New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

u-Constacyclic codes over $${\mathbb {F}}_p+ u{\mathbb {F}}_p$$Fp+ uFp and their applications of constr ucting new non-binary q uant um codes

Structural properties of u-constacyclic codes over the ring \({\mathbb {F}}_p+u{\mathbb {F}}_p\) are given, where p is an odd prime and \(u^2=1\). Under a special Gray map from \({\mathbb {F}}_p+u{\mathbb {F}}_p\) to \({\mathbb {F}}_p^2\), some new non-binary quantum codes are obtained by this class of constacyclic codes.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

Parallel Multi-Block ADMM with <Emphasis Type="Italic">o</Emphasis>(1 / <Emphasis Type="Italic">k</Emphasis>) Convergence

This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints:

$$\begin{aligned} \text {minimize} ~~&f_1(\mathbf{x}_1) + \cdots + f_N(\mathbf{x}_N)\\ \text {subject to}~~&A_1 \mathbf{x}_1 ~+ \cdots + A_N\mathbf{x}_N =c,\\&\mathbf{x}_1\in {\mathcal {X}}_1,~\ldots , ~\mathbf{x}_N\in {\mathcal {X}}_N, \end{aligned}$$

where \(N \ge 2\), \(f_i\) are convex functions, \(A_i\) are matrices, and \({\mathcal {X}}_i\) are feasible sets for variable \(\mathbf{x}_i\). Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices \(A_i\) are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices \(A_i\)). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that \(\Vert {\mathbf {x}}^{k+1} - {\mathbf {x}}^k\Vert _M^2\) converges at a rate of o(1 / k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported.

     

New asymmetric quantum codes over $$F_{q}$$

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length \(n=q^{m}-1\) over \(F_{q}\), where \(q\ge 5\) is a prime power. The second one is the asymmetric quantum codes with length \(n=3^{m}-1\). These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set \(T_{1}=T_{2}^{-q}\), then the real Z-distance of our asymmetric quantum codes are much larger than \(\delta _\mathrm{max}+1\), where \(\delta _\mathrm{max}\) is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature.     

Cost-efficient design of a quantum multiplier–accumulator unit

This paper proposes a cost-efficient quantum multiplier–accumulator unit. The paper also presents a fast multiplication algorithm and designs a novel quantum multiplier device based on the proposed algorithm with the optimum time complexity as multiplier is the major device of a multiplier–accumulator unit. We show that the proposed multiplication technique has time complexity \(O((3 {\hbox {log}}_{2}n)+1)\), whereas the best known existing technique has \(O(n{\hbox {log}}_{2} n)\), where n is the number of qubits. In addition, our design proposes three new quantum circuits: a circuit representing a quantum full-adder, a circuit known as quantum ANDing circuit, which performs the ANDing operation and a circuit presenting quantum accumulator. Moreover, the proposed quantum multiplier–accumulator unit is the first ever quantum multiplier–accumulator circuit in the literature till now, which has reduced garbage outputs and ancillary inputs to a great extent. The comparative study shows that the proposed quantum multiplier performs better than the existing multipliers in terms of depth, quantum gates, delays, area and power with the increasing number of qubits. Moreover, we design the proposed quantum multiplier–accumulator unit, which performs better than the existing ones in terms of hardware and delay complexities, e.g., the proposed (\(n\times n\))—qubit quantum multiplier–accumulator unit requires \(O(n^{2})\) hardware and \(O({\hbox {log}}_{2}n)\) delay complexities, whereas the best known existing quantum multiplier–accumulator unit requires \(O(n^{3})\) hardware and \(O((n-1)^{2} +1+n)\) delay complexities. In addition, the proposed design achieves an improvement of 13.04, 60.08 and 27.2% for \(4\times 4\), 7.87, 51.8 and 27.1% for \(8\times 8\), 4.24, 52.14 and 27% for \(16\times 16\), 2.19, 52.15 and 27.26% for \(32 \times 32\) and 0.78, 52.18 and 27.28% for \(128 \times 128\)-qubit multiplications over the best known existing approach in terms of number of quantum gates, ancillary inputs and garbage outputs, respectively. Moreover, on average, the proposed design gains an improvement of 5.62% in terms of area and power consumptions over the best known existing approach.     

Entanglement witness criteria of strong-<Emphasis Type="Italic">k</Emphasis>-separability for multipartite quantum states

Let \(H_{1}, H_{2},\ldots ,H_{n}\) be separable complex Hilbert spaces with \(\dim H_{i}\ge 2\) and \(n\ge 2\). Assume that \(\rho \) is a state in \(H=H_1\otimes H_2\otimes \cdots \otimes H_n\). \(\rho \) is called strong-k-separable \((2\le k\le n)\) if \(\rho \) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that \(\rho \) is not strong-k-separable if and only if there exist a k-division space \(H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}\) of H, a finite-rank linear elementary operator positive on product states \(\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})\) and a state \(\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})\), such that \(\mathrm {Tr}(W\rho )<0\), where \(W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}\) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

An interval extension of SMS method for computing weighted Moore–Penrose inverse

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore–Penrose inverse \(A^{\dagger }_{MN}\) along with its rigorous error bounds is proposed for given full rank \(m \times n\) complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing \(A^{\dagger }_{MN}\), this method generates a sequence of complex interval matrices each enclosing \(A^{\dagger }_{MN}\) and converging to it. A new method is developed for constructing initial complex interval matrix containing \(A^{\dagger }_{MN}\). Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where \(l \ge 2\). A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.     

The hierarchical Petersen network: a new interconnection network with fixed degree

Network cost and fixed-degree characteristic for the graph are important factors to evaluate interconnection networks. In this paper, we propose hierarchical Petersen network (HPN) that is constructed in recursive and hierarchical structure based on a Petersen graph as a basic module. The degree of HPN(n) is 5, and HPN(n) has \(10^n\) nodes and \(2.5 \times 10^n\) edges. And we analyze its basic topological properties, routing algorithm, diameter, spanning tree, broadcasting algorithm and embedding. From the analysis, we prove that the diameter and network cost of HPN(n) are \(3\log _{10}N-1\) and \(15 \log _{10}N-1\), respectively, and it contains a spanning tree with the degree of 4. In addition, we propose link-disjoint one-to-all broadcasting algorithm and show that HPN(n) can be embedded into FP\(_k\) with expansion 1, dilation 2k and congestion 4. For most of the fixed-degree networks proposed, network cost and diameter require \(O(\sqrt{N})\) and the degree of the graph requires O(N). However, HPN(n) requires O(1) for the degree and \(O(\log _{10}N)\) for both diameter and network cost. As a result, the suggested interconnection network in this paper is superior to current fixed-degree and hierarchical networks in terms of network cost, diameter and the degree of the graph.     

Hilbert–Schmidt quantum coherence in multi-qudit systems

Using Bloch’s parametrization for qudits (d-level quantum systems), we write the Hilbert–Schmidt distance (HSD) between two generic n-qudit states as an Euclidean distance between two vectors of observables mean values in \(\mathbb {R}^{\Pi _{s=1}^{n}d_{s}^{2}-1}\), where \(d_{s}\) is the dimension for qudit s. Then, applying the generalized Gell–Mann’s matrices to generate \(\mathrm{SU}(d_{s})\), we use that result to obtain the Hilbert–Schmidt quantum coherence (HSC) of n-qudit systems. As examples, we consider in detail one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied, and the contrasting behaviors of HSC, \(l_{1}\)-norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.     

The panpositionable panconnectedness of crossed cubes

In many parallel and distributed multiprocessor systems, the processors are connected based on different types of interconnection networks. The topological structure of an interconnection network is typically modeled as a graph. Among the many kinds of network topologies, the crossed cube is one of the most popular. In this paper, we investigate the panpositionable panconnectedness problem with respect to the crossed cube. A graph G is r-panpositionably panconnected if for any three distinct vertices x, y, z of G and for any integer \(l_1\) satisfying \(r \le l_1 \le |V(G)| - r - 1\), there exists a path \(P = [x, P_1, y, P_2, z]\) in G such that (i) \(P_1\) joins x and y with \(l(P_1) = l_1\) and (ii) \(P_2\) joins y and z with \(l(P_2) = l_2\) for any integer \(l_2\) satisfying \(r \le l_2 \le |V(G)| - l_1 - 1\), where |V(G)| is the total number of vertices in G and \(l(P_1)\) (respectively, \(l(P_2)\)) is the length of path \(P_1\) (respectively, \(P_2\)). By mathematical induction, we demonstrate that the n-dimensional crossed cube \(CQ_n\) is n-panpositionably panconnected. This result indicates that the path embedding of joining x and z with a mediate vertex y in \(CQ_n\) is extremely flexible. Moreover, applying our result, crossed cube problems such as panpositionable pancyclicity, panpositionably Hamiltonian connectedness, and panpositionable Hamiltonicity can be solved.     

Construction of quantum caps in projective space <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, 4) and quantum codes of distance 4

Constructions of quantum caps in projective space PG(r, 4) by recursive methods and computer search are discussed. For each even n satisfying \(n\ge 282\) and each odd z satisfying \(z\ge 275\), a quantum n-cap and a quantum z-cap in \(PG(k-1, 4)\) with suitable k are constructed, and \([[n,n-2k,4]]\) and \([[z,z-2k,4]]\) quantum codes are derived from the constructed quantum n-cap and z-cap, respectively. For \(n\ge 282\) and \(n\ne 286\), 756 and 5040, or \(z\ge 275\), the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in PG(r, 4) for \(r\ge 11\). These results concerning large caps provide improved lower bounds on the maximal sizes of caps in PG(r, 4) for \(r\ge 11\).     

Preparing large-scale maximally entangled <Emphasis Type="Italic">W</Emphasis> states in optical system

We propose an optical scheme to prepare large-scale maximally entangled W states by fusing arbitrary-size polarization entangled W states via polarization-dependent beam splitter. Because most of the currently existing fusion schemes are suffering from the qubit loss problem, that is the number of the output entangled qubits is smaller than the sum of numbers of the input entangled qubits, which will inevitably decrease the fusion efficiency and increase the number of fusion steps as well as the requirement of quantum memories, in our scheme, we design a effect fusion mechanism to generate \(W_{m+n}\) state from a n-qubit W state and a m-qubit W state without any qubit loss. As the nature of this fusion mechanism clearly increases the final size of the obtained W state, it is more efficient and feasible. In addition, our scheme can also generate \(W_{m+n+t-1}\) state by fusing a \(W_m\), a \(W_n\) and a \(W_t\) states. This is a great progress compared with the current scheme which has to lose at least two particles in the fusion of three W states. Moreover, it also can be generalized to the case of fusing k different W states, and all the fusion schemes proposed here can start from Bell state as well.     

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.     

Close to linear space routing schemes

Let \(G=(V,E)\) be an unweighted undirected graph with n vertices and m edges, and let \(k>2\) be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance \(\varDelta \) from s, routes a message from s to t on a path whose length is \(O(k\varDelta +m^{1/k})\). The total space used by our routing scheme is \(mn^{O(1/\sqrt{\log n})}\), which is almost linear in the number of edges of the graph. We present also a routing scheme with \(n^{O(1/\sqrt{\log n})}\) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every \(v\in V\) is at most \(kn^{O(1/\sqrt{\log n})}deg(v)\), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein (2009), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

Exact exponential algorithms for 3-machine flowshop scheduling problems

In this paper, we focus on the design of an exact exponential time algorithm with a proved worst-case running time for 3-machine flowshop scheduling problems considering worst-case scenarios. For the minimization of the makespan criterion, a Dynamic Programming algorithm running in \({\mathcal {O}}^*(3^n)\) is proposed, which improves the current best-known time complexity \(2^{{\mathcal {O}}(n)}\times \Vert I\Vert ^{{\mathcal {O}}(1)}\) in the literature. The idea is based on a dominance condition and the consideration of the Pareto Front in the criteria space. The algorithm can be easily generalized to other problems that have similar structures. The generalization on two problems, namely the \(F3\Vert f_\mathrm{max}\) and \(F3\Vert \sum f_i\) problems, is discussed.     

Quantum algorithm to solve function inversion with time–space trade-off

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time–space trade-off in this paper. The details are as follows. Let function \(f(x)=y\) have k solutions, where \(x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}\) for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability \(1-\left( 1-\frac{k}{2^{n}}\right) ^{L}\) for \(L\in Z^{+}\). The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time–space trade-off strategy, we improve the successful probability of algorithm.