Principles and symmetries of complexity in quantum field theory

Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it continuous} systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU($n$) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in $k$-local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schr\"{o}dinger's equation in isolated systems - the quantum state evolves by the process of minimizing "computational cost."Comment: Published version; added a short introduction on Finsler geometr

Fast Post-placement Rewiring Using Easily Detectable Functional Symmetries

Timing convergence problem arises when the estimations made during logic synthesis can not be met during physical design. In this paper, an efficient rewiring engine is proposed to explore maximal freedom after placement. The most important feature of this approach is that the existing placement solution is left intact throughout the optimization. A linear time algorithm is proposed to detect functional symmetries in the Boolean network and is used as the basis for rewiring. Integration with an existing gate sizing algorithm further proves the effectiveness of our technique. Experimental results are very promising

Methods for Solving Extremal Problems in Practice

During the 20 th century there has been an incredible progress in solving theoretically hard problems in practice. One of the most prominent examples is the DPLL algorithm and its derivatives to solve the Boolean satisfiability problem, which can handle instances with millions of variables and clauses in reasonable time, notwithstanding the theoretical difficulty of solving the problem. Despite this progress, there are classes of problems that contain especially hard instances, which have remained open for decades despite their relative small size. One such class is the class of extremal problems, which typically involve finding a combinatorial object under some constraints (e.g, the search for Ramsey numbers). In recent years, a number of specialized methods have emerged to tackle extremal problems. Most of these methods are applied to a specific problem, despite the fact there is a great deal in common between different problems. Following a meticulous examination of these methods, we would like to extend them to handle general extremal problems. Further more, we would like to offer ways to exploit the general structure of extremal problems in order to develop constraints and symmetry breaking techniques which will, hopefully, improve existing tools. The latter point is of immense importance in the context of extremal problems, which often hamper existing tools when there is a great deal of symmetry in the search space, or when not enough is known of the problem structure. For example, if a graph is a solution to a problem instance, in many cases any isomorphic graph will also be a solution. In such cases, existing methods can usually be applied only if the model excludes symmetries

Symmetric Arithmetic Circuits.

We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero

Hierarchical Strategies for Efficient Fault Recovery on the Reconfigurable PAnDA Device

A novel hierarchical fault-tolerance methodology for reconfigurable devices is presented. A bespoke multi-reconfigurable FPGA architecture, the programmable analogue and digital array (PAnDA), is introduced allowing fine-grained reconfiguration beyond any other FPGA architecture currently in existence. Fault blind circuit repair strategies, which require no specific information of the nature or location of faults, are developed, exploiting architectural features of PAnDA. Two fault recovery techniques, stochastic and deterministic strategies, are proposed and results of each, as well as a comparison of the two, are presented. Both approaches are based on creating algorithms performing fine-grained hierarchical partial reconfiguration on faulty circuits in order to repair them. While the stochastic approach provides insights into feasibility of the method, the deterministic approach aims to generate optimal repair strategies for generic faults induced into a specific circuit. It is shown that both techniques successfully repair the benchmark circuits used after random faults are induced in random circuit locations, and the deterministic strategies are shown to operate efficiently and effectively after optimisation for a specific use case. The methods are shown to be generally applicable to any circuit on PAnDA, and to be straightforwardly customisable for any FPGA fabric providing some regularity and symmetry in its structure