Given an arbitrary single-qubit operation, an important task is to efficiently decompose this operation into an (exact or approximate) sequence of fault-tolerant quantum operations. We derive a depth-optimal canonical form for single-qubit quantum circuits, and the corresponding rules for exactly reducing an arbitrary single-qubit circuit to this canonical form. We focus on the singlequbit univ...

A global robust output feedback dynamic compensator is proposed for stabilization and tracking of a class of systems that are globally di eomorphic into systems which are in generalized output-feedback canonical form. This form includes as special cases the standard outputfeedback canonical form and various other forms considered previously in the literature. Output-dependent nonlinearities are...

Determining the optimal implementation of a quantum gate is critical for designing a quantum computer. We consider the crucial task of efficiently decomposing a general single-qubit quantum gate into a sequence of fault-tolerant quantum operations. For a given single-qubit circuit, we construct an optimal gate sequence consisting of fault-tolerant Hadamard (H) and π/8 rotations (T ). Our scheme...

The notion of linear finite transducer (LFT) plays a crucial role in a family of cryptosystems introduced in the 80’s and 90’s. However, as far as we know, no study was ever conducted to count and enumerate these transducers, which is essential to verify if the size of the key space, of the aforementioned systems, is large enough to prevent an exhaustive search attack. In this paper, we determi...

The elliptic norm curve Qn in space S„_i admits a group G2ni of collineations, and in fact there is a single infinity of such curves which admit the same group. A particular Qn of the family is distinguished by a value of the parameter t , itself an elliptic modular function defined by the modular group congruent to identity (mod. n). In the group G2n2, there are certain involutory collineation...

Necessary and sufficient geometric conditions for the equivalence of a nonlinear time– delay system with one output, under bicausal change of coordinates and output transformation, to a linear weakly observable time–delay system up to output injection are given. These conditions are derived through the use of the Extended Lie Bracket operator recently introduced in the literature for dealing wi...

In this article, we design fast algorithms for the computation of approximant bases in shifted Popov normal form. We first recall the algorithm known as PM-Basis, which will be our second fundamental engine after polynomial matrix multiplication: most other fast approximant basis algorithms basically aim at efficiently reducing the input instance to instances for which PMBasis is fast. Such red...

Abstract: This paper shows that a large class of chaotic systems, introduced in (Čelikovský and Vaněček, 1994), (Vaněček and Čelikovský, 1996) as the generalized Lorenz system, can be further generalized to the hyperbolic-type generalized Lorenz system. While the generalized Lorenz system unifies both the famous Lorenz system and new Chen’s system (Ueta and Chen, 1999), (Chen and Ueta, 2000), t...

In this paper we present aane transformations as an extension of the Signal language for the speciication and validation of real-time applications. A Signal program is a system of equations which specify dependencies between program data and synchronization constraints on clock variables. In order to test if a program is functionally safe, the Signal compiler resolves the clock constraints and ...

We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions {Ci(P ) : P ∈ Q[x], i = 0, . . . , deg(P )} defined by Ci(P )(x) = { 0 if x ≤ α P (x) if x ≥ α where α is the i-th real root of the polynomial P . These functions will allow us to represent and manipul...