Computational model underlying the one-way quantum computer In this paper we present the computational model underlying the one-way quantum computer which we introduced recently [Phys. Rev. Lett. 86, 5188 (2001)]. The one-way quantum computer has the property that any quantum logic network can be simulated on it. Conversely, not all ways of quantum information processing that are possible with the one-way quantum computer can be understood properly in network model terms. We show that the logical depth is, for certain algorithms, lower than has so far been known for networks. For example, every quantum circuit in the Clifford group can be performed on the one-way quantum computer in a single step. Quantum computation has been formulated within different frameworks such as the quantum Turing machine [1] or the quantum logic network model [2] . In the latter it is particularly easy to establish a connection between physics and the processing of quantum information, since the building blocks of the quantum logic network -the quantum gates-are unitary transformations generated by suitably tailored Hamiltonians. More recently, the capability of projective (von Neumann) measurements to drive a quantum computation has been investigated [3] - [9] .. In [10] we have shown that universal quantum computation can be entirely built on one-qubit measurements on a certain class of highly entangled multi-qubit states, the cluster states [11] . In this scheme, a cluster state forms a resource for quantum computation and the set of measurements form the program. This scheme we called the "one-way quantum computer" since the entanglement in a cluster state is destroyed by the one-qubit measurements and therefore the cluster state can be used only once. To stress the importance of the cluster state for the scheme, we will use in the following the abbreviation QC C for "one-way quantum computer".. If a quantum logic network is simulated on a QC C , for many algorithms the number of computational steps scales more favorably with the input size than the number of steps in the original network does. To be specific, circuits which realize transformations in the Clifford group -which is generated by all the CNOT-gates, Hadamard-gates and π/2-phase shifts-can be performed by a QC C in a single time step, i.e. all the measurements to implement such a circuit can be carried out at the same time. Generally, in a simulation of a quantum logic network by a one-way quantum computer, the temporal ordering of the gates of the network is transformed into a spatial pattern of measurement bases for the individual qubits on the resource cluster state. For the temporal ordering of the measurements there is no counterpart in the network model. Therefore, the question of complexity of a quantum computation must be possibly revisited.. It should be understood that both the quantum logic network computer and the QC C can simulate each other efficiently. The fact that each quantum logic network can be simulated on the QC C has 2 Network picture of the QC C Before starting the description of the objects relevant for the computational model we would like to give a short summary of the one-way quantum computer. In [10] it was proved that the QC C is universal by showing that it can simulate any quantum logic network.. For the one-way quantum computer, the entire resource for the quantum computation is provided initially in the form of a specific entangled state -the cluster state [11] -of a large number of qubits. Information is then written onto the cluster, processed, and read out from the cluster by one-particle measurements only. The entangled state of the cluster thereby serves as a universal "substrate" for any quantum computation. Cluster states can be created efficiently in any system with a quantum Ising-type interaction (at very low temperatures) between two-state particles in a lattice configuration. More specifically, to create a cluster state |Φ C the qubits on a cluster C are at first all prepared in an individual state |+ = 1/ √ 2(|0 +|1 ) and then brought into a cluster state by switching on the Ising-type interaction H int for an appropriately chosen finite time span T . The time evolution operator generated by the Ising-type Hamiltonian which takes the initial product state to the cluster state is denoted by S. The unitary transformation S on a two-dimensional array of qubits, as used in this paper, has the form. = k∈C e −iπ/2 σ and u(k) the site of the upper neighbor of k, i.e. the site next following k in the y-direction. The interaction part of S is generated by an Ising Hamiltonian. In (1) there appear additional posterior local unitary transformations which have no influence on the entanglement properties of the states generated under S.. The quantum state |Φ C , the cluster state of a cluster C of neighbouring qubits provides in advance all entanglement that is involved in the subsequent quantum computation. It has been shown [11] that the cluster state |Φ C is characterized by a set of eigenvalue equations. where ngbh(a) specifies the sites of all qubits that interact with the qubit at site a ∈ C. The eigenvalues are specified by the distribution of the qubits on the lattice and encoded in κ a ∈ {0, 1}, a = 1, .., C . They can be altered e.g. by applying phase-flips σ z before or after the Ising interaction. For the special case of C = Z 2 , that is the case of an infinitely extended cluster, κ k = 0 ∀k ∈ C. The equations (2) are central for the described computation scheme. It is important to realize that information processing is possible even though the result of every individual measurement in any direction of the Bloch sphere is completely random. The reason for the randomness of the measurement results is that the reduced density operator for each qubit in the cluster state is 1 2 1. While the individual measurement results are irrelevant for the computation, the strict correlations between measurement results inferred from (2) are what makes the processing of quantum information on the QC C possible.. Let us for clarity emphasize that in the scheme of the QC C we distinguish between physical cluster qubits in C which are measured in the process of computation, and the logical qubits. The logical qubits constitute the quantum information being processed while the cluster qubits in the initial cluster state form an entanglement resource. Measurements of their individual one-qubit state drive the computation.. To process quantum information with this cluster, it suffices to measure its particles in a certain order and in a certain basis. Quantum information is thereby propagated through the cluster and processed. Measurements of σ z -observables effectively remove the respective lattice qubits from the cluster. Measurements of σ x are used for "wires" i.e. to propagate logical quantum bits through the cluster, and for the CNOT-gate between two logical qubits. Observables of the form cos(ϕ) σ x + sin(ϕ) σ y are measured to realize arbitrary rotations of logical qubits. Here, the angle ϕ specifies the measurement direction. For the one-qubit rotations, the basis in which a certain qubit is measured depends on the results of preceding measurements. This introduces a temporal ordering in which the measurements have to be performed. The processing is finished once all qubits except a last one on each wire have been measured. The remaining unmeasured qubits form the quantum register which is now ready for the readout. At this point, the results of previous measurements determine in which basis these "output" qubits need to be measured for the final readout, or if the readout measurements are in the σ x -, σ yor σ z -eigenbasis, how the readout measurements have to be interpreted. Without loss of generality, we assume in this paper that the readout measurements are performed in the σ z -eigenbasis.. For illustration and later reference we review two points of the universality proof for the QC C . First, the realization of the arbitrary one-qubit rotation and of the CNOT-gate as the elements of the universal set of gates. And second, the effect of the randomness of the individual measurement results and how to account for them.. An arbitrary rotation U Rot ∈ SU (2) can be achieved in a chain of 5 qubits. Consider a rotation in its Euler representation U Rot (ξ, η, ζ) = U x (ζ)U z (η)U x (ξ),. where the rotations about the x-and z-axis are. Initially, the first qubit is in some state |ψ in , which is to be rotated, and the other qubits are in |+ . After the 5 qubits are entangled by the time evolution operator S generated by the Ising-type Hamiltonian, the state |ψ in can be rotated by measuring qubits 1 to 4. At the same time, the state is also swapped to site 5. The qubits 1 . . . 4 are measured in appropriately chosen bases, viz.. B j (ϕ j,meas ) = |0 j + e iϕ j,meas |1 j √ 2 , |0 j − e iϕ j,meas |1 j √ 2. whereby the measurement outcomes s j ∈ {0, 1} for j = 1 . . . 4 are obtained. Here, s j = 0 means that qubit j is projected into the first state of B j (ϕ j,meas ). In (5) the basis states of all possible measurement bases lie on the equator of the Bloch sphere, i.e. on the intersection of the Bloch sphere with the x-y-plane. Therefore, the measurement basis for qubit j can be specified by a single parameter, the measurement angle ϕ j,meas . The measurement direction of qubit j is the vector on the Bloch sphere which corresponds to the first state in the measurement basis B j (ϕ j,meas ). Thus, the measurement angle ϕ j,meas is equal to the angle between the measurement direction at qubit j and the positive x-axis. For all of the so far constructed gates, the cluster qubits are either -if they are not required for the realization of the circuitmeasured in σ z , or -if they are required-measured in some measurement direction in the x-y-plane. In summary, the procedure to implement an arbitrary rotation U R (ξ, η, ζ), specified by its Euler angles ξ, η, ζ, is this:. 3. measure qubit 3 in. If the 5-qubit cluster state in Fig. 1 is created from a product state of all qubits in |+ via the interaction S of eq. (1), we have the specific values κ. Please note that in (6) we used the set κ. ., 5 to specify a cluster state |φ 5 on a chain of 5 qubits. We will use primed κ, {κ ′ a } whenever we describe a cluster state |φ CN on some cluster C N which consists only of those cluster qubits that are necessary for the implementation of some circuit. Further, the κ ′ associated with the input qubit 1 we denote by κ 5 . The reason for this notation will be discussed in Section 4.1. The unprimed κ, {κ a }, are reserved for the description of the cluster state |φ C on the whole cluster C.. Via the procedure (6) the rotation U ′ Rot is realized:. Therein, the random byproduct operator has the form. It can be corrected for at the end of the computation, as explained below. A CNOT-gate can be implemented on a cluster state of 15 qubits, as shown in Fig. 1 . All measurements can be performed simultaneously. Depending on the measurement results, the following gate is thereby realized:. Therein the byproduct operator U Σ,CN OT has the form these extra rotations can be propagated through the network to act upon the output state. By this we mean that the extra random rotations need not be corrected for after a gate. Instead, one just needs to keep track of them and delay correction until the end of the computation. Then, the extra rotations can be accounted for by properly interpreting the σ z -readout measurement results. The propagation relations (11) for the arbitrary rotation and (12) for the CNOT-gate differ with respect to which of the two unitary transformations -the gate or the byproduct operator U Σ -is modified on the right hand side of (11) and (12) . In the case of the propagation of a byproduct operator through a rotation (11), the gate is changed and the byproduct operator remains unchanged. It passes just through. Conversely, in the case of the propagation of a byproduct operator U Σ through a CNOT-gate (12) , it is the byproduct operator which is modified and the gate remains unchanged.. The measurement bases B(ϕ) and B(−ϕ) in (5) coincide for angles ϕ = 0 and for ϕ = ±π/2. For ϕ = 0 the measurement basis B(ϕ) is the eigenbasis of σ x , and for ϕ = ±π/2 the measurement basis B(ϕ) is the eigenbasis of σ y . In these cases, the choice of the measurement basis is not influenced by the results of measurements at other qubits. Therefore, rotations whose Euler angles ξ, η, ζ are in the set {0, ±π/2} can be realized simultaneously in the first round of measurements, that is no other cluster qubits need to be measured before. Among these rotations are the Hadamard gate and the π/2 phase shift. As displayed in Fig. 1 , the Hadamard gate and the π/2-phase shift are both realized by performing a pattern of σ x -and σ y -measurements on the cluster C. The byproduct operators which are thereby created are. Owing to the special Euler angles for the Hadamard-and the π/2-phase gate, the propagation relations for these rotations can also be written in a form resembling the propagation relation (12) for the CNOTgate. (a). Hadamard-gate π/2-phase gate Under propagation -via the propagation relations (11), (12) and (14)-the byproduct operators resulting from the implementation of the universal gates generate a subgroup of the group. z , i = 1..n, and products thereof (15) of all possible byproduct operators. U local ⊂ SU (2) ⊗n is a subset of the set of all multi-local unitary operations. Hence, it can be compensated for at the end of the computation by a local change of the measurement bases.. To summarize, any quantum logic network can be simulated on a one-way quantum computer. A set of universal gates can be realized by one-qubit measurements and the gates can be combined to circuits. Due to the randomness of the results of the individual measurements, unwanted byproduct operators are introduced. These byproduct operators can be accounted for by adapting measurement directions throughout the process of computation. In this way, a subset of qubits on the cluster C is prepared as the output register. The quantum state on this subset of qubits equals that of the quantum register of the simulated network up to the action of an accumulated byproduct operator. The byproduct operator determines how the measurements on the output register are to be interpreted. We have described the computational model underlying the one-way quantum computer, which is very different from the quantum logic network model.. The formal description of the QC C is based on primitive quantities of which the most important are the sets Q t ⊂ C of cluster qubits defining the temporal ordering of measurements on the cluster state, and the binary valued information flow vector I(t) which is the carrier of the algorithmic information.. The information processing with the QC C goes beyond mere emulation of quantum gates by sequences of measurements. The complete description of the computational process on the QC C contains the temporal order in which the measurements are performed, and the most efficient temporal order does not follow from the rules that apply for the temporal ordering of gates in the network model. Thus, the network picture is insufficient to describe computation with the QC C and is therefore abandoned. In the proposed scheme the unitary gates from some universal set can not be taken for the elementary building blocks of quantum computation.. As a practical implication of the new model, the logical depth of certain algorithms on the QC C is lower than for their corresponding network realizations. As shown for circuits of diagonal gates or of CNOT-, Hadamard-and π/2-phase gates, the logical depths for the QC C are constant in the number of gates and logical qubits, namely one and two. The best bounds for networks that have been known so far scale logarithmically. It therefore seems that, at least for the QC C , the question of temporal complexity must be revisited. The tools for this discussion together with first results have been provided in this paper.. A Proof: cone test and forward-backward cone relation. Here we prove the cone test (41) and the forward-backward cone relation (42). Considering the cone test, first note that whether a one-qubit rotation at some position in the network is about the z-axis or a about the x-axis can be identified by the potential byproduct operator produced when the rotation is implemented. This can be seen by inspecting (3), (7), (8) and the procedure to implement a general rotation as described in section 2. The x-rotations U x (ξ) and U x (ζ) of U R (ξ, η, ζ) in (3) are implemented by measurements on the qubits 2 and 4 of a 5-qubit chain. As can be seen from (8), they contribute to the byproduct operator U Σ of the rotation U R with σ x s2+s4 where s 2 and s 4 are the results of the measurements on qubits 2 and 4. Further, the rotation about the z-axis, U z (ζ), is implemented by measurement of qubit 3. The contribution to the byproduct operator which is thereby generated is, from (8), σ z s3 . We see that x-rotations only generate byproduct operators σ x and z-rotations only generate byproduct operators σ z .. A byproduct operator generated via the measurement on the cluster qubit k must be propagated either forward or backward to possibly reach the rotation on the logical qubit i implemented via the measurement on the cluster qubit j. Let be O K and O J two cuts through the network which intersect each logical qubit line only once. More specifically, O K intersects the qubit line i just before the rotation implemented by the measurement at cluster qubit k. O J intersects the qubit line i just before the rotation implemented by the measurement at cluster qubit j.. There are two cases which can occur. Either the cut O K is before the cut O J in the network N which we denote by O K ≤ O J , or O J is before the cut O K which we denote by O J ≤ O K . It can also be that both is true at the same time but it cannot be that neither of the two relations hold.. Case I: O K ≤ O J . The byproduct operator generated via the measurement at qubit k must be propagated forward to possibly affect the measurement at qubit j. It is not possible that the result of the measurement on qubit j has an effect on the measurement basis chosen at k.. Let us introduce a further cut O J ′ which is the same as O J , except for that it intersects the line of the logical qubit i in the network N just after the rotation implemented via the measurement on the cluster qubit j. The potential byproduct operator which is generated via the measurement on cluster qubit k and then propagated forward to the cuts O J and O J ′ , is denoted by U k | OJ and U k | O J ′ , respectively (the byproduct operators which are actually generated are (U k | OJ ) s k and U k | O J ′ s k ). Further, we denote the restriction of the byproduct operators U k | OJ and U k | O J ′ to the logical qubit i by [U k | OJ ] i and U k | O J ′ i . The two cuts differ only on the logical qubit i, and there only by the side of the respective cut on which the rotation is located. Therefore, using (11) , it follows that. If the rotation implemented via the measurement on cluster qubit j is about the x-axis, then the measurement on qubit j has to wait for the measurement on cluster qubit k iff [U k | OJ ] i contains a contribution σ z . The measurement on j itself produces a potential byproduct operator U j | O J ′ i = σ x . Similarly, if the rotation implemented via the measurement on j is about the z-axis then the measurement on j has to wait for the measurement on k iff [U k | OJ ] i contains a contribution σ x . The measurement on j itself produces a potential byproduct operator U j | O J ′ i = σ z . Because of (107) U k can as well be evaluated at the cut O J ′ instead of O J . The byproduct operator on the intersection of qubit line i and cut O J ′ resulting from the measurement on qubit j can be written in the form . formulation of the respective algorithm. Therefore, we must exclude the possibility that for the QC C the algorithmic complexity of a quantum computation is shuffled from the genuine quantum part of the computation to the classical pre-processing, and that this classical pre-processing may be exponentially hard. As will be shown below, such a case does not occur. All the classical pre-processing can be done in polynomial time.. To see this, we assume that the quantum algorithm on n logical qubits is given as a sequence of N elementary gates. For good quantum algorithms, N is polynomial in n, as is C , the number of physical qubits in the cluster C required to run the algorithm (see Section 5.3).. The layout of the measurement pattern requires to assign Q 0 measurement bases and C\Q 0 angles. Creating the pattern is for itself not a problem since it can be obtained by patching together the measurement patterns of the elementary gates which are available in block form. The temporal complexity for this step is thus O( C ).. To obtain the byproduct images we introduce N vertical cuts O i , i = 1, .., N to the network, one after each gate (such that O N = Ω) and compute the 2n × 2n-matrices C(N Oi→Ω ) for i = 1, .., N − 1, starting with i = N − 1. The operational effort for this is of the order O(n 3 N ). By use of these matrices the byproduct images for cluster qubits k ∈ C\Q 0,z can now be obtained via (29), which requires O(n 2 ) elementary operations per byproduct image. The way to obtain the byproduct images F g of the gates is the same. For k ∈ Q 0,z \O at most four byproduct images have to be added in (60), which requires O(n) operations. The computation of F k for k ∈ O is trivial. Thus, to compute a byproduct image requires at most O(n 2 ) operations per cluster qubit or gate such that the complexity to compute all of them is at most O n 2 ( C + N ) . The backward-and forward cones of the cluster qubits k ∈ C are computed using the temporal ordering of gates in a sequence representing the quantum logic network and the cone test (41). The number of cone tests that have to be performed in each case is N ( N − 1)/2 where the computational effort for each test scales like O(n). Thus, the complexity of this step is O(n N 2 ). The forward cones generate the anti-reflexive semi ordering "≺". The semi ordering can be computed from them in O( C 5 ) steps. For each set Q t there have to be Q (t) ≤ C test of the relation j ≺ k, j ∈ Q (t) performed to check whether some qubit k ∈ Q (t) is in Q t . Also, Q (t) qubits have to be checked for each Q t . At most C sets Q t exist such that the operational effort to obtain the these sets is O( C 3 ). As far as the stated upper bounds are conclusive, it looks as if the computation of the anti-reflexive semi ordering is the toughest part. However, as elementary a relation between the cluster qubits "≺" is, for the conversion of a quantum logic network into a QC C -algorithm it needs not be computed. Please note that the semi ordering is finally only needed to compute the sets Q t via (20). But instead of computing "≺" from the forward cones and the sets Q t from "≺", the sets Q t can also be computed from the forward cones directly. For this, please note that ∃j ∈ Q t | j ≺ k ∈ Q (t) ⇐⇒ ∃j ′ ∈ Q t | k ∈ fc(j ′ ). The direction "⇐=" is obvious with j = j ′ . The opposite direction, "=⇒", holds by an argument analogous to the one justifying (90). In fact, statement "=⇒" is the same as (90) with Q (1) replaced by Q (t) . Thus, eq. (20) can be replaced by. To set the algorithm angles via (71), (72) requires at most C + N additions per angle and there are at most C such angles. Hence, in total it takes O ( C ( C + N )) operations to set them. Finally, to initialize the information flow vector via (69) requires O (n( C + N )) operations. Thus we see that all classical processing requires only a polynomial overhead of elementary operations and can therefore be done in polynomial time. The discussion about the logical depth of certain algorithms with the QC C in Section 5 showed that there are ways of information processing possible with the QC C which cannot be explained adequately in network model terms. This made a computational model appropriately describing the QC C desirable. The computational model underlying the QC C that we found does not seem to have much in common with the network model. It is based on objects of a different sort which require an interpretation. In this section, we attempt to clarify the role of the binary valued information flow vector I(t) and that of the stepwise measured quantum state.. What is the role of the information flow vector I(t)? In every computational step except the final one the information flow vector I(t) is completely random. So one might ask whether it contains information at all. It does, since in every step except the last one it tells what has to be done next. After the final computational step at time t max the quantity I(t max ) contains the result of the computation. Thus, the quantity I(t) has a meaning in every computational step. In this sense, it represents the algorithmic information in the described scheme of quantum computation.. What is the role of the stepwise measured quantum state? To see that explicitly, let us consider the scenario where a quantum computation is halted in the middle and continued at a later time by another person who only knows which steps of the computation are left to perform but does not know what has been done so far. In analogy to a teleportation protocol where both the result from the Bell measurement and the quantum state at the receivers side are required to reconstruct the initial state, the halted computation can be successfully completed only if both pieces -the intermediate information flow vector I(t i ) and the half-measured quantum state-are stored until the computation proceeds. Thus, the quantum state cannot be neglected just because it does not appear in the formal description of the computational model. The quantum correlations in the stepwise measured state are what basically enables the described way of quantum information processing. However, the role of this state is a passive one. It serves as a resource that is used up during the course of computation.. For the description of the QC C there is no necessity to refer to the "qubit" as a basic notion of quantum information theory. As described above, the stepwise modified quantum state attains the role of a consumable resource but not that of the carrier of algorithmic information. It is thus sufficiently described in terms of standard quantum mechanics, namely as a quantum state of entangled two-level systems. The processed algorithmic information is classical.. Let us at the end of this discussion come back to the role which the randomness of the individual measurement results plays for the QC C . It may surprise that a set of classical binary numbers represents the algorithmic information in a scheme of quantum computation. In the network model the quantum state (of the quantum register) is usually considered to represent the processed information. For the QC C , the situation is different. There, the randomness of the individual measurement results makes it necessary to store classical steering information. The need to process this information has called for a novel information carrying quantity. What, in a network-like description of the QC C , has been regarded as a mere byproduct turns out to be the central quantity of information processing with the QC C .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Dynamics of information in the presence of deformation The entanglement of atomic system consists of two atoms interacts with a deformed cavity mode is quantified by the means of Bloch vectors and the cross dyadic of the traveling state inside the cavity. For large value of the deformation the amplitude of Bloch vectors decrease very fast and consequently the traveling state turns into mixed state quickly. The generated entangled state is used as quantum channel to implement quantum teleportation protocol. It is shown that both of the deformed parameter and the number of photons inside the cavity play a central role on controlling the fidelity of the transmitted information. Controlling quantum coherence is one of the most fundamental issues in quantum information processing [1] . From a practical point of view the investigating of the behavior of entanglement in the presences of noise is very important, where there are many studies that has been done in different directions. Among these noise is the deformation, which arise from the defects of devices [2, 3, 4] . So, in the presence of deformation, one can generate a deformed entangled state and consequently there will be a deformation on the information processing.. The quantization of the field in the presence of deformed operators has been investigated in different topics, where the relation between the deformed and non deformed operators have been obtained in [5, 6, 7] . As an applications of the deformed operators, Mancini and Manko have compared the dynamics of information which is coded in deformed cat state with non-deformed cat state [8] . Also, in [9] the holonomic quantum gates using isospectral deformations of an Ising model have been constructed. The survival of quantum coherence against dissipation provided to superpose distinguishable coherent states of suitable deformed field has been investigated in [10] . Recently, Metwally and et.al, have showed that the nonclassical properties of a two qubits state can be enhanced via deformed operators [11] .. These efforts motivate us to introduce a different application of the deformation. In this work, we consider a system consists of of two separable atoms interacts with a cavity mode with multiphotons. The operators which describe the field are assumed to be deformed. We investigate the behavior of entanglement which is generated between the two atoms. Also, the dynamics of the Bloch vectors are investigated for this deformed atomic system. Finally, we use the generated entangled state between the two atoms as quantum channel to implement the quantum teleportation protocol. The effect of the deformation on the fidelity of the transmitted information is investigated. This paper is organized as follows: In Sec.2, we introduce the model and its solution by means of the Bloch vectors and the cross dyadic. The entanglement and the dynamics of the Bloch vectors are studied in Sec.3. The quantum teleportation protocol is the subject of Sec.4. Finally, we summarize our results in Sec.5,. Also, the density operator of the field is given by,. Then using the Hamiltonian (1), the initial state of the atomic system (5), and the initial state of the field (9) one gets the time evolution of the density operator of the field and the atomic system, where λ 1 = λ 2 = λ, ω 1 = ω 2 = ω, and ω 0 − ω = ∆ = 0. After tracing out the state of the field, the density operator of the atomic system is given by. where,. n c * (1). and the elements of the cross dyadic are,. n c * (1). with,. where, In this contribution, a system of two atoms interacts with a deformed cavity mode is introduced. The solution of the system is introduced by means of the Bloch vectors and the cross dyadic of the travelling atoms inside the cavity. Due to the interaction, there are some entangled states have been generated. The amount of entanglement between the entangled atoms is quantified by using a measure depends on the Bloch vectors and the cross dyadic. Although, as one increases the deformity, the entanglement decreases more, but it is more robust, where it vanishes only on small interval of time. The deformation of entanglement appears clearly as one increases the number of photons inside the cavity, where the entanglement oscillates so fast in irregular behavior. Also, we investigate the dynamics of the Bloch vectors, where we show that as one increases the deformation parameter, the amplitudes of the Bloch vectors decrease much faster. One can obtain, entangled states of Werner types with high degree of entanglement.. The effect of the deformity on the accuracy of the transported information between two users is investigated. The amplitude of the fidelity decreases in the presence of the deformity. This means that the minimum value of the fidelity is improved as time goes on. For large values of the deformed parameter, the fidelity decreases and the oscillations of the amplitudes decrease more. If the number of photons inside the cavity increases, the fidelity flocculates very fast in small range of time and as time goes on the fidelity is almost constant. The deformation of the fidelity appears for larger values of the deformity parameter, where the revivals are very fast.. Finally, one can conclude that the deformation and the number of photons inside the cavity play an important role on the dynamics of entanglement, Bloch vectors and the fidelity of the teleported state. If the devices are defective manufacturing, then by increasing the number of photons within the cavity one can reduce the distortion resulting from such defects. The effect of deformation is different from other noise which leads to a sudden death of entanglement and consequently a sudden death of communication. So, we expect that these results are important in quantum communication and consequently in building quantum computers

O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O


Predecessor existence problems for finite discrete dynamical systems We study the predecessor existence problem for finite discrete dynamical systems. Given a finite discrete dynamical system S and a configuration C, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration C such that S has a transition from C to C. In addition to the decision version, we also study the following variants: the #-PREDECESSOR EXISTENCE (or #PRE) problem -counting the number of predecessors, the UNIQUE-PREDECESSOR EXISTENCE (or UPRE) problem -deciding whether there is a unique predecessor and the AMBIGUOUS-PREDECESSOR EXISTENCE (or APRE) problem -given a configuration C and a predecessor C of C, deciding whether there is a different predecessor C of C.General techniques are presented for simultaneously characterizing the computational complexity of the PRE problem and its three variants. Our hardness results are based on the concept of simultaneous reductions: single transformations that can be used to simultaneously prove the hardness of the different variants of the PRE problem for their respective complexity classes. Our easiness results are based on dynamic programming and they extend the previous results on PRE problem for one-dimensional cellular automata. The hardness results together with the easiness results provide a tight separation between easy and hard instances.Further, the results imply similar bounds for other classes of finite discrete dynamical systems including discrete Hopfield and recurrent neural networks, concurrent state machines, systolic networks and one-and two-dimensional cellular automata. Our results extend the earlier results of Green, Sutner and Orponen on the complexity of the predecessor existence problem and its variants. Computational aspects of CA have been studied by a number of researchers; see for example [43, 17, 23, 56, 72, 27, 26, 63] . Much of this work addresses decidability of properties for infinite CA. Barrett, Mortveit and Reidys [8, 47, 55] and Laubenbacher and Pareigis [41, 40] investigate the mathematical properties of sequential dynamical systems. Questions concerning the existence of fixed points and garden of Eden configurations in SDSs are addressed in [5, 21, 53, 65, 66] . Experimental results from the use of a SAT solver to analyze the properties of a CA are reported in [15] . See a recent comprehensive survey by Sarkar [61] for additional information.. The PRE problem was shown to be NP-complete for finite 2D-CA by Sutner [63] and Green [26] . Sutner also showed that the PRE problem for finite 1D-CA with a fixed neighborhood radius can be solved in polynomial time. As mentioned earlier, Green [26] studied generalized versions of the PRE problem for infinite CA. The problems were so formulated that his hardness results are also applicable to finite one-dimensional CA.. The results presented here are applicable to automated behavior analysis of large-scale distributed systems. The problem of finding predecessors and its variants (e.g. counting the number of predecessors) are closely related to the problem of detecting unreachable states in distributed systems [11] : a state is unreachable iff it has no predecessors (preimages). Our lower bound results show that deciding whether a given state is unreachable is intractable even for very simple classes of concurrent transition systems. In contrast, the results for treewidth bounded graphs show how graph theoretic structure can be exploited to obtain polynomial time results for detecting unreachable states. Moreover, the algorithm for treewidth bounded graphs allows us to specify any finite set of linear constraints on the states of the predecessors or on the target states. For example, when the underlying graph has bounded treewidth and the local transition functions are symmetric, our results lead to a polynomial algorithm for the following problem: Suppose A is the set of configurations of an SDS S such that for every configuration in A, at most five nodes have the state value 1. Count the number of predecessors of the configurations in A.. The predecessor existence problems for dynamical systems studied in this paper can also serve as a way of modeling problems such as the spread of influence in social networks [36] . In such problems, the general goal is to identify a suitable initial subset of individuals to publicize an idea so that the number of individuals to whom the idea will propagate over time through the underlying social network is maximized. Such an approach is also useful in marketing products (e.g. identifying individuals to whom free samples should be sent) and in studying the spread of infectious diseases (e.g. identifying a subset of individuals who should be vaccinated so as to minimize the rate at which a disease spreads). When a given social network is thought of as the underlying graph of an SDS (or a SyDS), the target of a marketing campaign can be modeled by a collection of configurations in the phase space of the SDS. (For example, if the marketing goal is to reach at least N individuals, then each configuration in which N or more state values are 1 is a suitable target for the campaign.) The initial group of individuals to be chosen can be then be thought of as finding not an immediate predecessor, but an ancestor that is t-generations early, for some suitable t.. Recently, there has also been substantial interest in the use of agent based models for understanding problems in social science [3, 48, 49, 19] . Several researchers (see e.g. [3, 4, 1, 50] ) have demonstrated that network topology, agent behavior and update order (schedule or activation regime) can have substantial impact on the overall dynamic behavior of multiagent systems. In particular, Axtell [3] points out the need for understanding the intrinsic computational intractability of basic social processes for making further progress in modeling such processes. Our results shed some light in this direction by identifying the effects of local functions, update schedules and network structure on the computational intractability of some basic problems.. Additional terminology: The remainder of this paper presents proofs of our results for the PRE problem and its variants. We now mention some terminology used in these proofs. Suppose C and C are configurations such that C is a predecessor of C. In our proofs, we use the terms "initial value" and "final value" of a node v to mean the state of v in C and C respectively. Proof. We present a reduction from the restricted version of PLANAR-3SAT (RP3SAT) problem defined in the Appendices A and B. In this restricted version, each clause has exactly three literals and each variable occurs in at most three clauses. We will first present the proof details for (BOOL, SYM, GRID)-SDSs and then indicate the necessary modifications for (BOOL, SYM, GRID)-SyDSs. Throughout this proof, the reader should bear in mind that for each node v, the neighborhood of v includes v itself.

B I O O O B I I O O O B I O O O B I I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Proposition for a sequential accelerator in future general-purpose manycore processors and the problem of migration-induced cache misses As the number of transistors on a chip doubles with every technology generation, the number of on-chip cores also increases rapidly, making possible in a foreseeable future to design processors featuring hundreds of general-purpose cores. However, though a large number of cores speeds up parallel code sections, Amdahl's law requires speeding up sequential sections too. We argue that it will become possible to dedicate a substantial fraction of the chip area and power budget to achieve high sequential performance. Current general-purpose processors contain a handful of cores designed to be continuously active and run in parallel. This leads to power and thermal constraints that limit the core's performance. We propose removing these constraints with a sequential accelerator (SACC). A SACC consists of several cores designed for ultimate sequential performance. These cores cannot run continuously. A single core is active at any time, the rest of the cores are inactive and power-gated. We migrate the execution periodically to another core to spread heat generation uniformly over the whole SACC area, thus addressing the temperature issue. The SACC will be viable only if it yields significant sequential performance. Migration-induced cache misses may limit performance gains. We propose some solutions to mitigate this problem. We also investigate a migration method using thermal sensors, such that the migration interval depends on the ambient temperature and the migration penalty is negligible under normal thermal conditions. The continued doubling of transistor density with every technology generation has been leading to a rapid increase in the number of on-chip cores. So far, general purpose multicores are symmetric, i.e., all cores are identical. But future manycores will likely be asymmetric, with some cores specialized for certain tasks. In particular, since performance on sequential applications and on sequential sections in parallel applications will remain a major issue, future manycores may feature some cores for providing high sequential performance.. On current multicores, the power and temperature walls forbid aggressive design and very high clock frequencies. However, as the number of on-chip cores increases, each core will consume a smaller fraction of the total chip power and area. For example if, in ten years from now, technology affords the silicon area and the power budget for 1000 EV6-like cores on the same chip, then it might be performance effective to implement only 900 of those cores and to dedicate the rest of the silicon area and power budget to any microarchitecture providing very high sequential performance.. In this paper, we propose a new direction for utilizing a large fraction of the chip area and power budget to achieve high sequential performance. We refer to our approach as sequential accelerator, or SACC. A SACC consists of several large power-hungry cores (LPH) designed for ultimate instantaneous sequential performance by using aggressive microarchitecture, circuits and technology, high voltage and clock frequency. In the design of the LPH, the fundamental thermal constraint that a core should be able to run continuously is removed. Only a single LPH is active at any time. The inactive LPHs are power-gated. The overall SACC power can be directed to the active LPH. Having several cores makes possible to spread the heat generation on the whole SACC area by migrating the execution periodically to another LPH, thus solving the temperature issue.. Activity migration for temperature has already been considered in the framework of small scale multicores [33, 13, 30] . Our thesis is that sacrificing the possibility of parallel execution in the SACC will allow us to specifically design an LPH able to consume a large instantaneous power, which will facilitate unprecedented levels of instantaneous sequential performance. We will also point out that migrating activity from one LPH to another LPH in the SACC is simpler than in a conventional multicore. The detailed implementation of the LPH is beyond the scope of this paper. We mostly focused on the global SACC architecture and on thermal aspects. In particular, even if the LPH achieves very high instantaneous performance, the SACC approach will be viable only if the SACC achieves high long-range performance. Migrations, especially migrationinduced cache misses, may incur a significant performance penalty. We propose some solutions for alleviating this impact. In particular, since only a single LPH is active at a time, we propose to use a write-back level-1 (L1) data cache and a write-through level-2 (L2) cache in each LPH. We show that the number of migration-induced L2 misses can be reduced by warming-up the L2 of next-to-be-active LPHs, which increases performance significantly when implementing a sensor-less migration scheme with a fixed migration interval. We also propose a migration method using thermal sensors that adapts the migration interval to the thermal conditions, while ensuring it cannot be shorter than a predefined value. The migration interval can vary with the ambient temperature, and we show that the performance loss due to migrations is negligible as long as the ambient temperature does not exceed the nominal value.. The paper is organized as follows. In Section 2, we argue that future manycores may use a substantial fraction of the chip area to implement one very large unit providing high sequential performance. We describe in this section the sequential accelerator, our proposition for such unit. Related work is mentioned in Section 3. In Section 4, we propose several schemes to mitigate the impact of migration-induced L2 misses. Our simulation results are presented and analyzed in Section 5. Finally, Section 6 concludes this study. In future general-purpose manycore processors, a large part of the chip area and power budget will likely be dedicated to high sequential performance. Microarchitecture and circuit design solutions that are deemed too power hungry today may help maximize the instantaneous sequential performance of a core, provided the core is not required to be continuously active. Based on this assumption, we proposed a sequential accelerator (SACC) consisting of several large power-hungry cores (LPHs), where a single LPH is active at a given time and other LPHs are power-gated. When needed, the execution is migrated to another LPH in order to spread the heat generation uniformly over the whole SACC area.. The effectiveness of a SACC may be significantly impaired by the migration penalty. We pointed out the potential impact of migration induced L2 misses on performance. We proposed and evaluated some schemes for decreasing this impact. We made a case for write-through L2 caches and write-back L1 data caches. We showed that warming-up the L2 of future LPHs on which we will migrate next is more interesting than fetching blocks from past LPHs. We showed that L2 warm-up may bring a significant performance gain for a sensor-less migration scheme with a fixed migration interval. We also defined a migration policy using thermal sensors which forces the migration interval to be longer than a predefined value. With this policy, the migration interval is variable and depends on the ambient temperature. As long as the ambient temperature stays below the nominal value, the migration interval is long and migrations incur little performance loss. However, when the ambient temperature exceeds the nominal value, performance degrades significantly if linear throttling is the only throttling method implemented.. Our study tackled only some of the questions that must be answered to make the sequential accelerator a viable solution. This work points to several new directions for future work, including microarchitectural design of LPHs, effective power-gating techniques, and effective power delivery to the active LPH.

O O O O O O O O B O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O B I O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O


Estimating the confidence of conditional functional dependencies Conditional functional dependencies (CFDs) have recently been proposed as extensions of classical functional dependencies that apply to a certain subset of the relation, as specified by a pattern tableau. Calculating the support and confidence of a CFD (i.e., the size of the applicable subset and the extent to which it satisfies the CFD) gives valuable information about data semantics and data quality. While computing the support is easier, computing the confidence exactly is expensive if the relation is large, and estimating it from a random sample of the relation is unreliable unless the sample is large.We study how to efficiently estimate the confidence of a CFD with a small number of passes (one or two) over the input using small space. Our solutions are based on a variety of sampling and sketching techniques, and apply when the pattern tableau is known in advance, and also the harder case when this is given after the data have been seen. We analyze our algorithms, and show that they can guarantee a small additive error; we also show that relative errors guarantees are not possible. We demonstrate the power of these methods empirically, with a detailed study using both real and synthetic data. These experiments show that it is possible to estimate the CFD confidence very accurately with summaries which are much smaller than the size of the data they represent. Conditional functional dependencies (CFDs) have recently been proposed to characterize the semantics of complex data and facilitate data cleaning [3, 4, 10, 11, 14, 15, 16, 18] . A CFD is a functional dependency (FD) that holds on a subset of the relation specified in an accompanying pattern tableau. The support of a CFD is the fraction of rows that match the tableau. Its confidence, defined formally below, is the fraction of rows satisfying the functional dependency amongst those that match the tableau. The support and confidence of a given tableau are key properties in understanding and exploring data quality. Consider a data warehouse containing large quantities of historical data. Careful analysis of a particular relation over time may have revealed various tableaux that have high support and confidence; checking the support and confidence on new data as they arrive hour by hour is an important quality check, and can reveal new trends if the support or confidence change suddenly. A second situation is when analysis has revealed a new tableau that has high support and confidence on the new data; it is then natural to ask over which historical data this tableau also had high support and confidence. Because of the quantity of the data, and the many different tableaux which may be considered, we seek efficient methods to find the support and confidence of CFDs in large relations.. In these settings, as advocated in [8] , although the full data may be stored exactly, it is desirable to also maintain various compact summaries as new data and/or new sources are integrated over time. Summaries enable rapid data auditing without having to run complex queries on massive fact tables, and when some data are physically stored at remote locations or archived on slow storage devices. We seek compact summaries for computing the support and confidence of CFDs in order to understand changes in data semantics. Ideally, these summaries should be easy to derive from the base data. They should require only one or a few passes over the data to compute, so that they can be computed as the data is loaded into a warehouse, without additional costly sorting steps. To make this precise, we describe CFDs in more detail, and go on to define the properties we desire from compact summaries for support and confidence estimation. Consider the FD {age_grp, education, occupation} → salary_grp on a census table, which states that individuals in the same age group that have the same education and occupation must have salaries in the same group. This rule is not expected to hold over the entire database; e.g., accountants working in different companies may earn different salaries despite having the same age group and education. It may, however, be true for certain government-regulated occupations with age-based salary scales, e.g., high school teachers with Masters' degrees and professors with Doctorate degrees. Table 1 illustrates the corresponding tableau. This indicates that the FD should hold on the set of tuples that match at least one of the listed patterns. The wildcard pattern "___" matches all possible attribute values.. In addition to limiting the scope of an FD, a tableau may specify tuple-level constraints. Consider a data warehouse of orders placed by an auto manufacturer to its suppliers. Table 2 illustrates a tableau for the FD {supplier, part, part_type} → price, i.e., a supplier charges a constant price for a part (of some type part_type). According to the first two patterns, this is true for tires and parts supplied by U.S. Auto Parts, respectively. According to the third pattern, a supplier of headlights for model 325 must charge a constant price, but not any constant price; it must be 50. Thus, constants in the left-hand-side attributes define the scope of the CFD and constants on the right-hand-side restrict attribute values. Note that patterns may overlap as U.S. Auto Parts may supply tires and/or 325_headlights.. Exceptions can occur in real data in the form of incorrect records (e.g., a supplier charges the wrong price of 325_headlights) or correct records that deviate from the assumed semantics (e.g., some professors in a given age group may earn lower or higher salaries than normal). Thus, even conditional functional dependencies may not hold exactly. Instead, we consider approximate CFDs. Following prior work [18, 19, 23] , we define the confidence of a CFD as the maximum fraction of tuples that may be retained, so that if all other tuples were deleted, the remaining ones would satisfy the CFD with no exceptions. The support is the fraction of tuples that match at least one pattern in the tableau. These quantities are fundamental to using CFDs to analyze data quality, and perform data auditing, and so our focus in this work is on problems related to estimating the confidence in various settings.. Given a supplied FD, one type of audit estimates the support and confidence of an existing tableau on new data. Another audit involves computing the support and confidence of new tableaux on historical data. For instance, the tableau from Table 1 may not hold on the latest census data due to changes in the job market. Instead, the tableau (___|Bachelors|Nurse ___) may be identified, in which case a natural question is to study how the support and confidence of this tableau has varied over time, and to identify when it first became significant. Similarly, the tableau from Table 2 may not hold on new data, perhaps because the contract with U.S. Auto Parts has been re-negotiated to allow variable prices, or because some prices are incorrectly recorded. These analyses may be time-consuming to compute exactly over archived data (and impossible if the data is not stored in full); instead, we look for more practical ways to answer these questions from pre-computed summaries.. For many query types, summarizing data with a random sample and evaluating the query over the sample is a good way to approximate the answer. However, while small random samples can give good estimates of simple frequency counts, such as the support of a CFD, estimating the confidence of a CFD is more complex. In fact, a uniform random sample of size Ω( √ N ) is required even to estimate whether a standard FD holds on a relation of size N with confidence above 1 − [23]. Thus we need a different approach to building summaries for CFD confidence estimation. We have introduced the problem of estimating the confidence of Conditional Functional Dependencies when the tableau is given upfront or specified after the fact. It is natural to extend our results to related problems; we briefly outline some extensions to which they apply. Given a tableau, it is sometimes important to know not just the global confidence of the CFD, but also the local confidence resulting from each row in isolation [18] . However, this can be thought of as a generalization of the variable CFD problem: each row of the tableau can be thought of as an entire tableau, and the relevant confidence estimated.. Given a CFD with less than perfect confidence, it is also informative to know for which subset of the data the CFD fails to hold. This is captured by a fail tableau, by analogy with the (hold) tableau on which the CFD holds with high confidence [18] . Given a fail tableau, it is straightforward for our methods to estimate its (low) confidence. Similarly, our algorithms apply to extended CFDs introduced in [3] , whose tableaux may include negations and disjunctions in addition to constants and wildcards. A last challenging problem is to summarize the data, and then to attempt to discover a high confidence tableau for a particular embedded FD from the summary [18] . A natural direction to study is to apply known tableau discovery algorithms directly to the data stored in the summaries we have defined here, and also use the summary to estimate the confidence of the candidate tableau rows. We hope to report on the quality of this approach in future work. Define G k as the set of groups with support close to r2 −k as:. we show that if we could draw a uniform random sample of the groups and find the corresponding Nx,y values exactly, we would have an estimator that is correct in expectation and with bounded variance. Let S k be the groups sampled at the kth level (so they are picked with probability α k ). Let Cx be the confidence of the group x as before, so that Cx = maxy Nx,y N and C φ (R) = P x Cx. Define the estimator X k as. The expectation and variance of X k satisfies. PROOF. For the expectation,. For the variance, define I x,k as the indicator variable that is 1 if x is chosen to be in the sample S k , and 0 otherwise. Then. This shows that if we could estimate each Cx value of the sampled groups accurately, then these could be combined into an estimate with low variance for the overall confidence. In particular, for X = P log N k=0 X k , we have Pr[|X − C φ (R)| > ] ≤ δ, by the Chebyshev inequality. We now show that by using the Count-Min sketch summary, each Cx is approximated with high accuracy. LEMMA 6. Using Count-Min sketches with b = 2 −3 log 1/δ buckets allows Nx and Nx,y to be estimated with additive error 3 α k N .. PROOF. Given any group x in S k , the two Count-Min sketches with b buckets promise to estimate Nx and Nx,y with error proportional to ∆ = F. is the frequency of all items in S k excluding the b most frequent items [13] . Note that if Nx or Nx,y is less than ∆, then estimating it by 0 will satisfy this requirement.. We now analyze F res(b) 1 (S k ), and argue that it is at most 10α k N log 1/δ with probability at least 1 − δ. First, consider groups x such that Nx ≥ 2 −k N -there can be at most 2 k of these. Moreover, in expectation, the random sampling picks at most α k 2 k = −2 such groups. Using the relative Chernoff bound, which states that for β > 2e − 1,. we have that the probability of picking more than b = −3 log 1 δ groups is at most 2. For groups with Nx < 2 −k N , we bound the total frequency that these contribute. Since these are sampled with probability α k , the expected sum of frequency of all such sampled items is at most α k N . Applying the relative Chernoff bound again, and using the fact that α k ≤ 1, the probability that F. for antecedent, consequent pairs Nx,y is similar. The difference arises since the sampling choice is based only on the antecedent x, so there are correlations between consequents in the same group. However, since the analysis is based on expectations, these do not affect the result, and again, we can conclude that the F res(b) 1 is bounded.. Using the sketches at level k, we recoverNx,y values for all x for which x ∈G k , whereG is defined as. The above lemma shows that these estimates are quite accurate. If we replace the random variable X k withX k , the same estimator built with the estimated frequencies, we obtain:. That is, the error comes from three places: error in the estimated frequencies on antecedents that both methods agree should be in the sample, plus groups missed by the sketching approach that should be included, and groups included by the sketching approach that should be omitted. For the first of these, we have. The last equality holds in expectation because for x ∈ G k , we have. Since each x is sampled into S k with probability α k , the expected value of this summation is P. Observe that if we take this sum over all x, then P x Nx/N ≤ 2 with constant probability, applying the Markov inequality.. For the errors due to misclassification of antecedents into groups, these affect only those x for which. For other values of Nx, the estimatedNx will place it in the correct G k by the guarantees on the sketch estimation. We can bound this quantity by observing that, based on the random choice of the parameter r,. So the contribution to the confidence of all such groups is at most 3 with constant probability, and errors on these groups can contribute at most this much to the overall error in the confidence estimate. In conclusion then, our final estimateĈ φ (R) formed as the sum of all estimatesX k satisfies:. with constant probability. We amplify this to arbitrarily small probability of failure by repeating the estimation process (with different random choices of the shift r) sufficiently many times, and taking the median of the estimates. Thus, by the bounds on X, we conclude (after appropriate rescaling of ) that:. THEOREM 3. The estimateĈ φ (R) produced by the multilevel algorithm satisfies The problem of estimating the confidence of CFDs is related to work on finding conditional functional dependencies, estimating the confidence of functional dependencies and other integrity constraints, and generating concise summaries of large data sets. Recent research on CFDs has followed three directions. The first involves reasoning about CFDs, that is, axiomatization, consistency, and implication [3, 4, 14, 16] . The second employs exact and approximate CFDs to characterize the semantics of data, for example, discovering and verifying the confidence of CFDs on a given relation instance [10, 14, 15, 18] . The third uses CFDs for data cleaning, such as identifying and "repairing" tuples in a given relation instance that violate a given set of CFDs [11] , which is usually performed after an appropriate set of CFDs has been identified and validated. Our work is closest to the second of these categories, although no prior work has addressed problems of estimating the confidence of CFDs directly. Our techniques for confidence estimation may be used in lieu of exact validation via an expensive SQL query, or when the entire data set cannot be stored or readily accessed.. The related question of determining whether an FD holds has been studied in [23] . In particular, it was shown that if a uniform random sample of size O( −1 √ N ) is collected, then any FD that does not hold perfectly on the sample cannot (with high probability) have a confidence of above 1 − . This was used to determine which FDs could hold exactly, by eliminating any which did not pass this randomized test; however, this approach does not obviously extend to estimating the confidence. Further, the space cost of this test is polynomial in the input size N and requires prior knowledge of N , whereas the solutions we seek are largely independent of this quantity, or depend only very weakly (logarithmically) on N . Estimating FDs from a sample was also considered in [20] , but no error bounds were given and the confidence metric was different (the number of distinct antecedent values divided by the number of distinct antecedent-consequent values).. There has also been work on estimating other types of integrity constraints from a (random) sample, such as association rules [26, 9] , algebraic relationships between attributes [7] , and universal first order logic sentences [22] . An association rule X → Y over a set of transactions (e.g., {bread, milk} → {butter}) asserts that whenever all the items in the set X appear in a transaction, so do all the items in the set Y . The confidence of an association rule is the count of transactions having X and Y divided by the number of transactions having X. Thus, approximating the confidence of an association rule requires an estimate of only these two counts, which can be obtained from a random sample with additive error. In contrast, CFD estimation involves counting the most frequent consequent value for each antecedent group. Similarly, [7] discovers correlations between attributes by estimating frequency counts of attribute values from a random sample, and [22] estimates the confidence of logical sentences by counting the proportion of tuples in the random sample that satisfy them.. Our work is related to summarizing large data sets. There has been recent work on maintaining bounded-size samples in a data warehouse that continually receives new data, but only random samples were considered [8, 17] . Our (one-pass and two-pass) algorithms can be thought of as constructing special summaries that may be stored in a data warehouse for CFD confidence estimation. These summaries are formed by careful combination of existing summaries such as reservoir sampling [27] , heavy-hitter summaries [25] , and count-min sketches [13] . Lastly, the CFD confidence estimation problem can be thought of as a special case of a "cascaded aggregate" [12] , formed by the summation of frequent items. In [12] , this problem is denoted as F1(F∞), for which solutions were not previously known. Similar multilevel techniques have been previously used to estimate other quantities over data streams [2] .

O O O O B I I B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Viscous flows in corner regions: Singularities and hidden eigensolutions Numerical issues arising in computations of viscous flows in corners formed by a liquidfluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the 'moving contact-line problem' and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite-element method in the context of the present problem are discussed. The ability of a numerical scheme to accurately approximate a physical problem in a domain containing corners is critical for the description of a number of phenomena, ranging from electromagnetic wave propagation in a waveguide [1] to die-swell effects in polymer extrusion [2] . Often, an analysis of such problems reveals singular behaviour of variables as the corner is approached, which requires special numerical treatment. Such problems are well known and have been thoroughly investigated in the setting of fracture mechanics, where one considers the propagation of a crack into a material [3] , and have also been studied in some fluid dynamics problems [4] .. Our interest here is the viscous flow in a corner formed between a liquid-fluid free surface and a solid boundary. Although a free surface is generally bent, to leading order as the corner is approached, it is often possible for the purpose of a local analysis to consider the flow domain as having a wedge shape. This is the case, for example, in dynamic wetting flows [5] , where the free surface and the solid boundary form what is referred to as the 'contact angle' and the liquid-fluid-solid 'contact line' moves with respect to the solid surface. This differs from the situation considered in some previous investigations on flow in a corner where the contact line is stationary with respect to the solid and motion is generated by disturbances in the far field [6, 2] .. It is well known that the classical fluid-mechanical approach when applied to dynamic wetting problems fails to provide an adequate description of the flow [7] . The conventional remedy to the problem is to relax the no-slip boundary condition on the solid surface and allow for 'slip' between the liquid and solid. A number of different forms for this slip behaviour have been examined in the literature (for a recent review see Ch. 3 of [5] ). Broadly, these split into conditions which (i) relate tangential stress to the slip velocity, such as the Navier condition [8] , or (ii) explicitly prescribe the velocity along the solid surface.. In this paper, we consider numerical problems arising in the second case where the velocity along the solid surface is a priori prescribed in a form which ensures that a solution exists, and that in the far field the usual no-slip condition is restored. This approach is appealing to some users due to its mathematical simplicity, and our goal here is to show numerical pitfalls one comes across in its numerical implementation and give a method of overcoming them which provides a framework for modelling this class of problems. A number of functions prescribing the fluid velocity on the solid surface have been proposed in the literature [9, 10, 11] and here we consider just one of these which captures all the main features of the problem. We have shown that accurate numerical calculation of viscous flows near corners of the flow domain in the framework of the finite-element method requires special treatment of the elements adjacent to the corners. In the case of zero-stress/prescribed velocity boundary conditions on the sides of the corner, the range of corner angles is split into two distinct regions. For the angles below the critical angle α c ≈ 128.7, it is sufficient to use a logarithmic basis function for the pressure, whereas for supercritical angles there appears a 'hidden' eigensolution which considerably complicates numerics. In order to obtain an acceptable (i.e. mesh-independent) solution, one has to incorporate this eigensolution into the code by altering the basis functions for the pressure in the elements adjacent to the corner point. Close to the critical angle it becomes necessary to use a two-term asymptotics in the numerical algorithm, including the leading terms of both the eigensolution and the solution of the inhomogeneous problem, as the finiteness of the mesh size could result in the algebraically and logarithmically singular terms having comparable values.. An issue that is now opened up for numerical and analytic investigation is how to generalize the developed methods for a three-dimensional case, i.e. in a situation where both the contact angle and the direction of velocity of the solid vary along a contact line. The first of these aspects becomes particularly challenging when the angle varies from subcritical to supercritical.

B I O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Central Limit Theorems for Super Ornstein-Uhlenbeck Processes Abstract Suppose that X = {X t : t ≥ 0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on− bx · ∇ as its underlying spatial motion and with branching mechanismLet P μ be the law of X with initial measure μ. Then the process W t = e −αt X t is a positive P μ -martingale. Therefore there isIn this paper we establish some spatial central limit theorems for X. Let P denote the function classFor each f ∈ P we define an integer γ (f ) in term of the spectral decomposition of f . In the small branching rate case α < 2γ (f )b, we prove that there is constant σ that, conditioned on no-extinction,where W * has the same distribution as W ∞ conditioned on no-extinction andIn the critical rate case α = 2γ (f )b,we prove that there is constant ρ 2 f ∈ (0, ∞) such that, conditioned on no-extinction,where W * has the same distribution as W ∞ conditioned on no-extinction and. Moreover W * and G 2 (f ) are independent. We also establish two central limit theorems in the large branching rate case α > 2γ (f )b.Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miłoś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miłoś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.

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