Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory

We study the +/- J random-plaquette Z_2 gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional +/- J random-bond Ising model (RBIM). The model is a pure Z_2 gauge theory in which randomly chosen plaquettes (occuring with concentration p) have couplings with the ``wrong sign'' so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional ``flux tubes'' that terminate at ``magnetic monopoles.'' Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of magnetic monopoles, or by a combination of the two. Like the RBIM, the RPGM has enhanced symmetry along a ``Nishimori line'' in the p-T plane (where T is the temperature). The critical concentration p_c of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line can be identified with the accuracy threshold for robust storage of quantum information using topological error-correcting codes: if qubit phase errors, qubit bit-flip errors, and errors in the measurement of local check operators all occur at rates below p_c, then encoded quantum information can be protected perfectly from damage in the limit of a large code block. Numerically, we measure p_{c0}, the critical concentration along the T=0 axis (a lower bound on p_c), finding p_{c0}=.0293 +/- .0002. We also measure the critical concentration of antiferromagnetic bonds in the two-dimensional RBIM on the T=0 axis, finding p_{c0}=.1031 +/-.0001. Our value of p_{c0} is incompatible with the value of p_c=.1093 +/-.0002 found in earlier numerical studies of the RBIM, in disagreement with the conjecture that the phase boundary of the RBIM is vertical (parallel to the T axis) below the Nishimori line.Comment: 16 pages, 11 figures, REVTeX, improved numerics and an additional autho

Clustering Boolean Tensors

Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required to be binary and we use Boolean algebra -- much of that hardness comes from the possibility of overlapping components. Yet, in many applications we are perfectly happy to partition at least one of the modes. In this paper we investigate what consequences does this partitioning have on the computational complexity of the Boolean tensor factorizations and present a new algorithm for the resulting clustering problem. This algorithm can alternatively be seen as a particularly regularized clustering algorithm that can handle extremely high-dimensional observations. We analyse our algorithms with the goal of maximizing the similarity and argue that this is more meaningful than minimizing the dissimilarity. As a by-product we obtain a PTAS and an efficient 0.828-approximation algorithm for rank-1 binary factorizations. Our algorithm for Boolean tensor clustering achieves high scalability, high similarity, and good generalization to unseen data with both synthetic and real-world data sets

The biology and behaviour of a free-living population of black rats (Rattus rattus)

A population of wild Rattus rattus living in the roofs of the laboratory buildings was studied by supplying food every evening and watching the behaviour of the animals at the feeding place. Some observations were also made on caged animals. The rats were predominantly of the black rattus variety but white-bellied greys appeared now and then. In breeding tests the grey colour behaved as though determined by a single recessive gene. The study covered two periods of approximately 9 months each, separated by an interval of 3 months during which a reduced quantity of food was provided and the rat population underwent a major decline. During the two periods of richer feeding the population first increased and then stabilized at a level where the animals remained in good condition and there was no starvation. In the first 9-month period, stabilization was achieved by emigration of young adults who colonized neighbouring buildings. Towards the end of the second period, stabilization was achieved by limitation of breeding. The rats accepted a wide variety of foods, including meat, and a number of instances of predation were seen. Small vertebrates as well as insects were killed and eaten. Small pieces of food were usually eaten in situ but large bits were taken up to the nests in the roof. Such differential treatment in relation to size may be a factor of some importance in the evolution of hoarding. The rats visiting the feeding place formed a unit with a definite social structure. A single dominant male and never more than one, was always present and in certain circumstances a linear male hierarchy was formed. There were usually two or three mutually tolerant top ranking females who were subordinate to the top male but dominant to all other members of the group. Within the group attacks were directed downwards in the social scale. An attacked subordinate either fled or appeased and serious fights therefore did not develop. The most essential component of the appease. ment appeared to be a mouth to mouth contact which may be derived from the infantile pattern of 'mouth suckling'. Appeasement permitted superior rats to maintain their status without the necessity of carrying attacks on subordinates to the point where actual hurt was inflicted. A group territory round the feeding place was defended against interlopers. Both sexes took part in chasing out intruders but since males showed inhibition in attacking females, the exclusion of strange females was due principally to the activities of the home females. The point at which pursuit of an intruder stopped was regarded as the territorial boundary. This was also the limit beyond which a group member would not allow himself to be chased but it was not a prison wall. When agonistic tendencies were not aroused the animals no longer always I turned back at the boundary and foraging beyond its limits allowed them to become familiar with an area larger than the territory. Although intruders were normally driven out, it was occasionally possible for a particularly determined animal of either sex to force its way in and ultimately become a member of the group. The patterns of behaviour seen are described, particularly those concerned with hostile encounters and with mating. Scent marking with urine drip trails was not seen but adults of both sexes marked by rubbing the cheeks and ventral surface on branches. The circumstances in which tooth gnashing was heard suggest that this behaviour is not a form of threat but a response to unfamiliar auditory or visual stimuli. There was some evidence that it functioned as an alarm signal within the group. Pilo-erection and a gait or posture with the hind legs much extended ('stegosauring') are considered to function as threats. Pilo-erection occurred in situations where there was little to suggest conflict and is considered to represent a form of threat which has undergone emancipation. Various forms of displacement and ambivalent behaviour were seen. Rapid vibration of the tail occurred in thwarting situations, either during mating or when a defeated opponent suddenly vanished. There was no evidence that it acted as a signal. The common form of amicable behaviour was social grooming. Another amicable action was sitting together with the bodies in contact. Animals reared in cages remained shy and wary and even hand reared young developed the usual alarm responses to movement and noises. Females had their first litters at ages of 3 to 5 months. For first litters gestation periods were 21 to 22 days but in females that were simultaneously lactating they ranged from 23 to 29 days. Eight was the commonest litter number and ten the highest recorded. At birth the tail is very much shorter than the body but has outstripped it by the time the youngster emerges from the nest. This was found to be the result of a period of extremely rapid tail growth immediately preceding emergence. In Rattus norvegicus the peak in tail growth rate was found to be later and less striking. The difference is interpreted as related to the importance of the tail in climbing in the more arboreal R. rattus. During the second week of life an edge response (retreat from a declivity) and a clinging response made their appearance: these have the function of preventing accidental falls from a nest situated above ground level. Mouth suckling was seen only during a period of a few days towards the end of lactation. Play developed within a few days of emergence from the nest: locomotor and fighting play were the common types. Older animals occasionally joined in play with the young. In problem solving tests, first solutions were not insightful but once a solution had been found, the successful technique was at once adopted and subsequently perfected. There was no evidence of learning by imitation but the rats did learn from each other's behaviour that food could be obtained at a certain location and thus the solution of a problem by one rat accelerated its independent solution by others. The reasons for the differences between the behaviour of the free living population and the caged animals studied by other authors are discussed

Clustering {Boolean} Tensors

Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required to be binary and we use Boolean algebra -- much of that hardness comes from the possibility of overlapping components. Yet, in many applications we are perfectly happy to partition at least one of the modes. In this paper we investigate what consequences does this partitioning have on the computational complexity of the Boolean tensor factorizations and present a new algorithm for the resulting clustering problem. This algorithm can alternatively be seen as a particularly regularized clustering algorithm that can handle extremely high-dimensional observations. We analyse our algorithms with the goal of maximizing the similarity and argue that this is more meaningful than minimizing the dissimilarity. As a by-product we obtain a PTAS and an efficient 0.828-approximation algorithm for rank-1 binary factorizations. Our algorithm for Boolean tensor clustering achieves high scalability, high similarity, and good generalization to unseen data with both synthetic and real-world data sets