Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1509.02629

Error Bounds for Finite-Dimensional Approximations of Input-Output Open Quantum Systems by Subspace Truncation and Adiabatic Elimination

Onvaree Techakesari, Hendra I. Nurdin

Published Sep 9, 2015• 0 citations
arXivHTMLPDF

Abstract

An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with error bounds in the finite-dimensional approximations of input-output open quantum systems defined on an infinite-dimensional Hilbert space. We develop a framework for developing error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. This framework is then applied to two approaches for obtaining finite-dimensional approximations: subspace truncation and adiabatic elimination. Applications of the bounds to some physical examples drawn from the literature are provided to illustrate our results.

DOI
10.48550/arxiv.1509.02629

Related papers

Error bounds on finite-dimensional approximations of input-output open quantum systems

Onvaree Techakesari, Hendra I. Nurdin

Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Application to a physical example drawn from the literature is provided to illustrate our results.

IEEE Conference on Decision and Control (CDC)•Dec 1, 2015•3 citations

A posteriori error estimates for the Lindblad master equation

Paul-Louis Etienney, Rémi Robin, Pierre Rouchon

We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.

arXiv:2501.09607•Quantum•Mar 16, 2026•0 citations

Quantum error correction on infinite-dimensional Hilbert spaces

Cédric Bény, Achim Kempf, David W. Kribs

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.

arXiv:0811.0421•Journal of Mathematical Physics•Jun 1, 2009•20 citations

Eternal adiabaticity in quantum evolution

Daniel Klaus Burgarth, Paolo Facchi A, Hiromichi Nakazato +2 more

We iteratively apply a recently formulated adiabatic theorem for the strong-coupling limit in finite-dimensional closed and open quantum systems. This allows us to improve approximations to a perturbed dynamics, beyond the standard approximation based on quantum Zeno dynamics and adiabatic elimination. The effective generators describing the approximate evolutions are endowed with the same block structure as the unperturbed part of the generator, and exhibit adiabatic evolutions. This iterative adiabatic theorem reveals that adiabaticity holds eternally, that is, the system evolves within each eigenspace of the unperturbed part of the generator, with an error bounded by $O(1/\ensuremath{\gamma})$ uniformly in time, where $\ensuremath{\gamma}$ is the strength of the unperturbed part of the generator. We prove that the iterative adiabatic theorem reproduces Bloch's perturbation theory in the unitary case, and is therefore a full generalization to open systems. We furthermore prove the equivalence of the Schrieffer-Wolff and des Cloizeaux approaches in the unitary case and generalize both to arbitrary open systems, showing that they share the eternal adiabaticity, and providing explicit error bounds. Finally we discuss the physical structure of the effective adiabatic generators and show that ideal effective generators for open systems do not exist in general.

arXiv:2011.04713•Physical Review A•Mar 17, 2021•14 citations

Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits

Evgeny Mozgunov, Daniel A. Lidar

We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the [Formula: see text]-dimensional qubit subspace of a circuit model of [Formula: see text] qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain [Formula: see text] as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a [Formula: see text] effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

arXiv:2011.08116•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 4, 2022•5 citations

Universal Error Bound for Constrained Quantum Dynamics

Zongping Gong, Nobuyuki Yoshioka, Naoyuki Shibata +1 more

It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking. Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models. We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law t^{d+1} in d dimensions. Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics.

arXiv:2001.03419•Physical Review Letters•May 27, 2020•14 citations
An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with error bounds in the finite-dimensional approximations of input-output open quantum systems defined on an infinite-dimensional Hilbert space. We develop a framework for developing error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. This framework is then applied to two approaches for obtaining finite-dimensional approximations: subspace truncation and adiabatic elimination. Applications of the bounds to some physical examples drawn from the literature are provided to illustrate our results.
Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Application to a physical example drawn from the literature is provided to illustrate our results.
We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.
We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.
We iteratively apply a recently formulated adiabatic theorem for the strong-coupling limit in finite-dimensional closed and open quantum systems. This allows us to improve approximations to a perturbed dynamics, beyond the standard approximation based on quantum Zeno dynamics and adiabatic elimination. The effective generators describing the approximate evolutions are endowed with the same block structure as the unperturbed part of the generator, and exhibit adiabatic evolutions. This iterative adiabatic theorem reveals that adiabaticity holds eternally, that is, the system evolves within each eigenspace of the unperturbed part of the generator, with an error bounded by $O(1/\ensuremath{\gamma})$ uniformly in time, where $\ensuremath{\gamma}$ is the strength of the unperturbed part of the generator. We prove that the iterative adiabatic theorem reproduces Bloch's perturbation theory in the unitary case, and is therefore a full generalization to open systems. We furthermore prove the equivalence of the Schrieffer-Wolff and des Cloizeaux approaches in the unitary case and generalize both to arbitrary open systems, showing that they share the eternal adiabaticity, and providing explicit error bounds. Finally we discuss the physical structure of the effective adiabatic generators and show that ideal effective generators for open systems do not exist in general.
We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the [Formula: see text]-dimensional qubit subspace of a circuit model of [Formula: see text] qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain [Formula: see text] as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a [Formula: see text] effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking. Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models. We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law t^{d+1} in d dimensions. Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics.