Optimal local unitary encoding circuits for the surface code

The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

Family of mappings with an equicontractive-type condition

In a real Banach space X and a complete metric space M, we consider a compact mapping C defined on a closed and bounded subset A of X with values in M and the operator $$T:A\times C(A) \rightarrow X$$ T : A × C ( A ) → X . Using a new type of equicontractive condition for a certain family of mappings and $$\beta $$ β -condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator $$x\mapsto T(x,C(x))$$ x ↦ T ( x , C ( x ) ) has a fixed point. The obtained results are applied to the initial value problem.

Thrombelastography and Thrombelastometry

The term thrombelastography / thrombelastometry was used to describe the trace produced from measurement of the viscoelastic changes associated with fibrin polymerization. The result of measurement is a compact mapping of the various stages of haemostasis. One of the first real clinical applications of this method was the haemostatic monitoring of liver transplantation and cardiac surgery using extracorporeal circulation. In trauma patients the thrombelastography /thrombelastometry was proved to predict early transfusion requirements. Another authors suggest thrombelastography /thrombelastometry as a possible tool for early identification of pregnant women at increased risk of fetal loss. This article provides overview on the development of thrombelastography / trombelastometry and its possible use in laboratory of haemostasis.

On characterizations of fixed points

We give some necessary and sufficient conditions for the existence of fixed points of a family of self mappings of a metric space and we establish an equivalent condition for the existence of fixed points of a continuous compact mapping of a metric space.

On a local degree for a class of multi-valued vector fields in infinite dimensional Banach spaces

This paper is devoted to the development of a local degree for multi-valued vector fields of the formf−F. Here,fis a single-valued, proper, nonlinear, Fredholm,C1-mapping of index zero andFis a multi-valued upper semicontinuous, admissible, compact mapping with compact images. The mappingsfandFare acting from a subset of a Banach spaceEinto another Banach spaceE1. This local degree is used to investigate the existence of solutions of a certain class of operator inclusions.