Yaxin Tu | Searcharxiv

arXiv2025

A modal approach towards substitutions

Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step substitutions is often adequate. Yet, for more complex cases, iterative substitutions become indispensable. In this article, our primary focus is to study logical frameworks that model both single-step and iterative substitutions. We explore a number of properties of these logics, including their expressive strength, Hilbert-style proof systems, and satisfiability problems. Additionally, we establish connections between our proposed frameworks and relevant existing ones in the literature. For instance, we precisely delineate the relationship between single-step substitutions and the standard syntactic replacements commonly found in many classical logics. Moreover, special emphasis is placed on iterative substitutions. In this context, we compare our proposed framework with existing ones involving iterative reasoning, thereby highlighting the advantages of our proposal.

math.LO↗

arXiv2023

LWE with Quantum Amplitudes: Algorithm, Hardness, and Oblivious Sampling

In this paper, we show new algorithms, hardness results and applications for $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ with real Gaussian, Gaussian with linear or quadratic phase terms, and other related amplitudes. Let $n$ be the dimension of LWE samples. Our main results are 1. There is a $2^{\tilde{O}(\sqrt{n})}$-time algorithm for $\sf{S|LWE\rangle}$ with Gaussian amplitude with \emph{known} phase, given $2^{\tilde{O}(\sqrt{n})}$ many quantum samples. The algorithm is modified from Kuperberg's sieve, and in fact works for more general amplitudes as long as the amplitudes and phases are completely \emph{known}. 2. There is a polynomial time quantum algorithm for solving $\sf{S|LWE\rangle}$ and $\sf{C|LWE\rangle}$ for Gaussian with quadratic phase amplitudes, where the sample complexity is as small as $\tilde{O}(n)$. As an application, we give a quantum oblivious LWE sampler where the core quantum sampler requires only quasi-linear sample complexity. This improves upon the previous oblivious LWE sampler given by Debris-Alazard, Fallahpour, Stehl\'{e} [STOC 2024], whose core quantum sampler requires $\tilde{O}(nr)$ sample complexity, where $r$ is the standard deviation of the error. 3. There exist polynomial time quantum reductions from standard LWE or worst-case GapSVP to $\sf{S|LWE\rangle}$ with Gaussian amplitude with small \emph{unknown} phase, and arbitrarily many samples. Compared to the first two items, the appearance of the unknown phase term places a barrier in designing efficient quantum algorithm for solving standard LWE via $\sf{S|LWE\rangle}$.

quant-ph↗