arXiv subjects

Gunther Dirr

Publications and source records attributed to Gunther Dirr.

18 recordsLinked to original sources

arXiv2025

On the convergence of the variational quantum eigensolver and quantum optimal control

When does a variational quantum algorithm converge to a globally optimal solution? Despite the large literature around variational approaches to quantum computing, the answer is largely unknown. We address this open question by developing a convergence theory for the variational quantum eigensolver (VQE). By leveraging the terminology of quantum control landscapes, we prove a sufficient criterion that characterizes when convergence to a ground state of a Hamiltonian can be guaranteed for almost all initial parameter settings. More specifically, we show that if (i) a parameterized unitary transformation allows for moving in all tangent-space directions (local surjectivity) in a bounded manner and (ii) the gradient descent used for the parameter update terminates, then the VQE converges to a ground state almost surely. We develop constructions that satisfy both aspects of condition (i) and analyze two commonly employed families of quantum circuit ans\"atze. Finally, we discuss regularization techniques for guaranteeing gradient descent to terminate, as for condition (ii), and draw connections to the halting problem.

quant-ph↗

arXiv2024

Randomized Gradient Descents on Riemannian Manifolds: Almost Sure Convergence to Global Minima in and beyond Quantum Optimization

We analyze convergence of gradient-descent methods on Riemannian manifolds. In particular, we study randomization of Riemannian gradient algorithms for minimizing smooth cost functions (of Morse-Bott type). We prove that randomized gradient descent methods, where the Riemannian gradient is replaced by a random projection of it, converge to a single local optimum almost surely despite the existence of saddle points. We consider both uniformly distributed and discrete random projections. We also discuss the time required to pass a saddle point. As a major application, we consider ground-state preparation through quantum optimization over the unitary group. In mathematical terms our randomized algorithm applied to the trace function $U \to \operatorname{tr}(AU\rho U^*)$ almost surely converges to its global minimum. The minimum corresponds to the smallest eigenvalue (ground state) of the selfadjoint operator $A$ (Hamiltonian) if $\rho$ is a rank-one projector (pure state). In this setting, one can efficiently replace the uniform random projections by implementing so-called discrete unitary 2-designs.

math.OC↗

arXiv2023

Reachability, Coolability, and Stabilizability of Open Markovian Quantum Systems with Fast Unitary Control

Open Markovian quantum systems with fast and full Hamiltonian control can be reduced to an equivalent control system on the standard simplex modelling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate applications to the cooling of Markovian quantum systems. We show that for certain tasks of interest, the control Hamiltonian can be chosen time-independent. -- The reduction picture is an example of dissipative interconversion between equivalence classes of states, where the classes are induced by fast controls.

quant-ph↗

arXiv2023

Reduced Control Systems on Symmetric Lie Algebras

For a symmetric Lie algebra $\mathfrak g=\mathfrak k\oplus\mathfrak p$ we consider a class of bilinear or more general control-affine systems on $\mathfrak p$ defined by a drift vector field $X$ and control vector fields $\mathrm{ad}_{k_i}$ for $k_i\in\mathfrak k$ such that one has fast and full control on the corresponding compact group $\mathbf K$. We show that under quite general assumptions on $X$ such a control system is essentially equivalent to a natural reduced system on a maximal Abelian subspace $\mathfrak a\subseteq\mathfrak p$, and likewise to related differential inclusions defined on $\mathfrak a$. We derive a number of general results for such systems and as an application we prove a simulation result with respect to the preorder induced by the Weyl group action.

math.OC↗

arXiv2023

Analytic, Differentiable and Measurable Diagonalizations in Symmetric Lie Algebras

We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix diagonalization, such as the eigenvalue decomposition of real symmetric or complex Hermitian matrices, and the real or complex singular value decomposition. Concretely, given a path of structured matrices with a certain smoothness, we study what kind of smoothness one can obtain for the corresponding diagonalization of the matrices.

math.RT↗

arXiv2023

Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of $d$-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.

quant-ph↗

arXiv2023

Exploring the Limits of Open Quantum Dynamics I: Motivation, New Results from Toy Models to Applications

Which quantum states can be reached by controlling open Markovian $n$-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature $T$. The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case $T=0$ (amplitude damping) we present new results for $0 <T < \infty$ using methods of $d$-majorisation.

quant-ph↗

arXiv2022

Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.

math.FA↗

arXiv2022

Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.

quant-ph↗

arXiv2022

The $d$-Majorization Polytope

We investigate geometric and topological properties of $d$-majorization -- a generalization of classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive a new, simplified characterization of $d$-majorization which allows us to work out a halfspace description of the corresponding $d$-majorization polytopes. That is, we write the set of all vectors which are $d$-majorized by some given vector $y \in \mathbb{R}^n$ as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type $Mx\leq b$. Here $b$ depends on $y$ while $M$ can be chosen independently of $y$. This description lets us prove continuity of the $d$-majorization polytope (jointly with respect to $d$ and $y$) and, furthermore, lets us fully characterize its extreme points. Interestingly, for $y\geq 0$ one of these extreme points classically majorizes every other element of the $d$-majorization polytope. Moreover, we show that the induced preorder structure on $\mathbb{R}^n$ admits minimal and maximal elements. While the former are always unique the latter are unique if and only if they correspond to the unique minimal entry of the $d$-vector.

math.CO↗

arXiv2020

Uniform and $L^q$-Ensemble Reachability of Parameter-dependent Linear Systems

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs $\big( A(θ),B(θ) \big)$ depending on a parameter $θ\in \p$ that is varying over a compact subset $\p$ of the complex plane. In particular, we investigate the following control task: Find an open-loop control which is {\it independent} of the parameter $θ\in \p$ and steers a given family of initial states $x_0(θ)$ arbitrarily close to a desired family of terminal states $f(θ)$ in finite time. Here, the maps $θ\mapsto x_0(θ)$ and $θ\mapsto f(θ)$ are assumed to lie in a common appropriately chosen {Banach space $X_n(\p)$ of $\C^n$-valued functions}. If this task is solvable for all initial and terminal states, the pair $\big( A(θ),B(θ) \big)$ is called {(completely)} ensemble controllable with respect to $X_n(\p)$. Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family $A(θ)$. Based on thses findings as well as approximation theory and cyclicity conditions of multiplications operators, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the Banach spaces of continuous functions and $L^q$-functions. In the last section, results on {averaged} controllability (reachability) for linear families $\big( A(θ),B(θ),C(θ) \big)$ are presented.

math.OC↗

arXiv2019

Reachable Sets from Toy Models to Controlled Markovian Quantum Systems

In the framework of bilinear control systems, we present reachable sets of coherently controllable open quantum systems with switchable coupling to a thermal bath of arbitrary temperature $T geq 0$. The core problem boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: (i) permutations within the simplex, (ii) contractions by a dissipative one-parameter semigroup. Our work illustrates how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. We completely characterize the case $T=0$ and present inclusions for $T>0$.

math.OC↗

arXiv2019

Von Neumann Type of Trace Inequalities for Schatten-Class Operators

We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix.

math.FA↗

arXiv2019

Reachability in Infinite Dimensional Unital Open Quantum Systems with Switchable GKS-Lindblad Generators

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite dimensional open quantum dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term $H_0$, finitely many bounded control Hamiltonians $H_j$ allowing for (at least) piecewise constant control amplitudes $u_j(t)\in{\mathbb R}$ plus a bang-bang (i.e. on-off) switchable noise term $\mathbfΓ_V$ in Kossakowski-Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite dimensional analogues.---The proof of the result is currently limited to the control Hamiltonians $ H_j$ being bounded and noise terms $\mathbfΓ_V$ with compact normal $V$.

quant-ph↗

arXiv2019

The C-Numerical Range for Schatten-Class Operators

We generalize the $C$-numerical range $W_C(T)$ from trace-class to Schatten-class operators, i.e. to $C\in\mathcal B^p(\mathcal H)$ and $T\in\mathcal B^q(\mathcal H)$ with $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin. For $q \in (1,\infty]$, this is equivalent to saying that the closure of the image of the unitary orbit of $T\in\mathcal B^q(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^q(\mathcal H))'$ is star-shaped with respect to the origin. For $q=1$, one has star-shapedness with respect to $\operatorname{tr}(T)W_e(L)$, where $W_e(L)$ denotes the essential range of $L$. Moreover, the closure of $W_C(T)$ is convex if $C$ or $T$ is normal with collinear eigenvalues. If $C$ and $T$ are both normal, then the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the closure of the convex hull of the $C$-spectrum. This closure coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.

math.FA↗

arXiv2019

Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups

We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single channel.

math-ph↗

arXiv2018

The C-Numerical Range in Infinite Dimensions

In infinite dimensions and on the level of trace-class operators $C$ rather than matrices, we show that the closure of the $C$-numerical range $W_C(T)$ is always star-shaped with respect to the set $\operatorname{tr}(C)W_e(T)$, where $W_e(T)$ denotes the essential numerical range of the bounded operator $T$. Moreover, the closure of $W_C(T)$ is convex if either $C$ is normal with collinear eigenvalues or if $T$ is essentially self-adjoint. In the case of compact normal operators, the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the convex hull of the closure of the $C$-spectrum. This convex hull coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.

math.FA↗

arXiv2007

The Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control--and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper (math-ph/0702005). We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.

math-ph↗