arXiv subjects

Francesca Albertini

Publications and source records attributed to Francesca Albertini.

18 recordsLinked to original sources

arXiv2026

Quantum Robust Control using Geometric Optimal Control Theory

In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field. We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches. We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.

quant-ph↗

arXiv2025

Neural Network-Driven Direct CBCT-Based Dose Calculation for Head-and-Neck Proton Treatment Planning

Accurate dose calculation on cone beam computed tomography (CBCT) images is essential for modern proton treatment planning workflows, particularly when accounting for inter-fractional anatomical changes in adaptive treatment scenarios. Traditional CBCT-based dose calculation suffers from image quality limitations, requiring complex correction workflows. This study develops and validates a deep learning approach for direct proton dose calculation from CBCT images using extended Long Short-Term Memory (xLSTM) neural networks. A retrospective dataset of 40 head-and-neck cancer patients with paired planning CT and treatment CBCT images was used to train an xLSTM-based neural network (CBCT-NN). The architecture incorporates energy token encoding and beam's-eye-view sequence modelling to capture spatial dependencies in proton dose deposition patterns. Training utilized 82,500 paired beam configurations with Monte Carlo-generated ground truth doses. Validation was performed on 5 independent patients using gamma analysis, mean percentage dose error assessment, and dose-volume histogram comparison. The CBCT-NN achieved gamma pass rates of 95.1 $\pm$ 2.7% using 2mm/2% criteria. Mean percentage dose errors were 2.6 $\pm$ 1.4% in high-dose regions ($>$90% of max dose) and 5.9 $\pm$ 1.9% globally. Dose-volume histogram analysis showed excellent preservation of target coverage metrics (Clinical Target Volume V95% difference: -0.6 $\pm$ 1.1%) and organ-at-risk constraints (parotid mean dose difference: -0.5 $\pm$ 1.5%). Computation time is under 3 minutes without sacrificing Monte Carlo-level accuracy. This study demonstrates the proof-of-principle of direct CBCT-based proton dose calculation using xLSTM neural networks. The approach eliminates traditional correction workflows while achieving comparable accuracy and computational efficiency suitable for adaptive protocols.

physics.med-ph↗

arXiv2024

Long Range Games

We consider $N$-player games, in continuous time, finite state space and finite time horizon, on a geometrical structure possessing a macroscopic limit in a suitable sense. This geometrical structure breaks the permutation invariance property that gives rise to mean field games. The corresponding limit game is a variant of mean field games that we call {\em long range game}. We prove that this asymptotic scheme satisfies the following key properties: a) the long range game admits al least one equilibrium; b) this equilibrium is unique under a suitable monotonicity condition; c) the feedback corresponding to any equilibrium of the long range game is a quasi-Nash equilibrium for the $N$-player games. We finally show that this scheme includes several examples of interaction mechanisms, in particular Kac-type interactions and interactions on generalized Erd\"{o}s-Renyi graphs.

math.OC↗

arXiv2021

Symmetric States and Dynamics of Three Quantum Bits

The unitary group acting on the Hilbert space of three quantum bits admits a Lie subgroup, of elements which permute with the symmetric group of permutations. Under the action of such Lie subgroup, the Hilbert space splits into three invariant subspaces of dimensions 4, 2 and 2 respectively, each corresponding to an irreducible representation of su(2). The subspace of dimension 4 is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called symmetric sector. We provide an analysis of pure states in the symmetric sector of three quantum bits for what concerns their entanglement properties, separability criteria and dynamics. We parametrize all the possible invariant two-dimensional subspaces and extend the previous analysis to these subspaces as well. We propose a physical set up for the states and dynamics we study which consists of a symmetric network of three spin 1/2 particles under a common driving electro-magnetic field. For such set up, we solve a control theoretic problem which consists of driving a separable state to a state with maximal distributed entanglement.

quant-ph↗

arXiv2020

Subspace controllability of multi-partite spin networks

In a network of spin 1/2 particles, controlled through an external electro-magnetic field, the gyromagnetic ratio of each spin is a parameter that characterizes the interaction of the spin with the external control field. Multipartite networks are such that the spins are divided into subsets according to their gyromagnetic ratio and spins in one set interact in the same way with all spins in another set. Due to the presence of symmetries in this type of systems, the underlying Hilbert state space splits into invariant subspaces for the dynamics. Subspace controllability is verified if every unitary evolution can be generated by the dynamics on these subspaces. We give an exact characterization, in term of graph theoretic conditions, of subspace controllability for multipartite quantum spin networks. This extends and unifies previous results.

quant-ph↗

arXiv2019

Subspace controllability of bipartite symmetric spin networks under global control

We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.

quant-ph↗

arXiv2018

Sub-Riemannian Geodesics on SU(n)/S(U(n-1)xU(1)) and Optimal Control of Three Level Quantum Systems

We study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.

quant-ph↗

arXiv2018

Controllability of Symmetric Spin Networks

We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between any two permutation invariant states. This class of states includes highly entangled states such as GHZ states and W states, which are of interest in quantum information.

quant-ph↗

arXiv2016

On Symmetries in Time Optimal Control, sub-Riemannian Geometries and the K-P Problem

The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit space M/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K-P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on SO(3).

math.OC↗

arXiv2015

Time optimal simultaneous control of two level quantum systems

In this paper, we solve the problem of simultaneously driving in minimum time to arbitrary final conditions, N two level quantum systems subject to independent controls. The solution of this problem is obtained via an explicit description of the reachable set of the associated control system on SU(2). The treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on Lie groups.

quant-ph↗

arXiv2014

Minimum Time Optimal Synthesis for a Control System on SU(2)

For the time optimal control on an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the maximum principle are explicit functions of time and the resulting differential equations can be explicitly integrated. We use this fact here to perform the optimal synthesis for these systems, i.e., find all optimal trajectories. As a consequence, we describe a simple method to find the minimum time control for every desired final condition. Although the Lie group SU(2) is three dimensional, optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. Our results are of interest to find the minimum time needed to achieve a given evolution of a two level quantum system.

quant-ph↗

arXiv2013

Exact Algebraic Conditions for Indirect Controllability in Quantum Coherent Feedback Schemes

In coherent quantum feedback control schemes, a target quantum system S is put in contact with an auxiliary system A and the coherent control can directly affect only A. The system S is controlled 'indirectly' through the interaction with A. The system S is said to be indirectly controllable if every unitary transformation can be performed on the state of S with this scheme. The indirect controllability of S will depend on the `dynamical Lie algebra' L characterizing the dynamics of the total system S+A and on the initial state of the auxiliary system A. In this paper we describe this characterization exactly. A natural assumption is that the auxiliary system A is minimal which means that there is no part of A which is uncoupled to S, and we denote by n_A the dimension of such a minimal A, which we assume to be fully controllable. We show that, if n_A is greater than or equal to 3, indirect controllability of S is verified if and only if complete controllability of the total system S+A is verified, i.e., L=su(n_Sn_A) or L=u(n_Sn_A), where n_S denotes the dimension of the system S. If n_A=2, it is possible to have indirect controllability without having complete controllability. The exact condition for that to happen is given in terms of a Lie algebra L_S which describes the evolution on the system S only. We prove that indirect controllability is verified if and only if L_S=u(n_S), and the initial state of the auxiliary system A is pure.

quant-ph↗

arXiv2010

Discrete-Time Controllability for Feedback Quantum Dynamics

Controllability properties for discrete-time, Markovian quantum dynamics are investigated. We find that, while in general the controlled system is not finite-time controllable, feedback control allows for arbitrary asymptotic state-to-state transitions. Under further assumption on the form of the measurement, we show that finite-time controllability can be achieved in a time that scales linearly with the dimension of the system, and we provide an iterative procedure to design the unitary control actions.

quant-ph↗

arXiv2010

Controllability of Quantum Walks on Graphs

In this paper, we consider discrete time quantum walks on graphs with coin focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. We then prove general results and criteria relating controllability to the algebraic and topological properties of the walk. As a consequence of these results, we prove that if the degree of the underlying graph is larger than $\frac{N}{2}$, where $N$ is the number of nodes, the quantum walk is always completely controllable, i.e., it is possible to having it to evolve according to an arbitrary unitary evolution. Another result is that controllability for decentralized models only depends on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.

quant-ph↗

arXiv2008

Non-stationary quantum walks on the cycle

We consider quantum walks on the cycle in the non-stationary case where the `coin' operation is allowed to change at each time step. We characterize, in algebraic terms, the set of possible state transfers and prove that, as opposed to the stationary case, it is possible to asymnptotically reach a uniform distribution among the nodes of the associated graph.

quant-ph↗

arXiv2006

Analysis and identification of quantum dynamics using Lie algebra homomorphisms and Cartan decompositions

In this paper, we consider the problem of model equivalence for quantum systems. Two models are said to be (input-output) equivalent if they give the same output for every admissible input. In the case of quantum systems, the output is the expectation value of a given observable or, more in general, a probability distribution for the result of a quantum measurement. We link the input-output equivalence of two models to the existence of a homomorphism of the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie algebra su(n) is useful to find such a homomorphism and to determine the classes of equivalent models. We consider in detail the important cases of two level systems with a Cartan structure and of spin networks. In the latter case, complete results are given generalizing previous results to the case of networks of spin particles with any value of the spin. In treating this problem, we prove some instrumental results on the subalgebras of su(n) which are of independent interest.

quant-ph↗

arXiv2005

Quantum Symmetries and Cartan Decompositions in Arbitrary Dimensions

We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd-even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimension.

quant-ph↗

arXiv2003

Model identification for spin networks

We consider the problem of determining the unknown parameters of the Hamiltonian of a network of spin 1/2 particles. In particular, we study experiments in which the system is driven by an externally applied electro-magnetic field and the expectation value of the total magnetization is measured. Under appropriate assumptions, we prove that, if it is possible to prepare the system in a known initial state, the above experiment allows to identify the parameters of the Hamiltonian. In the case where the initial state is itself an unknown parameter, we characterize all the pairs Hamiltonian-Initial State which give the same value of the magnetization for every form of the driving electro-magnetic field. The analysis is motivated by recent results on the isospectrality of Hamiltonians describing Magnetic Molecules.

quant-ph↗