arXiv subjects

Tal Gurfinkel

Publications and source records attributed to Tal Gurfinkel.

6 recordsLinked to original sources

arXiv2026

From Block-encoding to Generalized Quantum Signal Processing: Principles, Algorithms and Applications

Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.

quant-ph↗

arXiv2025

Benchmarking Lie-Algebraic Pretraining and Non-Variational QWOA for the MaxCut Problem

The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for achieving quantum advantage in combinatorial optimization on Near-Term Intermediate-Scale Quantum (NISQ) devices. However, random initialization of the variational parameters typically leads to vanishing gradients, rendering standard variational optimization ineffective. This paper provides a comparative performance analysis of two distinct strategies designed to improve trainability: Lie algebraic pretraining framework that uses Lie-algebraic classical simulation to find near-optimal initializations, and non-variational QWOA (NV-QWOA) that targets a restrict parameter subspace covered by 3 hyperparameters. We benchmark both methods on the unweighted Maxcut problem using a circuit depth of $p = 256$ across 200 Erd\H{o}s-R\'enyi and 200 3-regular graphs, each with 16 vertices. Both approaches significantly improve upon the standard randomly initialized QWOA. NV-QWOA attains a mean approximation ratio of 98.9\% in just 60 iterations, while the Lie-algebraic pretrained QWOA improves to 77.71\% after 500 iterations. That optimization proceeds more quickly for NV-QWOA is not surprising given its significantly smaller parameter space, however, that an algorithm with so few tunable parameters reliably finds near-optimal solutions is remarkable. These findings suggest that the structured parameterization of NV-QWOA offers a more robust training approach than pretraining on lower-dimensional auxiliary problems. Future work is needed to confirm scaling to larger problem sizes and to asses generalization to other problem classes.

quant-ph↗

arXiv2023

Rigidity of Travelling Times for Strictly Convex Obstacles in Riemannian Manifolds

Let $K$ and $L$ be two disjoint unions of strictly convex obstacles contained within a Riemannian manifold with boundary $S$ of dimension $m\geq 2$. The sets of travelling times $\mathcal{T}_K$ and $\mathcal{T}_L$ of $K$ and $L$, respectively, are composed of triples $(x,y,t)\in\partial S\times\partial S\times\mathbb{R}^+$ where $t$ is the length of a billiard trajectory with endpoints $x$ and $y$ that reflects elastically on $K$ (or $L$ for $(x,y,t)\in\mathcal{T}_L$). It has been shown (arXiv:2309.11141) that (under some natural curvature bounds on $S$) if $\mathcal{T}_K=\mathcal{T}_L$ and $K$ and $L$ were equivalent up to tangency then $K = L$. In this paper we remove this requirement for $K$ and $L$, and show that if $\mathcal{T}_K = \mathcal{T}_L$ then $K = L$ whenever $m\geq 3$.

math.DG↗

arXiv2023

Uniqueness of Obstacles in Riemannian Manifolds from Travelling Times

Suppose that $K$ and $L$ are two disjoint unions of strictly convex obstacles with the same set of travelling times, contained in an $n$-dimensional Riemannian manifold $M$ (where $n\geq2$). Under some natural curvature conditions on $M$, and provided that no geodesic intersects more than two components in $K$ or $L$, we show that $K = L$.

math.DG↗

arXiv2023

Recovering Obstacles from their Travelling Times

Noakes and Stoyanov (2021) introduced a method of recovering strictly convex planar obstacles from their set of travelling times. We provide an extension of this construction for obstacles on Riemannian surfaces under some general curvature conditions. It is required that no smooth geodesic intersect more than two obstacles.

math.DG↗

arXiv2020

Travelling Times in Scattering by Obstacles in Curved Space

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.

math.DG↗