Efficient computation of multiple environments ( reference implementation )

where the input parameters are specified as follows: tensorList is a 1× n cell array containing the n tensors which make up the tensor network. envList is a 1 × n array of integers. If an entry in envList is non-zero then the environment of the corresponding tensor is computed and returned in the specified output variable. For example, if the tensor list is {A,B,C,D,E} and envList is [0 2 0 0 1] then multienv() will return two tensors env1 and env2 corresponding to the environments of tensors E and B respectively. If an integer is repeated, the corresponding environments are added together. Thus an envList of [0 1 0 0 1] would cause multienv() to return a single tensor, being the sum of the environments of B and E. legLinks describes the tensor network using the leglabelling notation given in Ref. 1. In brief, the tensor network is represented using the customary diagrammatic notation (for which a summary may be found in Ref. 2) and an integer label is assigned to each index (represented in the diagram by a leg). Summed indices are associated with positive integer labels, while open indices are associated with negative integer labels. In the present context it is required that the tensor network have no open indices, so only positive integer labels are required. The variable legLinks is then a 1 × n cell array with each entry being a row vector whose entries are the integer labels associated with the corresponding tensor. The ordering of these labels matches the ordering of the indices on the corresponding tensor in Matlab. For example, Fig. 1(i) shows a closed diagram from the 3:1 1D MERA where all indices have been labelled with positive integers. A convention is adopted for relating the diagrammatic indices to indices in Matlab, whereby the indices of a specific Matlab tensor are associated with specific legs on the diagram. This is illustrated in Fig. 1(ii), where (for example) the topmost leg on tensor A is asFIG. 1. (i) A closed tensor network diagram arising in the optimization of the 3:1 1D MERA. (ii) Ordering of indices on the tensors of diagram (i). Ordering for B is the same as for A. Ordering for D and E is the same as for C. Ordering for H is the same as for G. Note that ordering for tensor F differs from C, D, and E because tensor F is customarily obtained from tensor C in the 3:1 1D MERA by complex conjugation and vertical reflection, and the process of reflection affects the leg ordering.