Notice: This is research code that will not necessarily be maintained to support further releases of Forest and other Rigetti Software. We welcome bug reports and PRs but make no guarantee about fixes or responses.

Representability

Representability is a library to work with linearly constrained mathematical programs over tensors.

It was designed as an easy to use framework for expressing semidefinite programs associated with varitional 2-RDM calculations.

At it's core, Representability is a collection of objects for Tensors and MultiTensor along with a methodology for building dual basis elements (linear relationships between elements of the tensors).

Features of Tensor()

• A general tensor object. Holds data without having to subclass numpy array.

• Has an associated basis which is a bijection from indices to tensor indexing. For example, a simple bijection might be the the {index:index} map. Another might be a {tuple: index} map where the tuple is some spin-adapted-geminal basis index. This is especially useful in the Fermionic case where the tensor objects are generally rank-4 tensors represented in matrix form--e.g. mathematically indexed by geminals instead of monomials

• The Tensor object is iterable and returns the basis indexing

• Tensor can be easily "indexed" into by returning the vectorized variable number. For example the matrix {{0, 1}, {2, 3}} vectorized in C-ordering turns into a vector {0, 1, 2, 3}. index_vectorized(1, 0) will return the vectorized index '2'. This is useful when defining a vector space of an aggregate of Tensor objects. When a basis is defined on the Tensor index_vectorized uses the basis.rev() to get the position in a symmetrized form.

Features of MultiTensor()

• As the name suggests this is an object that aggregates Tensor objects together to define a vector space.

• Vectorization is performed in the canonical way

• You can associate a DualBasis to a MultiTensor objects and synthesize the linear operator on the vector space as sparse operators.

Features of DualBasis()

• Container for DualBasisElements.

• Can be added together to create a larger set of operators.

Feature of DualBasisElement()

• Container for a single dual basis element.

• Each element can be extended and modified

• Simplification can be performed to avoid rewriting and minimizing if-else statements in the code.

RDM Projection

• Higham purification finds the closest positive semidefinite matrix with a fixed trace. This method is efficient because it does not require solving an SDP with a trace constraint. It involves a matrix diagonalization and a root finding step of monotonic function of the eigenvalues.

• iterative 2-RDM purification Iteratively purify a 2-RDM by Higham purification then mapping to other marginals and performing the Higham purification on those states.

• SDP 2-RDM purification. The method that finds the closest 2-RDM to a measured 2-RDM according the the rules of 2-positivity.

Information on all these techniques can be found in DOI:https://doi.org/10.1088/1367-2630/aab919

Installation

We recommend installing representability in a fresh environment built with a base python 3.6 installation. This can be accomplished with conda through the following commands

conda create -n rep_env python=3.6
source activate rep_env

You can install representability directly from the Python package manager pip using:

pip install -r requirements.txt
pip install representability -e .

The -e . reference the python package manager to the local directory for library source-code.

Current Roadmap

• Finish the Fermion module: This module performs variational 2-RDM code and the marginal reconstruction code for the measurement project. Need to implement the complete spin-adapted version and with T1, T2 T2', and sharp N-rep conditions.

• qubit module: Tomography from shot data or purification of qubit density matrices. Approximate reconstruction.

• Unify the density *_maps.py with Tensor objects and the appropriate dual basis object. DualBasis was designed to be general enough to perform this mapping.

• Boson representability

• Majorana representability

How to cite the representability

If you use the representability please cite the repository as follows:

bibTex:

@misc{rep2018.0.0.1,
  author = {Nicholas Rubin,
  title = {Representability},
  year = {2018},
  publisher = {GitHub},
  journal = {GitHub repository},
  howpublished = {\url{https://github.com/rigetticomputing},
  commit = {the commit you used}
}