Implementation of Fault-tolerant Quantum Logic Gates via Optimal Control

The implementation of fault-tolerant quantum gates on encoded logic qubits is considered. It is shown that transversal implementation of logic gates based on simple geometric control ideas is problematic for realistic physical systems suffering from imperfections such as qubit inhomogeneity or uncontrollable interactions between qubits. However, this problem can be overcome by formulating the task as an optimal control problem and designing efficient algorithms to solve it. In particular, we can find solutions that implement all of the elementary logic gates in a fixed amount of time with limited control resources for the five-qubit stabilizer code. Most importantly, logic gates that are extremely difficult to implement using conventional techniques even for ideal systems, such as the T-gate for the five-qubit stabilizer code, do not appear to pose a problem for optimal control.Comment: 18 pages, ioptex, many figure

Optimal Control of One-Qubit Gates

We consider the problem of carrying an initial Bloch vector to a final Bloch vector in a specified amount of time under the action of three control fields (a vector control field). We show that this control problem is solvable and therefore it is possible to optimize the control. We choose the physically motivated criteria of minimum energy spent in the control, minimum magnitude of the rate of change of the control and a combination of both. We find exact analytical solutions.Comment: 5 page

Complete controllability of quantum systems

Sufficient conditions for complete controllability of $N$-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.Comment: 8 pages, expanded and revised versio

Efficient Algorithms for Optimal Control of Quantum Dynamics: The "Krotov'' Method unencumbered

Efficient algorithms for the discovery of optimal control designs for coherent control of quantum processes are of fundamental importance. One important class of algorithms are sequential update algorithms generally attributed to Krotov. Although widely and often successfully used, the associated theory is often involved and leaves many crucial questions unanswered, from the monotonicity and convergence of the algorithm to discretization effects, leading to the introduction of ad-hoc penalty terms and suboptimal update schemes detrimental to the performance of the algorithm. We present a general framework for sequential update algorithms including specific prescriptions for efficient update rules with inexpensive dynamic search length control, taking into account discretization effects and eliminating the need for ad-hoc penalty terms. The latter, while necessary to regularize the problem in the limit of infinite time resolution, i.e., the continuum limit, are shown to be undesirable and unnecessary in the practically relevant case of finite time resolution. Numerical examples show that the ideas underlying many of these results extend even beyond what can be rigorously proved.Comment: 19 pages, many figure

The star formation rate history in the FORS Deep and GOODS South Fields

We measure the star formation rate (SFR) as a function of redshift z up to z \~4.5, based on B, I and (I+B) selected galaxy catalogues from the FORS Deep Field (FDF) and the K-selected catalogue from the GOODS-South field. Distances are computed from spectroscopically calibrated photometric redshifts accurate to (Delta_z / (z_spec+1)) ~0.03 for the FDF and ~0.056 for the GOODS-South field. The SFRs are derived from the luminosities at 1500 Angstroem. We find that the total SFR estimates derived from B, I and I+B catalogues agree very well (\lsim 0.1 dex) while the SFR from the K catalogue is lower by ~0.2 dex. We show that the latter is solely due to the lower star-forming activity of K-selected intermediate and low luminosity (L<L_*) galaxies. The SFR of bright (L>L_*) galaxies is independent of the selection band, i.e. the same for B, I, (I+B), and K-selected galaxy samples. At all redshifts, luminous galaxies (L>L_*) contribute only ~1/3 to the total SFR. There is no evidence for significant cosmic variance between the SFRs in the FDF and GOODs-South field, ~0.1 dex, consistent with theoretical expectations. The SFRs derived here are in excellent agreement with previous measurements provided we assume the same faint-end slope of the luminosity function as previous works (alpha ~ -1.6). However, our deep FDF data indicate a shallower slope of alpha=-1.07, implying a SFR lower by ~0.3 dex. We find the SFR to be roughly constant up to z ~4 and then to decline slowly beyond, if dust extinctions are assumed to be constant with redshift.Comment: 6 pages, 2 figures, Accepted for publication in ApJ

Entanglement Dissipation: Unitary and Non-unitary Processes

Dissipative processes in physics are usually associated with non-unitary actions. However, the important resource of entanglement is not invariant under general unitary transformations, and is thus susceptible to unitary "dissipation". In this note we discuss both unitary and non-unitary dissipative processes, showing that the former is ultimately of value, since reversible, and enables the production of entanglement; while even in the presence of the latter, more conventional non-unitary and non-reversible, process there exist nonetheless invariant entangled states.Comment: 9 pages, 2 figures, Symmetries in Science XV (Bregenz Symposiun 2011

Constraints on relaxation rates for N-level quantum systems

We study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open N-level systems. The Lindblad operators that govern the evolution of the system are expressed in terms of observable relaxation rates, explicit formulas for the decoherence rates due to population relaxation are derived, and it is shown that there are additional, non-trivial constraints on the pure dephasing rates for N>2. Explicit experimentally testable inequality constraints for the decoherence rates are derived for three and four-level systems, and the implications of the results are discussed for generic ladder-, Lambda- and V-systems, and transitions between degenerate energy levels.Comment: 10 pages, RevTeX, 4 figures (eps/pdf

Review of biorthogonal coupled cluster representations for electronic excitation

Single reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical (PT) order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison with a full (and hermitian) ISR method

Physics-based mathematical models for quantum devices via experimental system identification

We consider the task of intrinsic control system identification for quantum devices. The problem of experimental determination of subspace confinement is considered, and simple general strategies for full Hamiltonian identification and decoherence characterization of a controlled two-level system are presented.Comment: 15 pages, 8 figures, published in proceedings of workshop on Physics-based mathematical models of low-dimensional semi-conductor nanostructures (18-23 November, 2007, Banff International Research Station, Alberta, Canada

Physics-based mathematical models for quantum devices via experimental system identification

We consider the task of intrinsic control system identification for quantum devices. The problem of experimental determination of subspace confinement is considered, and simple general strategies for full Hamiltonian identification and decoherence characterization of a controlled two-level system are presented.Comment: 15 pages, 8 figures, published in proceedings of workshop on Physics-based mathematical models of low-dimensional semi-conductor nanostructures (18-23 November, 2007, Banff International Research Station, Alberta, Canada