Current approaches to fault-tolerant quantum computation will not enable useful quantum computation on near term devices of 50 to 100 qubits. Leading proposals, such as the color code and surface code schemes, must devote a large fraction of their physical quantum bits to quantum error correction. Building from recent quantum machine learning techniques, we propose an alternative approach to quantum error correction aimed at reducing this overhead, which can be implemented in existing quantum hardware and on a myriad of quantum computing architectures. This method aims to optimize the average fidelity of encoding and recovery circuits with respect to the actual noise in the device, as opposed to that of an artificial or approximate noise model. The quantum variational error corrector (QVECTOR) algorithm employs a quantum circuit with parameters that are variationally-optimized according to processed data originating from quantum sampling of the device, so as to learn encoding and error-recovery gate sequences. We develop this approach for the task of preserving quantum memory and analyze its performance with simulations. We find that, subject to phase damping noise, the simulated QVECTOR algorithm learns a three-qubit encoding and recovery which extend the effective T 2 of a quantum memory six-fold. Subject to a continuous-time amplitude-plus phase-damping noise model on five qubits, the simulated QVECTOR algorithm learns encoding and decoding circuits which exploit the coherence among Pauli errors in the noise model to outperform the five-qubit stabilizer code and any other scheme that does not leverage such coherence. Both of these schemes can be implemented with existing hardware.

For positive integers m, n, the greatest number of colors that can appear in an edge coloring of K m,n which avoids rainbow cycles is m + n − 1. Here these colorings are constructively characterized; it turns out that these colorings can be encoded by certain vertex labelings of full binary trees with m + n leafs.

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Philip Hall's famous theorem on systems of distinct representatives and its not‐so‐famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list‐colorings or list‐multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall‐Halmos‐Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut‐vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut‐vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000