arXiv subjects

T. Schulte-Herbrueggen

Publications and source records attributed to T. Schulte-Herbrueggen.

24 records · Page 2Linked to original sources

arXiv2008

Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups

Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squares distance $$\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}.$$ Connections of the results to different pure and applied areas are discussed.

math.NA↗

arXiv2008

Gradient Flows for Optimisation and Quantum Control: Foundations and Applications

For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such as the full unitary group to closed subgroups like partitionings by factorisation into tensor products, where the finest partitioning consists of purely local unitary operations. Moreover, the common framework is kept sufficiently general and allows for setting up gradient flows on (sub-)manifolds, Lie (sub-)groups, quotient groups, and reductive homogeneous spaces. Relevant convergence conditions are discussed meant to serve as justification for recent and new achievements, and as foundation for further research. Exploiting the differential geometry of quantum dynamics under different scenarios helps to provide highly useful algorithms: (a) On an abstract level, gradient flows may establish the exact upper bounds of pertinent quality functions, i.e. upper bounds reachable within the underlying manifold of the state space dynamics; (b) in a second stage referring to a concrete experimental setting, gradient flows on the manifold of piecewise constant control amplitudes $\R^m$ may be set up to provide (approximations to) optimal control of quantum devices under realistic conditions. Illustrative examples and new applications are given relating to distance measures of pure-state entanglement. We establish the correspondence to best rank-1 approximations of higher-order tensors.

quant-ph↗

arXiv2007

Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.

math-ph↗

arXiv2006

Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks

Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multi-qubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins-$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{-1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\in\mathbf{SU}(2)^{\otimes n}$ comprises two classes: in type-I, $K_1$ and $K_2$ are inverse to one another, while in type-II they are not. {Type-I} consists of one-parameter groups that can jointly be inverted for all times $t\in\R{}$ because their Hamiltonian generators satisfy $K H K^{-1} = \Ad K (H) = -H$. As all the Hamiltonians generating locally invertible unitaries of type-I are spanned by the eigenspace associated to the eigenvalue -1 of the {\em local} conjugation map $\Ad K$, this eigenspace can be given in closed algebraic form. The relation to the root space decomposition of $\mathfrak{sl}(N,\C{})$ is pointed out. Special cases of type-I invertible Hamiltonians are of $p$-quantum order and are analysed by the transformation properties of spherical tensors of order $p$. Effective multi-qubit interaction Hamiltonians are characterised via the graphs of their coupling topology. {Type-II} consists of pointwise locally invertible propagators, part of which can be classified according to the symmetries of their matrix representations. Moreover, we show gradient flows for numerically solving the decision problem whether a propagator is type-I or type-II invertible or not by driving the least-squares distance $\norm{K_1 e^{-itH} K_2 - e^{+itH}}^2_2$ to zero.

quant-ph↗

arXiv2005

Optimal Control of Coupled Josephson Qubits

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 [T. Yamamoto et al., Nature 425 (2003), 941] 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 Lie-algebraically closed.

quant-ph↗

arXiv2005

Optimal Control-Based Efficient Synthesis of Building Blocks of Quantum Algorithms Seen in Perspective from Network Complexity towards Time Complexity

In this paper, we demonstrate that optimal control algorithms can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristicratio of the time scales for local controls {\em vs} non-local ({\em i.e.} coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as {\em e.g.} in NMR), time-optimised realisations are shown for the quantum Fourier transform (QFT) and the multiply controlled {\sc not}-gate ({\sc c$^{n-1}$not}) in various coupling topologies of $n$ qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and {\sc swap} gates, while the {\sc c$^{n-1}$not}-gate for completely coupled network of six qubits is nearly seven times faster.

quant-ph↗