768 research outputs found
Koopman Kernel Regression
Many machine learning approaches for decision making, such as reinforcement
learning, rely on simulators or predictive models to forecast the
time-evolution of quantities of interest, e.g., the state of an agent or the
reward of a policy. Forecasts of such complex phenomena are commonly described
by highly nonlinear dynamical systems, making their use in optimization-based
decision-making challenging. Koopman operator theory offers a beneficial
paradigm for addressing this problem by characterizing forecasts via linear
time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix
multiplication. Though there exists a variety of learning approaches, they
usually lack crucial learning-theoretic guarantees, making the behavior of the
obtained models with increasing data and dimensionality unclear. We address the
aforementioned by deriving a universal Koopman-invariant reproducing kernel
Hilbert space (RKHS) that solely spans transformations into LTI dynamical
systems. The resulting Koopman Kernel Regression (KKR) framework enables the
use of statistical learning tools from function approximation for novel
convergence results and generalization error bounds under weaker assumptions
than existing work. Our experiments demonstrate superior forecasting
performance compared to Koopman operator and sequential data predictors in
RKHS.Comment: Accepted to the thirty-seventh Conference on Neural Information
Processing Systems (NeurIPS 2023
Quantum-Classical hybrid systems and their quasifree transformations
The focus of this work is the description of a framework for quantum-classical hybrid systems.
The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case.
Here, we are going to solve two main tasks:
The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure.
Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined.
We start with a general introduction to operator algebras and algebraic quantum theory.
Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems.
Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism.
The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system.
Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz:
We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system.
Then, we present solutions for our initial tasks.
We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem.
After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra.
While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose.
They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture.
The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity.
We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations.
All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected
Locality and Exceptional Points in Pseudo-Hermitian Physics
Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices.
An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory.
A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iγ, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when γ increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase
of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states.
Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case
Convergence of Dynamics on Inductive Systems of Banach Spaces
Many features of physical systems, both qualitative and quantitative, become
sharply defined or tractable only in some limiting situation. Examples are
phase transitions in the thermodynamic limit, the emergence of classical
mechanics from quantum theory at large action, and continuum quantum field
theory arising from renormalization group fixed points. It would seem that few
methods can be useful in such diverse applications. However, we here present a
flexible modeling tool for the limit of theories: soft inductive limits
constituting a generalization of inductive limits of Banach spaces. In this
context, general criteria for the convergence of dynamics will be formulated,
and these criteria will be shown to apply in the situations mentioned and more.Comment: Comments welcom
How Haag-tied is QFT, really?
Haag's theorem cries out for explanation and critical assessment: it sounds the alarm that something is (perhaps) not right in one of the standard way of constructing interacting fields to be used in generating predictions for scattering experiments. Viewpoints as to the precise nature of the problem, the appropriate solution, and subsequently-called-for developments in areas of physics, mathematics, and philosophy differ widely. In this paper, we develop and deploy a conceptual framework for critically assessing these disparate responses to Haag's theorem. Doing so reveals the driving force of more general questions as to the nature and purpose of foundational work in physics
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Quantum Detector and Process Tomography: Algorithm Design and Optimisation
This thesis develops new algorithms and investigates optimisation in quantum detector tomography (QDT) and quantum process tomography (QPT).
QDT is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. We design optimal probe states based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the lower bounds of the UMSE and the condition number for the probe states, and provide concrete examples that can achieve these lower bounds. In order to enhance the estimation precision, we also propose a two-step adaptive QDT and present a sufficient condition on when the infidelity scales where is the number of state copies.
We then utilize regularization to improve the QDT accuracy whenever the probe states are informationally complete or informationally incomplete. We discuss different regularization forms and prove the mean squared error scales as or tends to a constant with state copies under the static assumption. We also characterize the ideal best regularization for the identifiable parameters.
QPT is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. We firstly study the identification of time-varying decoherence rates for open quantum systems. We expand the unknown decoherence rates into Fourier series and take the expansion coefficients as optimisation variables. We then convert it into a minimax problem and apply sequential linear programming technique to solve it.
For general QPT, we propose a two-stage solution (TSS) for both trace-preserving and non-trace-preserving QPT. Using structure simplification, our algorithm has computational complexity where is the dimension of the quantum system and , are the type numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number.
A quantum optical experiment test shows that a suitable regularization form can reach a lower mean squared error in QDT and the testing on IBM quantum machine demonstrates the effectiveness of our TSS algorithm for QPT
Quantum-Classical Hybrid Systems and their Quasifree Transformations
We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous
Quantum-Classical Hybrid Systems and their Quasifree Transformations
We study continuous variable systems, in which quantum and classical degrees
of freedom are combined and treated on the same footing. Thus all systems,
including the inputs or outputs to a channel, may be quantum-classical hybrids.
This allows a unified treatment of a large variety of quantum operations
involving measurements or dependence on classical parameters. The basic
variables are given by canonical operators with scalar commutators. Some
variables may commute with all others and hence generate a classical subsystem.
We systematically study the class of "quasifree" operations, which are
characterized equivalently either by an intertwining condition for phase-space
translations or by the requirement that, in the Heisenberg picture, Weyl
operators are mapped to multiples of Weyl operators. This includes the
well-known Gaussian operations, evolutions with quadratic Hamiltonians, and
"linear Bosonic channels", but allows for much more general kinds of noise. For
example, all states are quasifree. We sketch the analysis of quasifree
preparation, measurement, repeated observation, cloning, teleportation, dense
coding, the setup for the classical limit, and some aspects of irreversible
dynamics, together with the precise salient tradeoffs of uncertainty, error,
and disturbance. Although the spaces of observables and states are infinite
dimensional for every non-trivial system that we consider, we treat the
technicalities related to this in a uniform and conclusive way, providing a
calculus that is both easy to use and fully rigorous.Comment: 63 pages, 6 figure
Noncommutative Geometry and Gauge theories on AF algebras
Non-commutative geometry (NCG) is a mathematical discipline developed in the
1990s by Alain Connes. It is presented as a new generalization of usual
geometry, both encompassing and going beyond the Riemannian framework, within a
purely algebraic formalism. Like Riemannian geometry, NCG also has links with
physics. Indeed, NCG provided a powerful framework for the reformulation of the
Standard Model of Particle Physics (SMPP), taking into account General
Relativity, in a single "geometric" representation, based on Non-Commutative
Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient
framework to study various possibilities to go beyond the SMPP, such as Grand
Unified Theories (GUTs). This thesis intends to show an elegant method recently
developed by Thierry Masson and myself, which proposes a general scheme to
elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs
based on approximately finite -algebras (AF-algebras), using either
derivations of the algebra or spectral triples to build up the underlying
differential structure of the Gauge Theory. The inductive sequence defining the
AF-algebra is used to allow the construction of a sequence of NCGFTs of
Yang-Mills Higgs types, so that the rank can represent a grand unified
theory of the rank . The main advantage of this framework is that it
controls, using appropriate conditions, the interaction of the degrees of
freedom along the inductive sequence on the AF algebra. This suggests a way to
obtain GUT-like models while offering many directions of theoretical
investigation to go beyond the SMPP
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