arXiv subjects

Antonio Falco

Publications and source records attributed to Antonio Falco.

10 recordsLinked to original sources

arXiv2026

On the encoding complexity of quantum numerical integration: an angle-structure characterization

We study numerical integration on $[0,1]$ by quantum amplitude estimation (QAE), with emphasis on the cost of constructing the amplitude oracle. We introduce a hierarchy of grid functions $\mathcal{G}_n^{(d)}$ whose angle map $\Theta_g:\{0,1\}^n\to[0,\pi]$ is multilinear of degree at most $d$. Membership is classically checkable in $O(n2^n)$ time by the Walsh--Hadamard transform, and each $g\in\mathcal{G}_n^{(d)}$ admits a canonical encoding circuit with $\sum_{k=0}^d\binom{n}{k}$ multi-controlled $R_Y$ gates. Combining this circuit bound with classical discretisation estimates, we obtain a depth-versus-accuracy trade-off: for $g\in C^\alpha[0,1]$, total gate count $O((\log(1/\varepsilon))^d\varepsilon^{-1})$ suffices for $\varepsilon$-accuracy with constant probability; in the affine case $d=1$ this is $O(\varepsilon^{-1}\log(1/\varepsilon))$ at fixed discretisation. We also show that encoding degree and Sobolev smoothness are independent: for every $s\in(0,1/2)$, $\mathcal{G}_n^{(1)}$ contains restrictions of functions in $W^{s',2}(0,1)$ for all $s'<s$ but not in $W^{s,2}(0,1)$. Experiments on the SpinQ Triangulum (NMR) and IBM Kingston (superconducting) processors at $n=2$ validate the predicted hierarchy: affine encodings run reliably on both platforms, while quadratic encodings exceed the Triangulum coherence budget but execute on Kingston.

quant-ph↗

arXiv2026

A Rigorous and Self--Contained Proof of the Grover--Rudolph State Preparation Algorithm

We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum_k\sqrt{p_k}\ket{k}$ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $\eta$ changes the output distribution by at most $\min(1,n\eta)$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge\log_2(2n\pi/\varepsilon)$ bits and $S\ge 2^{n+1}\log(2/\delta)/\varepsilon^2$ shots suffice to achieve accuracy $\varepsilon$ with confidence $1-\delta$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.

quant-ph↗

arXiv2026

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

quant-ph↗

arXiv2025

Geometric Measures of Complexity for Open and Closed Quantum Systems

The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion originally developed by Michael Nielsen (Circa, 2007). However, for nonunitary dynamics, it is unclear how one can recover a completely geometric characterization of Nielsen-like geometric complexity. The main obstacle to overcome is that nonunitary dynamics cannot be characterized by Lie groups (which are Riemannian manifolds), as is the case for unitary dynamics. Building on Nielsen's work, we present a definition of geometric complexity for a fairly generic family of quantum channels. These channels are useful for modeling noise in quantum circuits, among other things, and analyze the geometric complexity of these quantum channels.

quant-ph↗

arXiv2024

Finite-time quantum equilibration for continuous variables

Leveraging the techniques found in the literature on Quantum Equilibration for finite dimensional systems, we develop the theory of Quantum Equilibration for the case of infinite-dimensional systems, particularly the cases where the dynamics-generating Hamiltonians have continuous spectrum. The main goal of this paper will be to propose a framework to extend the results obtained by Short in, where estimates for the equilibration-on-average and effective equilibration for the case of Hamiltonians with continuous spectrum are derived. We will show that in the latter setting, it is compulsory to constrain ourselves to finite time equilibration; we then develop estimates analogous to the main results in the proposed setting.

quant-ph↗

arXiv2021

Monitoring weeder robots and anticipating their functioning by using advanced topological data analysis

The present paper aims at analyzing the topological content of the complex trajectories that weeder-autonomous robots follow in operation. We will prove that the topological descriptors of these trajectories are affected by the robot environment as well as by the robot state, with respect to maintenance operations. Topological Data Analysis will be used for extracting the trajectory descriptors, based on homology persistence. Then, appropriate metrics will be applied in order to compare that topological representation of the trajectories, for classifying them or for making efficient pattern recognition.

cs.RO↗

arXiv2018

Tree-based tensor formats

The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of the tree, and prove the existence of best approximations from sets of tensors with bounded tree-based rank, under some assumptions on the norms weaker than in the existing results.

math.NA↗

arXiv2017

Principal bundle structure of matrix manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

math.DG↗

arXiv2015

Geometric Structures in Tensor Representations (Final Release)

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.

math.NA↗

arXiv2011

Proper Generalized Decomposition for Nonlinear Convex Problems in Tensor Banach Spaces

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called Proper Generalized Decompositions methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

math.NA↗