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In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.

We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, {\it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group $S_n$. Because of the symmetry, the underlying Hilbert space, ${\cal H}=(\mathbb{C}^d)^{\otimes n}$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(d^n)$, denoted here by $u^{S_n}(d^n)$. The dynamical Lie algebra ${\cal L}$, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra $u^{S_n}(d^n)$. If ${\cal L}$ acts as $su\left( \dim(V) \right)$ on each of the invariant subspaces $V$, the system is called {\it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space ${\cal H}$ coincides with the {\it Clebsch-Gordan} splitting of $(\mathbb{C}^d)^{\otimes n}$ into {\it irreducible representations} of $su(d)$. In this view, $u^{S_n}(d^n)$, is the direct sum of certain $su(n_j)$ for some $n_j$'s we shall specify, and its {\it center} which is the Abelian (Lie) algebra generated by the {\it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general $n$ and $d$ but we recast previous results on $n$ qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of $n=3$, $d=3$, that is, {\it three qutrits}.

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.

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.

We provide a rigorous analysis of the quantum optimal control problem in the setting of a linear combination $s(t)B+(1-s(t))C$ of two noncommuting Hamiltonians $B$ and $C$. This includes both quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA). The target is to minimize the energy of the final ``problem'' Hamiltonian $C$, for a time-dependent and bounded control schedule $s(t)\in [0,1]$ and $t\in \mc{I}:= [0,t_f]$. It was recently shown, in a purely closed system setting, that the optimal solution to this problem is a ``bang-anneal-bang'' schedule, with the bangs characterized by $s(t)= 0$ and $s(t)= 1$ in finite subintervals of $\mc{I}$, in particular $s(0)=0$ and $s(t_f)=1$, in contrast to the standard prescription $s(0)=1$ and $s(t_f)=0$ of quantum annealing. Here we extend this result to the open system setting, where the system is described by a density matrix rather than a pure state. This is the natural setting for experimental realizations of QA and QAOA. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of $s(0)=0$ and $s(t_f)=1$. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which $s(t_f)=1$, and even this result is not recovered in the fully Markovian limit. The analysis, which we carry out entirely within the geometric framework of Pontryagin Maximum Principle, simplifies using the density matrix formulation compared to the state vector formulation., Comment: 21 pages (incl. 9 page appendix), 3 figures

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., Comment: 14 pages, 3 figures

The problem of finding minimizing geodesics for a manifold M with a sub-Riemannian structure is equivalent to the time optimal control of a driftless system on M with a bound on the control. We consider here a class of sub-Riemannian problems on the classical Lie groups G where the dynamical equations are of the form \dot x=\sum_j X_j(x) u_j and the X_j=X_j(x) are right invariant vector fields on G and u_j:=u_j(t) the controls. The vector fields X_j are assumed to belong to the P part of a Cartan K-P decomposition. These types of problems admit a group of symmetries K which act on G by conjugation. Under the assumption that the minimal isotropy group in K is discrete, we prove that we can reduce the problem to a Riemannian problem on the regular part of the associated quotient space G/K. On this part we define the corresponding quotient metric. For the special cases of the K-P decomposition of SU(n) of type AIII we prove that the assumption on the minimal isotropy group is verified. Moreover, under the assumption that the quotient space G/K with the given metric has negative curvature we prove that the cut locus has to belong to the singular part of G. As an example of applications of these techniques we characterize the cut locus for a problem on SU(2) of interest in the control of quantum systems., Comment: Different shorter version to appear in the Proceedings of the European Control Conference 2019

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries which allows us to reduce the problem of control to a quotient space where the control system is `fully actuated'. As a result we can prescribe a desired trajectory which is, to some extent, arbitrary and derive the corresponding control. We discuss the application to the simultaneous control of two non-interacting spin 1/2 particles with different gyromagnetic ratios in zero field NMR in detail. Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration with smooth controls.

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.

We derive the explicit solution of the problem of time-optimal control by a common magnetic fields for two independent spin-$\frac{1}{2}$ particles. Our approach is based on the Pontryagin Maximum Principle and a novel symmetry reduction technique. We experimentally implement the optimal control using zero-field nuclear magnetic resonance. This reveals an average gate error of $1\%$ and a $70 \%$ to $80$ $\%$ decrease in the experiment duration as compared to existing methods. This is the first analytical solution and experimental demonstration of time-optimal control in such a system and it provides a route to achieve time optimal control in more general quantum systems.

We describe a method to analyze and decompose the dynamics of a control system on a Lie group subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This is the case for quantum mechanical multipartite systems, such as spin networks, where each factor of the tensor product represents the state of one of the component systems. We present several examples of applications and indicate directions for future research., Comment: 24 pages

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.

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.

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).

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.

In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin Maximum Principle, and provide a complete characterization of singular and non-singular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator., Comment: 37 pages, 10 figures; Latex

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.

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., Comment: Expanded version of the previously posted paper where what was before mentioned as a conjecture is proved

We consider a control scheme where a quantum system S is put in contact with an auxiliary quantum system A and the control can affect A only, while S is the system of interest. The system S is then controlled indirectly through the interaction with A. Complete controllability of S +A means that every unitary state transformation for the system S +A can be achieved with this scheme. Indirect controllability means that every unitary transformation on the system S can be achieved. We prove in this paper, under appropriate conditions and definitions, that these two notions are equivalent in finite dimension. We use Lie algebraic methods to prove this result.

A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S+A, extending previous results, and prove that complete controllability of S+A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states., Comment: Paper to appear in final version in IEEE Transactions on Automatic Control, Special issue on Quantum Control

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks., Comment: 12 pages, 1 figure

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.

For a right-invariant system on a compact Lie group G, I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate assumptions, achieves exact control to the target but requires estimation of the `size' of a neighborhood of the identity in G. The second method, does not involve any mathematical difficulty, and obtains control to a desired target with arbitrary accuracy. A third method is then given combining the main ideas of the previous methods. This is also very simple in its formulation and turns out to be generically more efficient as illustrated by one of the examples we consider. The methods described in the paper provide arbitrary constructive control for any right-invariant system on a compact Lie group. I give examples including closed multilevel quantum systems and lossless electrical networks. In particular, the results can be applied to the coherent control of general multilevel quantum systems.

I obtain the dynamics of the continuous time quantum walk on a $d$-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of arXiv:quant-ph/0606050 which proved this limit for the infinite line. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.

We study generalized concurrences as a tool to detect the entanglement of bipartite quantum systems. By considering the case of 2 X 4 states of rank 2, we prove that generalized concurrences do not, in general, give a necessary and sufficient condition of separability. We identify a set of entangled states which are undetected by this method., Comment: It replaces two previous versions. The new version incorporates important references on the topic. It corrects a mistake. Title has changed