Invariant Subspace Theory Research Articles

Dense outputs from quantum simulations

The quantum dense output problem is the process of evaluating time-accumulated observables from time-dependent quantum dynamics using quantum computers. This problem arises frequently in applications such as quantum control and spectroscopic computation. We present a range of algorithms designed to operate on both early and fully fault-tolerant quantum platforms. These methodologies draw upon techniques like amplitude estimation, Hamiltonian simulation, quantum linear Ordinary Differential Equation (ODE) solvers, and quantum Carleman linearization. We provide a comprehensive complexity analysis with respect to the evolution time T and error tolerance ϵ. Our results demonstrate that the linearization approach can nearly achieve optimal complexity O(T/ϵ) for a certain type of low-rank dense outputs. Moreover, we provide a linearization of the dense output problem that yields an exact and finite-dimensional closure which encompasses the original states. This formulation is related to the Koopman Invariant Subspace theory and may be of independent interest in nonlinear control and scientific machine learning.

Relative controllability of nonlinear switched fractional systems

AbstractThis article investigates relative controllability of nonlinear switched fractional systems (SFSs). With the aid of Mittag‐Leffler functions and invariant subspace theory, two sufficient and necessary conditions for corresponding linear SFSs are first established. Then, piecewise continuous control functions and a novel nonlinear operator are constructed to overcome the difficulties arising from switching rules, nonlinearity, and fractional derivatives. Under different constraints on nonlinear functions, two controllability conditions depending on system parameters for nonlinear SFSs are proposed by applying Schauder's and Banach's fixed point theorems, respectively. The obtained criteria may well show the influence of coefficient matrices, fractional derivatives, and switching rules on relative controllability. In addition, our proposed method is also applicable for integer‐order switched systems. Finally, two simulated examples are worked out to verify the theoretical results.

General Framework for Randomized Benchmarking

Randomized benchmarking refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors. Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing. In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions. Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays. We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data. We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations. In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols. These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.2 MoreReceived 8 December 2020Revised 23 December 2021Accepted 4 March 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020357Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum benchmarkingQuantum computationQuantum gatesQuantum Information

The role of the algebraic structure in Wold-type decomposition

Abstract In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗ \ast -rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer * {*} -rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer * {*} -rings.

An Efficient Eigenvalue Tracking Method for Time-Delayed Power Systems Based on Continuation of Invariant Subspaces

Time delays are inevitable in the power systems with wide-area control signals, which may compromise the control pe-rformance and thus jeopardize the stability of the power systems. To analyze the impacts of system parameters on critical oscillation modes of a time-delayed power system (TDPS), a new method for eigenvalue trajectory tracking of TDPS is proposed in this paper based on the continuation theory of invariant subspaces. When small changes are imposed on system parameters, such as parameters of wide-area damping controllers (WADCs) and time delays, the invariant subspaces are continuously predicted and corrected, followed by the eigenvalues and eigenvectors. In the process, the predictor and corrector are computed by solving linear equations sparsely and in blocks, which is highly scalable and computationally efficient for large-scale power system applications. The 16-generator 68-bus test system and two real-world power grids in China are used to test and demonstrate the proposed method. The results show that the proposed method can accurately and efficiently track eigenvalue trajectories for TDPS, especially for large-scale power grids. In addition, the sensitivities of eigenvalues and eigenvectors with respect to WADCs' parameters and time delays are obtained as by-products of the approach.

Polycyclic codes as invariant subspaces

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.

The role of algebraic structure in the invariant subspace theory

Following a recent work of authors [1], we establish the algebraic analogs of three major decomposition theorems of Nagy-Foias-Langer, Halmos-Wallen and Fishel in Baer ⁎-rings. This approach not only provides purely algebraic and shorter proofs of these decomposition theorems, but also suggests the possibility of eliminating the role of analytic structure in results related to the invariant subspace theory.

On the additive cyclic structure of quasi-cyclic codes

An index ℓ, length mℓ quasi-cyclic code can be viewed as a cyclic code of length m over the field Fqℓ via a basis of the extension Fqℓ∕Fq. However, this cyclic code is only linear over Fq, making it an additive cyclic code, or an Fq-linear cyclic code, over the alphabet Fqℓ. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have Fqℓ-linear cyclic images under a basis of the extension Fqℓ∕Fq. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.

Halmos problems and related results in the theory of invariant subspaces

This paper contains a survey of main results in the theory of invariant subspaces that have a direct or intermediate connection with the well-known list of problems in operator theory posed by Halmos in 1970. For most results we present brief proofs, schemes of proofs, or basic ideas of proofs. Bibliography: 124 titles.

Halmos problems and related results in the theory of invariant subspaces

Работа содержит обзор основных результатов теории инвариантных подпространств, которые имеют прямую или опосредованную связь с известным списком проблем теории операторов, составленным П. Халмошем в 1970 г. Для большинства результатов приводятся краткие доказательства, схемы или основные идеи доказательств. Библиография: 124 названия.

Invariant subspaces for H2 spaces of σ-finite algebras

We show that a Beurling type theory of invariant subspaces of noncommutative $H^2$ spaces holds true in the setting of subdiagonal subalgebras of $\sigma$-finite von Neumann algebras. This extends earlier work of Blecher and Labuschagne for finite algebras, and complements more recent contributions in this regard by Bekjan, and Chen, Hadwin and Shen in the finite setting, and Sager in the semifinite setting. We then also introduce the notion of an analytically conditioned algebra, and go on to show that in the class of analytically conditioned algebras this Beurling type theory is part of a list of properties which all turn out to be equivalent to the maximal subdiagonality of the given algebra.

Switching Stabilization for a Class of Slowly Switched Systems

In this technical note, the problem of switching stabilization for slowly switched linear systems is investigated. In particular, the considered systems can be composed of all unstable subsystems. Based on the invariant subspace theory, the switching signal with mode-dependent average dwell time (MDADT) property is designed to exponentially stabilize the underlying system. Furthermore, sufficient condition of stabilization for switched systems with all stable subsystems under MDADT switching is also given. The correctness and effectiveness of the proposed approaches are illustrated by a numerical example.

Invariant subspaces with no generator and a problem of H. Helson

In the almost periodic context, any $H_0^2-$space cannot be generated by one of its elements. Together with cocycle argument, this derives that there exist all kinds of invariant subspaces without single generator, from which we can answer some questions on invariant subspace theory.

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.

Applications of fixed point theorems in the theory of invariant subspaces

We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.MSC:47A15, 47H10.

Applications of fixed point theorems in the theory of invariant subspaces

Applications of fixed point theorems in the theory of invariant subspaces

Operator-valued Bergman inner functions as transfer functions

An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discrete time linear systems. This points to a new interaction between the fields of invariant subspace theory and mathematical systems theory.

Self-adjoint time operators and invariant subspaces

The question of existence of self-adjoint time operators for unitary evolutions in classical and quantum mechanics is revisited on the basis of Halmos-Helson theory of invariant subspaces, Sz.-Nagy-Foias dilation theory and Misra-Prigogine-Courbage theory of irreversibility. It is shown that the existence of self-adjoint time operators is equivalent to the intertwining property of the evolution plus the existence of simply invariant subspaces or rigid operator-valued functions for its Sz.-Nagy-Foias functional model. Similar equivalent conditions are given in terms of intrinsic randomness in the context of statistical mechanics. The rest of the contents are mainly a unifying review of the subject scattered throughout an unconnected literature. A well-known extensive set of equivalent conditions is derived from the above results; such conditions are written in terms of Schrrdinger couples, the Weyl commutation relation, incoming and outgoing subspaces, innovation processes, Lax-Phillips scattering, translation and spectral representations, and spectral properties. Also the natural procedure dealing with symmetric time operators in standard quantum mechanics involving their self-adjoint extensions is illustrated by considering the quantum Aharonov-Bohm time-of-arrival operator.

Reflexivity for Subnormal Systems With Dominating Spectrum in Product Domains

The question whether every subnormal tuple \({S}\,=\,({S}_1, \ldots , {S}_n)\) on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain \(U = U_1 \times \ldots \times U_n \subset {\mathbb{C}}^{n}\) is reflexive.

Nonunitary similarity transformation of conservative to dissipative evolutions: Intertwining without time operator

Reversible evolutions are usually expressed in terms of unitary groups on separable Hilbert spaces, whereas irreversible ones are described by contraction semigroups. In the theory of nonunitary similarity transformations intertwining unitary groups and contraction semigroups, proposed initially in the context of statistical mechanics as part of an exact theory of irreversibility, the unitary groups with such intertwining property have been qualified by the existence of an internal time operator. This work tackles the question of existence of internal time operators for unitary groups with the intertwining property. Equivalent conditions to the existence of internal time operators for such unitary groups are given on the basis of the Sz.-Nagy–Foiaş [Harmonic Analysis of Operators on Hilbert Spaces (North-Holland, Amsterdam, 1970)] dilation theory and the theory of shift invariant subspaces. These conditions permit us to solve the inverse intertwining problem in the negative: there are unitary groups with the intertwining property which do not admit internal time operator. A representative family of such unitary groups is given.