Given an m−qubit Φ0 and an (n, m)−quantum code C, let Φ be the n−qubit that results from the C−encoding of Φ0. Suppose that the state Φ is affected by an isotropic error (decoherence), becoming Ψ, and that the corrector circuit of C is applied to Ψ, obtaining the quantum state Φ. Alternatively, we analyze the effect of the isotropic error without using the quantum code C. In this case the error transforms Φ0 into Ψ0. Assuming that the correction circuit does not introduce new errors and that it does not increase the execution time, we compare the fidelity of Ψ, Φ and Ψ0 with the aim of analyzing the power of quantum codes to control isotropic errors. We prove that F (Ψ0) ≥ F ( Φ) ≥ F (Ψ). Therefore the best option to optimize fidelity against isotropic errors is not to use quantum codes.

In this article, we focus on structural properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus (undirected) trees. For some of these properties, we give the matrix version, regarding nearly reducible matrices. We give bounds for the coefficients of the characteristic polynomial corresponding to the adjacency matrix of trees, and we conjecture bounds for MSDs. We also propose two different representations of an MSD in terms of trees (the union of a spanning tree and a directed forest; and a double directed tree whose vertices are given by the contraction of connected Hasse diagrams).

The sum of quantum computing errors is the key element both for the estimation and control of errors in quantum computing and for its statistical study. In this article we analyze the sum of two independent quantum computing errors, X_1 and X_2, and we obtain the formula of the variance of the sum of these errors: V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}. We conjecture that this result holds true for general quantum computing errors and we prove the formula for independent isotropic quantum computing errors.

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This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after suppressing the ares of the cycle. By this process and by the contraction of all SCs into single vértices we obtain a Hasse diagram from the MSD. Among other properties, we show that any SC conformed by more than one vértex (non trivial SC) has at least one linear vértex (a vértex with indegree and outdegree equal to 1) in the MSD (Theorem 1); that in the Hasse diagram at least one linear vértex exists for each non trivial maximal (resp. minimal) vértex (Theorem 2); that if an SC contains a number X of vértices of the cycle then it contains at least X linear vértices in the MSD (Theorem 3); and, finally, that given a cycle of length q contained in the MSD, the number a of linear vértices contained in the MSD satisfies a > | _(q + 1)/2J (Theorem4).