Papers by Thomas Schulte-Herbrüggen
arXiv (Cornell University), Jul 31, 2023
Open Markovian quantum systems with fast and full Hamiltonian control can be reduced to an equiva... more 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.
Reduction of multiplet complexity in COSY-type NMR spectra
Molecular Physics, Mar 1, 1991
... By T. SCHULTE-HERBRI]GGEN, ZL MADI, OW SORENSEN and RR ERNST Laboratorium f/Jr Physikalische ... more ... By T. SCHULTE-HERBRI]GGEN, ZL MADI, OW SORENSEN and RR ERNST Laboratorium f/Jr Physikalische Chemie, Eidgen6ssische Technische Hochschule, 8092 Z/irich, Switzerland ... 0026-8976/91 $3.00 9 1991 Taylor & Francis Ltd Page 2. 848 T. Schulte-Herbriiggen et al. ...
Relative<b><i>C</i></b>-numerical ranges for applications in quantum control and quantum information
Linear & Multilinear Algebra, 2008
Aspects and prospects of high-resolution NMR
ABSTRACT Gramicidin-A, a linear pentadecapeptide (with four tryptophans) forming simplest ion cha... more ABSTRACT Gramicidin-A, a linear pentadecapeptide (with four tryptophans) forming simplest ion channels in lipids, is studied as a cesium-complex in an organic solvent and as an SDS-micellar complex in aqueous solution. Whereas the former is a double-stranded helical dimer, the latter forms a single-stranded head-to-head dimer mimicking ion channels. NMR relaxation gives similar order parameters for tryptophan internal motion in either type of structure, and an explorative MD simulation is consistent with Gaussian axial fluctuations on the subnanosecond timescale. Planar Mixing, a new pulse sequence derived by symmetry selection rules, is designed to give utmost cross-peak- multiplet simplicity for extracting J coupling constants. It comes in two variants: By preserving coherence order, zero-quantum planar mixing selects the antiecho part of the spectrum, while the two-quantum analogue retains the echo part. In contrast to isotropic mixing, the former also allows for monitoring spin waves through spin chains. Novel Concepts of Optimisation are introduced in order to get utmost sensitivity in coherent ensemble spectroscopy. The general question of how to maximise unitary transfers from coherent signals observed in one dimension to those detected in another one relates to many other fields, e.g. the C-numerical radius in mathematics. Our operator- gradient-based computer algorithm is the first one to determine the value of the C-numerical radius and its optimum unitary operators leading to optimised NMR pulse sequences. The general theoretical framework characterises unitary controllability of spin systems and invites further impact of optimal control techniques in coherent ensemble spectroscopy.
Physical Review A, Jan 2, 2007
This paper is dedicated to the memory of Martti Salomaa. Quantum optimal control theory is applie... more This paper is dedicated to the memory of Martti Salomaa. Quantum optimal control theory is applied to two and three coupled Josephson charge qubits. It is shown that by using shaped pulses a cnot gate can be obtained with a trace fidelity > 0.99999 for the two qubits, and even when including higher charge states, the leakage is below 1%. Yet, the required time is only a fifth of the pioneering experiment [1] for otherwise identical parameters. The controls have palindromic smooth time courses representable by superpositions of a few harmonics. We outline schemes to generate these shaped pulses such as simple network synthesis. The approach is easy to generalise to larger systems as shown by a fast realisation of Toffoli's gate in three linearly coupled charge qubits. Thus it is to be anticipated that this method will find wide application in coherent quantum control of systems with finite degrees of freedom whose dynamics are Liealgebraically closed.
Physical Review Letters, Mar 2, 2009
A central challenge for implementing quantum computing in the solid state is decoupling the qubit... more A central challenge for implementing quantum computing in the solid state is decoupling the qubits from the intrinsic noise of the material. We investigate the implementation of quantum gates for a paradigmatic, non-Markovian model: A single qubit coupled to a two-level system that is exposed to a heat bath. We systematically search for optimal pulses using a generalization of the novel open systems Gradient Ascent Pulse Engineering (GRAPE) algorithm. We show and explain that next to the known optimal bias point of this model, there are optimal shapes which refocus unwanted terms in the Hamiltonian. We study the limitations of controls set by the decoherence properties. This can lead to a significant improvement of quantum operations in hostile environments.
IEEE Transactions on Automatic Control, Aug 1, 2012
We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian ... more We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of quantum channels. These cones are the counterpart of the infinitesimal generator of a single one-parameter semigroup. They comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics. Such a differential characterisation is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations.
Linear Algebra and its Applications, 2013
The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is ... more The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum many-body systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properties and efficient computations. The variational ansatz for this numerical approach leads to the minimization of the Rayleigh quotient. The Alternating Least Squares technique is then applied to break down the eigenvector computation to problems of appropriate size, which can be solved by classical methods. Efficient computations require fast computation of the matrix-vector product and of the inner product of two decomposed vectors. To this end, both appropriate representations of vectors and efficient contraction schemes are needed. Here approaches from many-body quantum physics for one-dimensional and two-dimensional systems (Matrix Product States and Projected Entangled Pair States) are treated mathematically in terms of tensors. We give the definition of these concepts, bring some results concerning uniqueness and numerical stability and show how computations can be executed efficiently within these concepts. Based on this overview we present some modifications and generalizations of these concepts and show that they still allow efficient computations such as applicable contraction schemes. In this context we consider the minimization of the Rayleigh quotient in terms of the parafac (CP) formalism, where we also allow different tensor partitions. This approach makes use of efficient contraction schemes for the calculation of inner products in a way that can easily be extended to the mps format but also to higher dimensional problems.
Semidefinite Programming Relaxations Applied to Determining Upper Bounds of C-numerical Ranges
2006 IEEE International Conference on Control Applications, Oct 1, 2006
In this contribution the global optimal upper bounds of the C-numerical range of an arbitrary squ... more In this contribution the global optimal upper bounds of the C-numerical range of an arbitrary square matrix A is investigated. In general the geometry of the C-numerical range is quite complicated and can be yet only partially understood. However, quadratically constrained quadratic programs (QQPs), as an important modelling tool, are used to describe this optimization problem, where the quadratic constraints are in this case the unitary matrix condition U+U = I und its seemingly redundant unitary matrix condition UU+ = I. Generally the QQPs are NP-hard and numerically intractable. However the semidefinite programming (SDP) relaxations to the QQPs, based upon the Positivstellensatz, can be solved in a numerically stable way and then offer sharp approximate solutions to these optimization problems. Numerical results for some physical benchmark examples are presented which indicate that the proposed method yields at least competitive upper bounds of the C-numerical ranges in comparison with other methods
arXiv (Cornell University), Apr 29, 2015
What can one do with a given tunable quantum device? We provide complete symmetry criteria decidi... more What can one do with a given tunable quantum device? We provide complete symmetry criteria deciding whether some effective target interaction(s) can be simulated by a set of given interactions. Symmetries lead to a better understanding of simulation and permit a reasoning beyond the limitations of the usual explicit Lie closure. Conserved quantities induced by symmetries pave the way to a resource theory for simulability. On a general level, one can now decide equality for any pair of compact Lie algebras just given by their generators without determining the algebras explicitly. Several physical examples are illustrated, including entanglement invariants, the relation to unitary gate membership problems, as well as the central-spin model.
A new type ofC -numerical range arising in quantum computing
Proceedings in applied mathematics & mechanics, Dec 1, 2006
We consider a new type of numerical range motivated by recent applications in quantum computing. ... more We consider a new type of numerical range motivated by recent applications in quantum computing. We term the object of interest local C ‐numerical rangeWloc(C, A) of A. It is obtained by replacing the special unitary group in the definition of the C ‐numerical range by the so‐called local subgroup of SU (2N ), i.e. by the N ‐fold tensor product SU (2) ⊗ · · · ⊗ SU(2) of unitary (2 × 2)‐matrices. First, it is shown that the local C ‐numerical range has rather unusual geometric properties compared to the ordinary one, e.g. it is in general neither star‐shaped nor simply connected. Then two numerical algorithms, a Newton and a conjugate gradient method on the Lie group SU (2) ⊗ · · · ⊗ SU (2), are demonstrated to maximize the real part of Wloc(C, A) which also gives a Euclidean measure of the so‐called pure‐state entanglement in quantum computing. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
arXiv (Cornell University), Jul 25, 2023
For a symmetric Lie algebra g = k ⊕ p we consider a class of bilinear or more general control-aff... more For a symmetric Lie algebra g = k ⊕ p we consider a class of bilinear or more general control-affine systems on p defined by a drift vector field X and control vector fields ad k i for k i ∈ k such that one has fast and full control on the corresponding compact group 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 a ⊆ p, and likewise to related differential inclusions defined on 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.
Bounds on spin dynamics tightened by permutation symmetry application to coherence transfer in I<sub>2</sub>S and I<sub>3</sub>S spin systems
Molecular Physics, Aug 20, 1995
The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfe... more The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfer between pairs of Hermitian or non-Hermitian spin operators is investigated. It is shown that spin permutation symmetry may tighten significantly the quantum mechanical bounds hitherto known to apply for coherence transfer processes. This feature is demonstrated for the heteronuclear coherence transfer processes Sx → Fx, S
Open Systems & Information Dynamics
Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups o... more 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 [Formula: see text]-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajoris...
Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources
arXiv (Cornell University), Mar 3, 2023
arXiv (Cornell University), Dec 1, 2022
We generalize several important results from the perturbation theory of linear operators to the s... more 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.
Molecular dynamics of peptides and proteins investigated by NMR
Proteins, 1991
Physical Review Letters, 2009
A central challenge for implementing quantum computing in the solid state is decoupling the qubit... more A central challenge for implementing quantum computing in the solid state is decoupling the qubits from the intrinsic noise of the material. We investigate the implementation of quantum gates for a paradigmatic, non-Markovian model: A single qubit coupled to a two-level system that is exposed to a heat bath. We systematically search for optimal pulses using a generalization of the novel open systems Gradient Ascent Pulse Engineering (GRAPE) algorithm. We show and explain that next to the known optimal bias point of this model, there are optimal shapes which refocus unwanted terms in the Hamiltonian. We study the limitations of controls set by the decoherence properties. This can lead to a significant improvement of quantum operations in hostile environments.
Nature, 2014
Hybrid quantum systems seek to combine the strength of its constituents to master the fundamental... more Hybrid quantum systems seek to combine the strength of its constituents to master the fundamental conflicting requirements of quantum technology: fast and accurate systems control together with perfect shielding from the environment, including the measurements apparatus, to achieve long quantum coherence. Excellent examples for hybrid quantum systems are heterogeneous spin systems where electron spins are used for readout and control [1-10] while nuclear spins are used as long-lived quantum bits [11-13]. Here we show that joint initialization, projective readout and fast local and non-local gate operations are no longer conflicting requirements in those systems, even under ambient conditions. We demonstrate high-fidelity initialization of a whole spin register (99%) and single-shot readout of multiple individual nuclear spins by using the ancillary electron spin of a nitrogen-vacancy defect in diamond. Implementation of a novel non-local gate generic to our hybrid electron-nuclear quantum register allows to prepare entangled states of three nuclear spins, with fidelities exceeding 85%. An important tool for scalable quantum computation is quantum error correction [14-20]. Combining, for the first time, optimal-control based error avoidance with error correction, we realize a three-qubit phase-flip error correction algorithm. Utilizing optimal control, all of the above algorithms achieve fidelities approaching fault tolerant quantum operation, thus paving the way to large scale integrations. Our techniques can be used to improve scaling of quantum networks relying on diamond spins [8, 10], phosphorous in silicon [12] or other spin systems like quantum dots [21], silicon carbide [4] or rare earth ions in solids [13].
Bounds on spin dynamics tightened by permutation symmetry application to coherence transfer in I2S and I3S spin systems
Molecular Physics, 1995
The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfe... more The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfer between pairs of Hermitian or non-Hermitian spin operators is investigated. It is shown that spin permutation symmetry may tighten significantly the quantum mechanical bounds hitherto known to apply for coherence transfer processes. This feature is demonstrated for the heteronuclear coherence transfer processes Sx → Fx, S
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Papers by Thomas Schulte-Herbrüggen