arXiv subjects

Tanay Saha

Publications and source records attributed to Tanay Saha.

5 recordsLinked to original sources

arXiv2026

Nearest structured matrix having an eigenvalue with prescribed lower bounds of algebraic and geometric multiplicities

We study the problem of perturbing a given matrix X belonging to a subspace S of linearly structured real matrices of order n to its nearest counterpart Y=X+D, where D belongs to S and Y possesses an eigenvalue with prescribed lower bounds on its algebraic multiplicity and geometric multiplicity. The proposed framework takes care of both cases where the target eigenvalue is known and where it is treated as an unknown decision variable to be determined jointly with the structured perturbation. We establish necessary and sufficient conditions for the existence of such matrices by expressing the feasibility constraints through Jordan-chain relations in structural coordinates. This allows us to formulate the problem as a nonlinear constrained nested optimization that minimizes the Frobenius norm of the perturbation. To solve this, we implemented a two-level strategy, utilizing MATLAB's fmincon for continuous inner optimization via a multi-start Sequential Quadratic Programming approach, while the outer level solves a discrete optimization problem. Comprehensive numerical experiments on various classes of structured matrices, including comparisons with existing methods, demonstrate the effectiveness and accuracy of the proposed approach.

math.SP↗

arXiv2025

Sublogarithmic Distillation in all Prime Dimensions using Punctured Reed-Muller Codes

Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate $\epsilon$ is $O(\log^{\gamma} (\epsilon^{-1}))$ where $\gamma$ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., $\gamma < 1$) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension $p>2$ can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension $p$. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as $p$ increases, and the asymptotic yield parameter approaches $\frac{1}{\ln p}$ as $p \to \infty$. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a $[[519,106,5]]_5$ code with $\gamma=0.99$.

quant-ph↗

arXiv2024

Low Overhead Qutrit Magic State Distillation

We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of $[[9m-k, k, 2]]_3$ triorthogonal qutrit error-correcting codes for any positive integers $m$ and $k$ with $k \leq 3m-2$ that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate $\epsilon$ is $O(\log^\gamma \epsilon^{-1})$, where the yield parameter $\gamma$ characterizes the overhead cost. For $k=3m-2$, our codes have $\gamma = \log_2 (2+\frac{6}{3 m-2})$, which tends to $1$ as $m \to \infty$. Moreover, the $[[20,7,2]]_3$ qutrit code that arises from our construction when $m=3$ already has a yield parameter of $1.51$ which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.

quant-ph↗

arXiv2023

Repeated quantum game as a stochastic game: Effects of the shadow of the future and entanglement

We present a systematic investigation of the quantum games, constructed using a novel repeated game protocol, when played repeatedly ad infinitum. We focus on establishing that such repeated games -- by virtue of inherent quantum-mechanical randomness -- can be mapped to the paradigm of stochastic games. Subsequently, using the setup of two-player--two-action games, we explore the pure reactive strategies belonging to the set of reactive strategies, whose support in the quantum games is no longer countably finite but rather non-denumerably infinite. We find that how two pure strategies fare against each other is crucially dependent on the discount factor (the probability of occurrence of every subsequent round) and how much entangled the quantum states of the players are. We contrast the results obtained with the corresponding results in the classical setup and find fundamental differences between them: e.g, when the underlying game is the prisoner's dilemma, in the quantum game setup, always-defect strategy can be beaten by the tit-for-tat strategy for high enough discount factor.

quant-ph↗

arXiv2019

Structured real radius of controllability for higher order LTI systems

In this paper, we consider the problem of computing the nearest uncontrollable (C-uncontrollable) system to a given higher order system. The distance to the nearest uncontrollable system, also termed as the radius of controllability, is a good measure of gauging the numerical robustness of the given system with respect to controllability. Here, we invoke the equivalence of C-controllability of a higher order system with full rank property of a certain Toeplitz structured matrix. This enables us to pose the problem of computing the radius of controllability as equivalent to the problem of computing the nearest structured low rank approximation of this Toeplitz structured matrix. Through several numerical examples and comparison with the benchmark numerical problem, we illustrate that our approach works well.

math.OC↗