On a connection between the switching separability of a graph and that of its subgraphs

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: if removing any one or two vertices of a graph always results in a switching separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order greater than 4, there is a graph that is not switching separable, but removing any vertex always results in a switching separable subgraph. We show a connection with similar facts on the separability of Boolean functions and reducibility of $n$-ary quasigroups. Keywords: two-graph, reducibility, separability, graph switching, Seidel switching, graph connectivity, $n$-ary quasigroupComment: english: 9 pages; russian: 9 page

n-Ary quasigroups of order 4

We characterize the set of all N-ary quasigroups of order 4: every N-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an N-ary quasigroup can be represented as a composition of K-ary and (N-K+1)-ary quasigroups for some K from 2 to N-1, where the order of arguments in the representation can differ from the original order. The set of semilinear N-ary quasigroups has a characterization in terms of Boolean functions. Keywords: Latin hypercube, n-ary quasigroup, reducibilityComment: 10pp. V2: revise

Asymptotics for the number of n-quasigroups of order 4

The asymptotic form of the number of n-quasigroups of order 4 is $3^{n+1} 2^{2^n +1} (1+o(1))$. Keywords: n-quasigroups, MDS codes, decomposability, reducibility.Comment: 15 p., 3 fi

On the volumes and affine types of trades

A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\le i\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami--Tokura of such functions with small number of ones, it is known that any simple $[t]$-trade of volume at most $2\cdot2^t$ belongs to one of two affine types, called Type\,(A) and Type\,(B) where Type\,(A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type\,(B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\cdot 2^t$, extending the known result for volumes less than $2\cdot 2^t$. We also give a characterization of "small" $[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified $t$, $v$ is given. The result of the implementation of the algorithm for $t\le4$, $v\le7$ is reported.Comment: 30 pages, final version, to appear in Electron. J. Combi

On the structure of non-full-rank perfect codes

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.Comment: 8 page

Photon storage in Lambda-type optically dense atomic media. IV. Optimal control using gradient ascent

We use the numerical gradient ascent method from optimal control theory to extend efficient photon storage in Lambda-type media to previously inaccessible regimes and to provide simple intuitive explanations for our optimization techniques. In particular, by using gradient ascent to shape classical control pulses used to mediate photon storage, we open up the possibility of high efficiency photon storage in the non-adiabatic limit, in which analytical solutions to the equations of motion do not exist. This control shaping technique enables an order-of-magnitude increase in the bandwidth of the memory. We also demonstrate that the often discussed connection between time reversal and optimality in photon storage follows naturally from gradient ascent. Finally, we discuss the optimization of controlled reversible inhomogeneous broadening.Comment: 16 pages, 7 figures. V2: As published in Phys. Rev. A. Moved most of the math to appendices or removed altogether. Switched order of Sections II and III. Shortened abstract. Added reference

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