Burgarth group
University of Erlangen–Nuremberg, Erlangen, Germany
- Tracked papers
- 40 (2008–2025)
Topics
The most frequent field-map topics among the PI' tracked papers.
- Dynamical decoupling 13
- Controllability 10
- Gates 7
- Many-body 6
- Speed limits 4
- Open systems 4
- Feedback 4
- Bosonic 4
Recent papers
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On the convergence of the variational quantum eigensolver and quantum optimal control
Abstract
When do variational quantum algorithms converge to a globally optimal solution? Despite extensive work, this question remains largely open. We develop a convergence theory for the variational quantum eigensolver (VQE), using quantum control landscape terminology to prove a sufficient criterion guaranteeing convergence to a Hamiltonian's ground state for almost all initial parameters. Specifically, we show that if (i) the parameterized unitary transformation allows movement in all tangent-space directions in a bounded manner (local surjectivity), and (ii) the gradient descent terminates, then VQE converges almost surely to a ground state. Under similar assumptions, we also guarantee almost-sure convergence to a global optimum on a smaller, closed unitary Lie subgroup, with representation theory providing a criterion for when this optimum is a ground state. We construct examples satisfying condition (i) and analyze two common circuit ansatz families, then discuss regularization techniques that ensure gradient descent terminates and connect to the halting problem.
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Quantum Speed Limits from Symmetries in Quantum Control
Abstract
In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary $U$ and on the other hand we bound the time that is needed to implement the dynamics of a target Hamiltonian $H$ in the worst case. We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.
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Qualitative differences in the robust controllability of model two-qubit systems
Abstract
The precise implementation and manipulation of quantum gates is key to extracting advantages from future quantum technologies. Achieving this requires very accurate control over the quantum system. If one has complete knowledge about a Hamiltonian, accurate manipulation of the system is possible. However, in real scenarios, there will often be some uncertainty in the parameters of the Hamiltonian, which makes full control of the system either difficult or impossible. In this paper we consider two model Hamiltonians with a continuous parameter that is partly unknown. We assess robust controllability against this parameter uncertainty using existing theoretical frameworks and take a numerical route by discretizing the unknown parameter in the cases where we cannot predict controllability. Furthermore, we introduce a penalty term into the fidelity function to optimize control pulses, enhancing robustness against the influence of parameter fluctuations. Within our framework, we analyze the qualitative differences in the robust controllability of the two systems.
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Bath Dynamical Decoupling with a Quantum Channel
Abstract
Bang-bang dynamical decoupling protects an open quantum system from decoherence due to its interaction with the surrounding bath/environment. In its standard form, this is achieved by strongly kicking the system with cycles of unitary operations, which average out the interaction Hamiltonian. In this paper, we generalize the notion of dynamical decoupling to repeated kicks with a quantum channel, which is applied to the bath. We derive necessary and sufficient conditions on the employed quantum channel and find that bath dynamical decoupling works if and only if the kick is ergodic. Furthermore, we study in which circumstances CPTP kicks on a mono-partite quantum system induce quantum Zeno dynamics with its Hamiltonian cancelled out. This does not require the ergodicity of the kicks, and the absence of decoherence-free subsystems is both necessary and sufficient. While the standard unitary dynamical decoupling is essentially the same as the quantum Zeno dynamics, our investigation implies that this is not true any more in the case of CPTP kicks. To derive our results, we prove some spectral properties of ergodic quantum channels, that might be of independent interest. Our approach establishes an enhanced and unified mathematical understanding of several recent experimental demonstrations and might form the basis of new dynamical decoupling schemes that harness environmental noise degrees of freedom.
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Efficiency of Dynamical Decoupling for (Almost) Any Spin-Boson Model
Abstract
Dynamical decoupling is a technique aimed at suppressing the interaction between a quantum system and its environment by applying frequent unitary operations on the system alone. In the present paper, we analytically study the dynamical decoupling of a two-level system coupled with a structured bosonic environment initially prepared in a thermal state. We find sufficient conditions under which dynamical decoupling works for such systems, and, most importantly, we find bounds for the convergence speed of the procedure. Our analysis is based on a new Trotter theorem for multiple Hamiltonians and involves a rigorous treatment of the evolution of mixed quantum states via unbounded Hamiltonians. A comparison with numerical experiments shows that our bounds reproduce the correct scaling in various relevant system parameters. Furthermore, our analytical treatment allows for quantifying the decoupling efficiency for boson baths with infinitely many modes, in which case a numerical treatment is unavailable.
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Creation and manipulation of surface code defects with quantum optimal control
Abstract
The surface code is a spin-1/2 lattice system that can exhibit non-trivial topological order when defects are punctured in the lattice and thus can be used as a stabiliser code. The protocols developed to create defects in the system have previously relied on adiabatic dynamics. In this work we use techniques of quantum optimal control to overcome the requirement for adiabaticity and achieve defect creation and implemention of other important operations required for quantum computation at much faster timescales.
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Open loop linear control of quadratic Hamiltonians with applications
Abstract
The quantum harmonic oscillator is one of the most fundamental objects in physics. We consider the case where it is extended to an arbitrary number modes and includes all possible terms that are bilinear in the annihilation and creation operators, and assume we also have an arbitrary time-dependent drive term that is linear in those operators. Such a Hamiltonian is very general, covering a broad range of systems including quantum optics, superconducting circuit QED, quantum error correcting codes, Bose-Einstein condensates, atomic wave packet transport beyond the adiabatic limit and many others. We examine this situation from the point of view of quantum control, making use of optimal control theory to determine what can be accomplished, both when the controls are arbitrary and when they must minimize some cost function. In particular we develop a class of analytical pulses. We then apply our theory to a number of specific topical physical systems to illustrate its use and provide explicit control functions, including the case of the continuously driven conditional displacement gate.
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Exact and lower bounds for the quantum speed limit in finite dimensional systems
Abstract
A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups and differential geometry, and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian.
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Scalable quantum control and non-abelian anyon creation in the Kitaev honeycomb model
Abstract
The Kitaev honeycomb model is a system allowing for experimentally realisable quantum computation with topological protection of quantum information. Practical implementation of quantum information processing typically relies on adiabatic, i.e. slow dynamics. Here we show that the restriction to adiabatic dynamics can be overcome with optimal control theory, enabled by an extension of the fermionization of the Kitaev honeycomb model to the time-dependent case. Moreover we present a quantum control method that is applicable to large lattice models due to sub-exponential scaling.
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Stability and convergence of dynamical decoupling with finite amplitude control
Abstract
Dynamical decoupling is a key method to mitigate errors in a quantum mechanical system, and we studied it in a series of papers dealing in particular with the problems arising from unbounded Hamiltonians. The standard bangbang model of dynamical decoupling, which we also used in those papers, requires decoupling operations with infinite amplitude, which is strictly speaking unrealistic from a physical point of view. In this paper we look at decoupling operations of finite amplitude, discuss under what assumptions dynamical decoupling works with such finite amplitude operations, and show how the bangbang description arises as a limit, hence justifying it as a reasonable approximation.