Unsupervised approaches to learning in neural networks are of substantial interest for furthering artificial intelligence, both because they would enable the training of networks without the need for large numbers of expensive annotations, and because they would be better models of the kind of general-purpose learning deployed by humans. However, unsupervised networks have long lagged behind the performance of their supervised counterparts, especially in the domain of large-scale visual recognition. Recent developments in training deep convolutional embeddings to maximize non-parametric instance separation and clustering objectives have shown promise in closing this gap. Here, we describe a method that trains an embedding function to maximize a metric of local aggregation, causing similar data instances to move together in the embedding space, while allowing dissimilar instances to separate. This aggregation metric is dynamic, allowing soft clusters of different scales to emerge. We evaluate our procedure on several large-scale visual recognition datasets, achieving state-of-the-art unsupervised transfer learning performance on object recognition in ImageNet, scene recognition in Places 205, and object detection in PASCAL VOC.

We propose and analyze a new approach for quantum state transfer between remote spin qubits. Specifically, we demonstrate that coherent quantum coupling between remote qubits can be achieved via certain classes of random, unpolarized (infinite temperature) spin chains. Our method is robust to coupling strength disorder and does not require manipulation or control over individual spins. In principle, it can be used to attain perfect state transfer over arbitrarily long range via purely Hamiltonian evolution and may be particularly applicable in a solid-state quantum information processor. As an example, we demonstrate that it can be used to attain strong coherent coupling between Nitrogen-Vacancy centers separated by micrometer distances at room temperature. Realistic imperfections and decoherence effects are analyzed.PACS numbers: 03.67. Lx, 03.67.Hk, 05.50.+q, 75.10.Dg In addition to diverse applications ranging from quantum key distribution to quantum teleportation [1, 2], reliable quantum state transfer between distant qubits forms an essential ingredient of any scalable quantum information processor [3]. However, most direct qubit interactions are short-range and the corresponding interaction strength decays rapidly with physical separation. For this reason, most of the feasible approaches that have been proposed for quantum computation rely upon the use of quantum channels which serve to connect remote qubits; such channels include: electrons in semiconductors [4], optical photons [5][6][7][8], and the physical transport of trapped ions [9]. Coupled quantum spin chains have also been extensively studied [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. A key advantage of such spin chain quantum channels is the ability to manipulate, transfer, and process quantum information utilizing the same fundamental hardware [25]; indeed, both quantum memory and quantum state transfer can be achieved in coupled spin chain arrays [26], eliminating the requirement for an external interface between the quantum channel and the quantum register. Prior work on spin chain quantum channels has focused on three distinct regimes, in which the spin chain is either initialized [10][11][12][13]24], engineered [15,27,29] or dynamically controlled [19,28,[30][31][32]].An important application of spin-chain mediated coherent coupling is in the context of realizing a room temperature quantum information processor based upon localized spins in the solid-state [33]. In this case, it is difficult to envision mechanical qubit transport, while other coupling mechanisms are often not available or impose additional prohibitive requirements such as cryogenic cooling [8]. At the same time, long spin chains are generally difficult to polarize, impossible to control with single-spin resolution, and suffer from imperfect spin-positioning [21,22]; such imperfections can cause both on-site and coupling disorder, resulting in localization [34]. For these reasons, a detailed understanding of quantum coherence and state transfer in rando...

The insertion-deletion channel takes as input a bit string \mathbf x \in \{0,1\}^{n} , and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover \mathbf x from many independent outputs (called "traces") of the insertion-deletion channel applied to \mathbf x . We show that if \mathbf x is chosen uniformly at random, then (O(\mathrm {log}^{1/3} n)) traces suffice to reconstruct \mathbf x with high probability. For the deletion channel with deletion probability q < 1/2 the earlier upper bound was exp (O(\mathrm {log}^{1/2}n)) . The case of q \geq 1/2 or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., exp (O(n^{1/3})) . We also show that our reconstruction algorithm runs in n^{1+o(1)} time. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of \mathbf x . The alignment is done by viewing the strings as random walks and comparing the increments in the walk associated with the input string and the trace, respectively.

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n g is the number of numerical semigroups of genus g, we prove thatwhere S is a constant, resolving several related conjectures concerning the growth of n g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f < 3m approaches 1 as g → ∞, where m is the multiplicity and f is the Frobenius number.

The deletion channel takes as input a bit string x ∈ {0, 1} n , and deletes each bit independently with probability q, yielding a shorter string. The trace reconstruction problem is to recover an unknown string x from many independent outputs (called "traces") of the deletion channel applied to x.We show that if x is drawn uniformly at random and q < 1/2, then e O(log 1/2 n) traces suffice to reconstruct x with high probability. The previous best bound, established in 2008 by Holenstein, Mitzenmacher, Panigrahy, and Wieder [5], uses n O(1) traces and only applies for q less than a smaller threshold (it seems that q < 0.07 is needed). Our algorithm combines several ideas: 1) an alignment scheme for "greedily" fitting the output of the deletion channel as a subsequence of the input; 2) a version of the idea of "anchoring" used in [5]; and 3) complex analysis techniques from recent work of Nazarov and Peres [9]