Optimal fermion-qubit mappings

Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. The key characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians. Improvements in the locality of fermion-qubit mappings have traditionally come at higher resource costs elsewhere, such as in the form of a significant number of additional qubits. We present a new way to design fermion-qubit mappings by making use of the extra degree of freedom: the choice of numbering scheme for the fermionic modes, a feature all mappings must have. This allows us to minimse the average Pauli weight of a qubit Hamiltonian -- its average number of Pauli matrices per term. Our approach leads to a rigorous notion of optimality by viewing fermion-qubit mappings as functions of their enumeration schemes. Furthermore, finding the best enumeration scheme allows one to increase the locality of the target qubit Hamiltonian without expending any additional resources. Minimising the average Pauli weight of a mapping is an NP-complete problem in general. We show how one solution, Mitchison and Durbin's enumeration pattern, leads to a qubit Hamiltonian for simulating the square fermionic lattice consisting of terms with an average Pauli weight 13.9% shorter than previously any previously known. Adding just two ancilla qubits, we can reduce the average Pauli weight of Hamiltonian terms by 37.9% on square lattices compared to previous methods. Lastly, we demonstrate the potential of our techniques to polynomially reduce the average Pauli weight by exhibiting $n$-mode fermionic systems where optimisation yields patterns that achieve $n^{\frac{1}{4}}$ improvement in average Pauli weight over na\"ive enumeration schemes.Comment: 29 pages, 30 figure

A correction on Shiloach's algorithm for minimum linear arrangement of trees

More than 30 years ago, Shiloach published an algorithm to solve the minimum linear arrangement problem for undirected trees. Here we fix a small error in the original version of the algorithm and discuss its effect on subsequent literature. We also improve some aspects of the notation.Comment: A new introductory paragraph has been added; error solutions and notation improvements are discussed with more dept

Multiscale approach for the network compression-friendly ordering

We present a fast multiscale approach for the network minimum logarithmic arrangement problem. This type of arrangement plays an important role in a network compression and fast node/link access operations. The algorithm is of linear complexity and exhibits good scalability which makes it practical and attractive for using on large-scale instances. Its effectiveness is demonstrated on a large set of real-life networks. These networks with corresponding best-known minimization results are suggested as an open benchmark for a research community to evaluate new methods for this problem

Bandwidth of trees of diameter at most 4

For a graph G, let γ:V(G)→1,⋯,|V(G)| be a one-to-one function. The bandwidth of γ is the maximum of |γ(u)-γ(v)| over uv∈E(G). The bandwidth of G, denoted b(G), is the minimum bandwidth over all embeddings γ, b(G)=min γmax|γ(u)-γ(v) |:uv∈E(G). In this paper, we show that the bandwidth computation problem for trees of diameter at most 4 can be solved in polynomial time. This naturally complements the result computing the bandwidth for 2-caterpillars. © 2012 Elsevier B.V. All rights reserved

Exact and Approximate Digraph Bandwidth

In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively. - Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree. - Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor. - Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n). - Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n)

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis