Renato Portugal

The Invar tensor package

Abstract The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a four-dimensional metric-compatible connection. The package can maximally simplify any polynomial containing tensor products of up to seven Riemann tensors within seconds. It has been implemented both in Mathematica and Maple algebraic systems. Program summary Program title: Invar Tensor Package Catalogue identifier: ADZK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 136 240 No. of bytes in distributed program, including test data, etc.: 2 711 923 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 5.2 or Maple versions 9 and 10 Operating system: Linux, Unix, Windows XP RAM: 30 Mb Word size: 64 or 32 bits Classification: 5 External routines: The Mathematica version requires the xTensor and xPerm packages. These are freely available at http://metric.iem.csic.es/Martin-Garcia/xAct Nature of problem: Manipulation and simplification of tensor expressions. Special attention on simplifying scalar polynomial expressions formed from the Riemann tensor on a four-dimensional metric-compatible manifold. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Restrictions: The present versions do not fully address the problem of reducing differential invariants or monomials of the Riemann tensor with free indices. Running time: Less than a second to fully reduce a monomial of the Riemann tensor of degree 7 in terms of independent invariants.

The staggered quantum walk model

There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work, we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs.

An algorithm to simplify tensor expressions

Abstract The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations and the renaming of dummy indices. The tensor indices are split into classes and a natural place for them is defined. The canonical form is the closest configuration to the natural configuration. In the second part, the Grobner basis method is used to simplify tensor expressions which obey the linear identities that come from cyclic symmetries (or more general tensor identities, including nonlinear identities). The algorithm is suitable for implementation in general purpose computer algebra systems. Some timings of an experimental implementation over the Riemann package are shown.

The Invar tensor package: Differential invariants of Riemann☆

Abstract The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ⋅ 10 23 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6 ⋅ 10 5 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica ) and Canon (for Maple ). Program summary Program title: Invar Tensor Package v2.0 Catalogue identifier: ADZK_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3 243 249 No. of bytes in distributed program, including test data, etc.: 939 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11 Operating system: Linux, Unix, Windows XP, MacOS RAM: 100 Mb Word size: 64 or 32 bits Supplementary material: The new database of relations is much larger than that for the previous version and therefore has not been included in the distribution. To obtain the Mathematica and Maple database files click on this link. Classification: 1.5, 5 Does the new version supersede the previous version?: Yes. The previous version (1.0) only handled algebraic invariants. The current version (2.0) has been extended to cover differential invariants as well. Nature of problem: Manipulation and simplification of scalar polynomial expressions formed from the Riemann tensor and its covariant derivatives. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Reasons for new version: With this new version, the user can manipulate differential invariants of the Riemann tensor. Differential invariants are required in many physical problems in classical and quantum gravity. Summary of revisions: The database of syzygies has been expanded by a factor of 30. New commands were added in order to deal with the enlarged database and to manipulate the covariant derivative. Restrictions: The present version only handles scalars, and not expressions with free indices. Additional comments: The distribution file for this program is over 53 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. Running time: One second to fully reduce any monomial of the Riemann tensor up to degree 7 or order 10 in terms of independent invariants. The Mathematica notebook included in the distribution takes approximately 5 minutes to run.

Spatial search on a honeycomb network

The spatial search problem consists of minimising the number of steps required to find a given site in a network under the restriction that only oracle queries or translations to neighbouring sites are allowed. We propose a quantum algorithm for the spatial search problem on a honeycomb lattice with N sites and torus-like boundary conditions. The search algorithm is based on a modified quantum walk on an hexagonal lattice and the general framework proposed by Ambainis, Kempe and Rivosh (Ambainis et al. 2005) is employed to show that the time complexity of this quantum search algorithm is

Mixing times in quantum walks on the hypercube

O(\sqrt{N \log N})

Establishing the equivalence between Szegedy's and coined quantum walks using the staggered model

.

Staggered Quantum Walks on Graphs

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time

The QWalk simulator of quantum walks

O(n\text{ }\text{log}\text{ }n)

One-dimensional coinless quantum walks

. We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution