characterization of Z-bias-preserving gates based on their decomposition as a linear combination of Pauli operators. (2) We show that any Z-bias-preserving gate can be approximated arbitrarily well using only gates from the set {X,R_z(θ),CNOT,CCNOT}, where θis any irrational multiple of 2π. (3) We prove, by drawing a connection with coherence resource theory, that any Z-bias-preserving logical operator acting on the logical qubits of a Calderbank-Shor-Steane (CSS) code can be realized by applying Z-bias-preserving gates on the physical qubits. Along the way, we also demonstrate that Z-bias-preserving gates are far from being universal for quantum computation.

quant-ph↗

arXiv2025

Codeword Stabilized Codes from m-Uniform Graph States

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d \ge m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]_2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[2^{2r}-1,2^{2r}-2r-3,3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2 - 32r-7, 5]]_2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.

quant-ph↗

arXiv2025

On the Coverage Required for Diploid Genome Assembly

The repeat content and heterozygosity rate of a target genome are important factors in determining the feasibility of achieving a complete telomere-to-telomere assembly. The mathematical relationship between the required coverage and read length for the purpose of unique reconstruction remains unexplored for diploid genomes. We investigate the information-theoretic conditions that the given set of sequencing reads must satisfy to achieve the complete reconstruction of the true sequence of a diploid genome. We also analyze the standard greedy and de-Bruijn graph-based assembly algorithms. Our results show that the coverage and read length requirements of the assembly algorithms are considerably higher than the lower bound because both algorithms require the double repeats in the genome to be bridged. Finally, we derive the necessary conditions for the overlap graph-based assembly paradigm.

cs.IT↗

arXiv2024

Error-Resilient Weakly Constrained Coding via Row-by-Row Coding

A weakly constrained code is a collection of finite-length strings over a finite alphabet in which certain substrings or patterns occur according to some prescribed frequencies. Buzaglo and Siegel (ITW 2017) gave a construction of weakly constrained codes based on row-by-row coding, that achieved the capacity of the weak constraint. In this paper, we propose a method to make this row-by-row coding scheme resilient to errors.

cs.IT↗

arXiv2024

Degree-$M$ Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix

The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-$M$ Bethe permanents, which are based on degree-$M$ covers of the underlying factor graph. In this paper, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.

math.CO↗

arXiv2024

Estimating the Weight Enumerators of Reed-Muller Codes via Sampling

This paper develops an algorithmic approach for obtaining estimates of the weight enumerators of Reed-Muller (RM) codes. Our algorithm is based on a technique for estimating the partition functions of spin systems, which in turn employs a sampler that produces codewords according to a suitably defined Gibbs distribution. We apply our method to moderate-blocklength RM codes and derive approximate values of their weight enumerators. We observe that the rates of the weight enumerator estimates returned by our method are close to the true rates when these rates are either known or computable by brute-force search; in other cases, our computations provide provably robust estimates. As a byproduct, our sampling algorithm also allows us to obtain estimates of the weight spectra of RM codes. We illustrate our methods by providing estimates of the hitherto unknown weight enumerators of the RM$(11,5)$ code and the exact weight spectra of the RM$(10,3)$ and RM$(10,4)$ codes.

cs.IT↗

arXiv2023

Sampling-Based Estimates of the Sizes of Constrained Subcodes of Reed-Muller Codes

This paper develops an algorithmic approach for obtaining approximate, numerical estimates of the sizes of subcodes of Reed-Muller (RM) codes, all of the codewords in which satisfy a given constraint. Our algorithm is based on a statistical physics technique for estimating the partition functions of spin systems, which in turn makes use of a sampler that produces RM codewords according to a Gibbs distribution. The Gibbs distribution is designed so that it is biased towards codewords that respect the constraint. We apply our method to approximately compute the sizes of runlength limited (RLL) subcodes and obtain estimates of the weight distribution of moderate-blocklength RM codes. We observe that the estimates returned by our method are close to the true sizes when these sizes are either known or computable by brute-force search; in other cases, our computations provide provably robust estimates. As an illustration of our methods, we provide estimates of the weight distribution of the RM$(9,4)$ code.

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arXiv2023

Wiretap Secret Key Agreement Via Secure Omniscience

In this paper, we explore the connection between secret key agreement and secure omniscience within the setting of the multiterminal source model with a wiretapper who has side information. While the secret key agreement problem considers the generation of a maximum-rate secret key through public discussion, the secure omniscience problem is concerned with communication protocols for omniscience that minimize the rate of information leakage to the wiretapper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience, $R_{\mathop{\mathrm{L}}}$, in terms of the wiretap secret key capacity, $C_{\mathop{\mathrm{W}}}$. Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear wiretapper. Duality also holds for any FLS model in which $C_{\mathop{\mathrm{W}}}$ is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.

cs.IT↗

arXiv2023

Coding Schemes Based on Reed-Muller Codes for $(d,\infty)$-RLL Input-Constrained Channels

The paper considers coding schemes derived from Reed-Muller (RM) codes, for transmission over input-constrained memoryless channels. Our focus is on the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s be separated by at least $d$ $0$s. In our study, we first consider $(d,\infty)$-RLL subcodes of RM codes, taking the coordinates of the RM codes to be in the standard lexicographic ordering. We show, via a simple construction, that RM codes of rate $R$ have linear $(d,\infty)$-RLL subcodes of rate $R\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$. We then show that our construction is essentially rate-optimal, by deriving an upper bound on the rates of linear $(d,\infty)$-RLL subcodes of RM codes of rate $R$. Next, for the special case when $d=1$, we prove the existence of potentially non-linear $(1,\infty)$-RLL subcodes that achieve a rate of $\max\left(0,R-\frac38\right)$. This, for $R > 3/4$, beats the $R/2$ rate obtainable from linear subcodes. We further derive upper bounds on the rates of $(1,\infty)$-RLL subcodes, not necessarily linear, of a certain canonical sequence of RM codes of rate $R$. We then shift our attention to settings where the coordinates of the RM code are not ordered according to the lexicographic ordering, and derive rate upper bounds for linear $(d,\infty)$-RLL subcodes in these cases as well. Finally, we present a new two-stage constrained coding scheme, again using RM codes of rate $R$, which outperforms any linear coding scheme using $(d,\infty)$-RLL subcodes, for values of $R$ close to $1$.

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