Automated Synthesis of Dynamically Corrected Quantum Gates

We address the problem of constructing dynamically corrected gates for non-Markovian open quantum systems in settings where limitations on the available control inputs and/or the presence of control noise make existing analytical approaches unfeasible. By focusing on the important case of singlet-triplet electron spin qubits, we show how ideas from optimal control theory may be used to automate the synthesis of dynamically corrected gates that simultaneously minimize the system's sensitivity against both decoherence and control errors. Explicit sequences for effecting robust single-qubit rotations subject to realistic timing and pulse-shaping constraints are provided, which can deliver substantially improved gate fidelity for state-of-the-art experimental capabilities.Comment: 5 pages; further restructure and expansio

Christoffel words and the Calkin-Wilf tree

In this note we present some results on the Calkin-Wilf tree of irreducible fractions, giving an insight on the duality relating it to the Stern-Brocot tree, and proving noncommutative versions of known results relating labels of the Calkin-Wilf trees to hyperbinary expansions of positive integers. The main tool is the Christoffel tree introduced in a paper by Berstel and de Luca

Homomorphisms between Solomon's descent algebras

In a previous paper (see A. Garsia and C. Reutenauer (Adv. in Math. 77, 1989, 189–262)), we have studied algebraic properties of the descent algebras Σn, and shown how these are related to the canonical decomposition of the free Lie algebra corresponding to a version of the Poincaré-Birkhoff-Witt theorem. In the present paper, we study homomorphisms between these algebras Σn. The existence of these homomorphisms was suggested by properties of some directed graphs that we constructed in the previous paper (reference above) describing the structure of the descent algebras. More precisely, examination of the graphs suggested the existence of homomorphisms Σn→Σn−s and Σn→Σn+s. We were then able to construct, for any s (0<s<n), a surjective homomorphism Δs: Σn→Σn−s and an embedding Γs:Σn−s→Σn, which reflects these observations. The homomorphisms Δs may also be defined as derivations of the free associative algebra Q〈t1,t2,…> which sends ti on ti−s, if one identifies the basis element D⊆S of Σn with some word (coding S) on the alphabet T={t1, t2,…}. We show that this mapping is indeed a homomorphism, using the combinatorial description of the multiplication table of Σn given in the previous paper (reference above)

Algebraic structure of stochastic expansions and efficient simulation

We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein we: show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.Comment: 19 page

Sofic-Dyck shifts

We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift

Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra

We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(xi), are quadratic equations S^i G = G xi^i G in concatenation of correlations. The Schwinger-Dyson operator S^i is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger-Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.Comment: 13 pages, added discussion & references, title changed, minor corrections, published versio

The finite index basis property

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on

Quantifying molecular oxygen isotope variations during a Heinrich stadial

International audienceδ 18 O of atmospheric oxygen (δ 18 O atm) undergoes millennial-scale variations during the last glacial period, and systematically increases during Heinrich stadials (HSs). Changes in δ 18 O atm combine variations in biospheric and water cycle processes. The identification of the main driver of the millennial variability in δ 18 O atm is thus not straightforward. Here, we quantify the response of δ 18 O atm to such millennial events using a freshwater hosing simulation performed under glacial boundary conditions. Our global approach takes into account the latest estimates of isotope frac-tionation factor for respiratory and photosynthetic processes and make use of atmospheric water isotope and vegetation changes. Our modeling approach allows to reproduce the main observed features of a HS in terms of climatic conditions , vegetation distribution and δ 18 O of precipitation. We use it to decipher the relative importance of the different processes behind the observed changes in δ 18 O atm. The results highlight the dominant role of hydrology on δ 18 O atm and confirm that δ 18 O atm can be seen as a global integrator of hydrological changes over vegetated areas