cotengra.core

Core contraction tree data structure and methods.

Classes

SliceInfo
AdderWithMaybeExponentStripped	Object that ddds two arrays, or tuples of (array, exponent) together in
ContractionTree	Binary tree representing a tensor network contraction.
ContractionTreeCompressed	A contraction tree for compressed contractions. Currently the only
PartitionTreeBuilder	Function wrapper that takes a function that partitions graphs and

Functions

cached_node_property(name)	Decorator for caching information about nodes.
legs_union(legs_seq)	Combine a sequence of legs into a single set of legs, summing their
legs_without(legs, ind)	Discard ind from legs to create a new set of legs.
get_with_default(k, obj, default)
get_slice_strides(sliced_inds)	Compute the 'strides' given the (ordered) dictionary of sliced indices.
_reconfigure_tree(tree, *args, **kwargs)
_slice_and_reconfigure_tree(tree, *args, **kwargs)
_get_tree_info(tree)
_describe_tree(tree[, info])
jitter(x, strength, rng)
jitter_dict(d, strength[, seed])
separate(xs, blocks)	Partition xs into n different list based on the corresponding

Module Contents

cotengra.core.cached_node_property(name)

Decorator for caching information about nodes.

cotengra.core.legs_union(legs_seq)

Combine a sequence of legs into a single set of legs, summing their appearances.

cotengra.core.legs_without(legs, ind)

Discard ind from legs to create a new set of legs.

cotengra.core.get_with_default(k, obj, default)

class cotengra.core.SliceInfo

inner: bool

ind: str

size: int

project: int | None

property sliced_range

cotengra.core.get_slice_strides(sliced_inds)

Compute the ‘strides’ given the (ordered) dictionary of sliced indices.

class cotengra.core.AdderWithMaybeExponentStripped

Object that ddds two arrays, or tuples of (array, exponent) together in a stable and branchless way. It also internally caches the backend on the first call.

__slots__ = ('backend', 'namespace', 'need_to_cast')

backend = None

namespace = None

need_to_cast = False

__call__(x, y)

class cotengra.core.ContractionTree(inputs, output, size_dict, track_childless=False, track_flops=False, track_write=False, track_size=False, objective=None, nodeops='auto')

Binary tree representing a tensor network contraction.

Parameters:

• inputs (sequence of str) – The list of input tensor’s indices.

• output (str) – The output indices.

• size_dict (dict[str, int]) – The size of each index.

• track_childless (bool, optional) – Whether to dynamically keep track of which nodes are childless. Useful if you are ‘divisively’ building the tree.

• track_flops (bool, optional) – Whether to dynamically keep track of the total number of flops. If False You can still compute this once the tree is complete.

• track_write (bool, optional) – Whether to dynamically keep track of the total number of elements written. If False You can still compute this once the tree is complete.

• track_size (bool, optional) – Whether to dynamically keep track of the largest tensor so far. If False You can still compute this once the tree is complete.

• objective (str or Objective, optional) – An default objective function to use for further optimization and scoring, for example reconfiguring or computing the combo cost. If not supplied the default is to create a flops objective when needed.

children

Mapping of each node to two children.

Type:

dict[node, tuple[node]]

info

Information about the tree nodes. The key is the set of inputs (a set of inputs indices) the node contains. Or in other words, the subgraph of the node. The value is a dictionary to cache information about effective ‘leg’ indices, size, flops of formation etc.

Type:

dict[node, dict]

inputs

output

N

appearances

preprocessing

children

nodeops

info

root

track_childless = False

_track_flops = False

_track_write = False

_track_size = False

already_optimized

multiplicity = 1

sliced_inds

sliced_inputs

contraction_cores

_default_objective = None

set_state_from(other)

Set the internal state of this tree to that of other.

copy()

Create a copy of this ContractionTree.

set_default_objective(objective)

Set the objective function for this tree.

get_default_objective()

Get the objective function for this tree.

get_default_combo_factor()

Get the default combo factor for this tree.

get_score(objective=None)

Score this tree using the default objective function.

property nslices

Simple alias for how many independent contractions this tree represents overall.

property nchunks

The number of ‘chunks’ - determined by the number of sliced output indices.

input_to_node(i)

Create a node from a single input index, i.e. the subgraph that only contains the input tensor i.

Parameters:

i (int) – The input index.

Returns:

node

Return type:

node_type

node_to_input(node)

Assuming node has one element, i.e. is a leaf, return the corresponding input index.

Parameters:

node (node_type) – The node to convert.

Returns:

i

Return type:

int

node_to_terms(node)

Turn a node into the corresponding terms a sequence of leaf legs, corresponding to input indices.

gen_leaves()

Generate the nodes representing leaves of the contraction tree, i.e. of size 1 each corresponding to a single input tensor.

get_incomplete_nodes()

Get the set of current nodes that have no children and the set of nodes that have no parents. These are the ‘childless’ and ‘parentless’ nodes respectively, that need to be contracted to complete the tree. The parentless nodes are grouped into the childless nodes that contain them as subgraphs.

Returns:

groups – A mapping of childless nodes to the list of parentless nodes are beneath them.

Return type:

dict[node_type, list[node_type]]

See also

autocomplete

autocomplete(**contract_opts)

Contract all remaining node groups (as computed by tree.get_incomplete_nodes) in the tree to complete it.

Parameters:

contract_opts – Options to pass to tree.contract_nodes.

See also

get_incomplete_nodes, contract_nodes

classmethod from_path(inputs, output, size_dict, *, path=None, ssa_path=None, edge_path=None, optimize='auto', autocomplete='auto', check=False, **kwargs)

Create a (completed) ContractionTree from the usual inputs plus a standard contraction path or ‘ssa_path’ - you need to supply one.

Parameters:

• inputs (Sequence[Sequence[str]]) – The input indices of each tensor, as single unicode characters.

• output (Sequence[str]) – The output indices.

• size_dict (dict[str, int]) – The size of each index.

• path (Sequence[Sequence[int]], optional) – The contraction path, a sequence of pairs of tensor ids to contract. The ids are linear indices into the list of temporary tensors, which are recycled as each contraction pops a pair and appends the result. One of path, ssa_path or edge_path must be supplied.

• ssa_path (Sequence[Sequence[int]], optional) – The contraction path, a sequence of pairs of indices to contract. The indices are single use, as if the result of each contraction is appended to the end of the list of temporary tensors without popping. One of path, ssa_path or edge_path must be supplied.

• edge_path (Sequence[str], optional) – The contraction path, a sequence of indices to contract in order. One of path, ssa_path or edge_path must be supplied.

• optimize (str, optional) – If a contraction within the path contains 3 or more tensors, how to optimize this subcontraction into a binary tree.

• autocomplete ("auto" or bool, optional) – Whether to automatically complete the path, i.e. contract all remaining nodes. If “auto” then a warning is issued if the path is not complete.

• check (bool, optional) – Whether to perform some basic checks while creating the contraction nodes.

Return type:

ContractionTree

classmethod from_info(info, **kwargs)

Create a ContractionTree from an opt_einsum.PathInfo object.

classmethod from_eq(eq, size_dict, **kwargs)

Create a empty ContractionTree directly from an equation and set of shapes.

Parameters:

• eq (str) – The einsum string equation.

• size_dict (dict[str, int]) – The size of each index.

get_eq()

Get the einsum equation corresponding to this tree. Note that this is the total (or original) equation, so includes indices which have been sliced.

Returns:

eq

Return type:

str

get_shapes()

Get the shapes of the input tensors corresponding to this tree.

Returns:

shapes

Return type:

tuple[tuple[int]]

get_inputs_sliced()

Get the input indices corresponding to a single slice of this tree, i.e. with sliced indices removed.

Returns:

inputs

Return type:

tuple[tuple[str]]

get_output_sliced()

Get the output indices corresponding to a single slice of this tree, i.e. with sliced indices removed.

Returns:

output

Return type:

tuple[str]

get_eq_sliced()

Get the einsum equation corresponding to a single slice of this tree, i.e. with sliced indices removed.

Returns:

eq

Return type:

str

get_shapes_sliced()

Get the shapes of the input tensors corresponding to a single slice of this tree, i.e. with sliced indices removed.

Returns:

shapes

Return type:

tuple[tuple[int]]

classmethod from_edge_path(edge_path, inputs, output, size_dict, optimize='auto', autocomplete='auto', check=False, **kwargs)

Create a ContractionTree from an edge elimination ordering.

_add_node(node, check=False, **kwargs)

Add a node to this tree, specified either directly as a existing node type, or as a subgraph (i.e. a sequence of input positions) which is then converted to a node with the corresponding extent and subgraph information.

Note if “ssa” nodes are used, then adding two equivalent subgraphs will result in two new nodes, since the node labels do not themselves encode the subgraph information.

Parameters:

• node (node_type or Sequence[int]) – The node to add, either directly as a node type, or as a subgraph specified by the sequence of input positions it contains.

• check (bool, optional) – Whether to perform some basic checks on the node and tree state before adding the node.

• kwargs (dict, optional) – Additional information to cache about this node, for example its ‘extent’ or ‘subgraph’. If it is being specified as a sequence of input positions, these two will be injected automatically.

Returns:

node – The node that was added, which may be different from the input if the input was specified as a sequence of input positions.

Return type:

node_type

_remove_node(node)

Remove node from this tree and update the flops and maximum size if tracking them respectively, as well as input pre-processing.

compute_leaf_legs(i)

Compute the effective ‘outer’ indices for the ith input tensor. This is not always simply the ith input indices, due to A) potential slicing and B) potential preprocessing.

has_preprocessing()

has_hyper_indices()

Check if there are any ‘hyper’ indices in the contraction, i.e. indices that don’t appear exactly twice, when considering the inputs and output.

get_extent(node)

Get the number of input tensors contained in the subgraph represented by node.

Parameters:

node (node_type) – The node to compute the extent of.

Returns:

extent

Return type:

int

get_subgraph(node) → tuple[int, Ellipsis]

Get the sequence of input tensors contained in subgraph represented by node.

Parameters:

node (node_type) – The node to compute the subgraph of.

Returns:

subgraph – The input tensor indices contained in this subgraph.

Return type:

tuple[int]

get_legs(node)

Get the effective ‘outer’ indices for the collection of tensors in node.

get_involved(node)

Get all the indices involved in the formation of subgraph node.

get_size(node)

Get the tensor size of node.

get_flops(node)

Get the FLOPs for the pairwise contraction that will create node.

get_can_dot(node)

Get whether this contraction can be performed as a dot product (i.e. with tensordot), or else requires einsum, as it has indices that don’t appear exactly twice in either the inputs or the output.

get_inds(node)

Get the indices of this node - an ordered string version of get_legs that starts with tree.inputs and maintains the order they appear in each contraction ‘ABC,abc->ABCabc’, to match tensordot.

get_tensordot_axes(node)

Get the axes arg for a tensordot ocontraction that produces node. The pairs are sorted in order of appearance on the left input.

get_tensordot_perm(node)

Get the permutation required, if any, to bring the tensordot output of this nodes contraction into line with self.get_inds(node).

get_einsum_eq(node)

Get the einsum string describing the contraction that produces node, unlike get_inds the characters are mapped into [a-zA-Z], for compatibility with numpy.einsum for example.

is_leaf(node)

Check if node is a leaf node in this tree.

Parameters:

node (node_type) – The node to check.

Returns:

is_leaf

Return type:

bool

is_root(node)

Check if node is the root node in this tree.

Parameters:

node (node_type) – The node to check.

Returns:

is_root

Return type:

bool

is_descendant(node, ancestor)

Check if node is a descendant of ancestor in this tree.

Parameters:

• node (node_type) – The node to check.

• ancestor (node_type) – The potential ancestor node.

Returns:

is_descendant

Return type:

bool

get_peak_size(node)

Get the peak size for all but only the contractions required to produce node. The value for the root note will be the peak size of the entire contraction.

Parameters:

node (node_type) – The node to compute the peak size of.

Returns:

peak_size

Return type:

int

reorder_for_peak_size()

This reorders the depth first traversal of the tree to minimize the peak size of the contraction.

get_centrality(node)

total_flops(dtype=None, log=None)

Sum the flops contribution from every node in the tree.

Parameters:

dtype ({'float', 'complex', None}, optional) – Scale the answer depending on the assumed data type.

total_write()

Sum the total amount of memory that will be created and operated on.

combo_cost(factor=DEFAULT_COMBO_FACTOR, combine=sum, log=None)

total_cost

max_size(log=None)

The size of the largest intermediate tensor.

max_contraction_size(log=None)

The maximum size of a single contraction in the tree. This includes the size of the two input tensors and the output tensor, and can be a more practical measure of the peak memory required.

Parameters:

log (float, optional) – If provided, return the log of the size to this base.

Returns:

size – The maximum size of a single contraction in the tree, or its log.

Return type:

int or float

peak_size(order=None, log=None)

Get the peak concurrent size of tensors needed - this depends on the traversal order, i.e. the exact contraction path, not just the contraction tree.

contract_stats(force=False)

Simulteneously compute the total flops, write and size of the contraction tree. This is more efficient than calling each of the individual methods separately. Once computed, each quantity is then automatically tracked.

Returns:

stats – The total flops, write and size.

Return type:

dict[str, int]

arithmetic_intensity()

The ratio of total flops to total write - the higher the better for extracting good computational performance.

contraction_scaling()

This is computed simply as the maximum number of indices involved in any single contraction, which will match the scaling assuming that all dimensions are equal.

contraction_cost(log=None)

Get the total number of scalar operations ~ time complexity.

naive_cost(log=None)

Get the naive cost of performing this contraction as a single einsum summation, without any intermediate contractions. This is given the as product of the size of all indices.

Parameters:

log (float, optional) – If provided, return log of the cost to this base.

speedup(log=None)

Speedup compared to naive summation.

Parameters:

log (float, optional) – If provided, return log of the speedup to this base.

contraction_width(log=2)

Get log2 of the size of the largest tensor.

compressed_contract_stats(chi=None, order='surface_order', compress_late=None)

total_flops_compressed(chi=None, order='surface_order', compress_late=None, dtype=None, log=None)

Estimate the total flops for a compressed contraction of this tree with maximum bond size chi. This includes basic estimates of the ops to perform contractions, QRs and SVDs.

contraction_cost_compressed

total_write_compressed(chi=None, order='surface_order', compress_late=None, log=None)

Compute the total size of all intermediate tensors when a compressed contraction is performed with maximum bond size chi, ordered by order. This is relevant maybe for time complexity and e.g. autodiff space complexity (since every intermediate is kept).

combo_cost_compressed(chi=None, order='surface_order', compress_late=None, factor=None, log=None)

total_cost_compressed

max_size_compressed(chi=None, order='surface_order', compress_late=None, log=None)

Compute the maximum sized tensor produced when a compressed contraction is performed with maximum bond size chi, ordered by order. This is close to the ideal space complexity if only tensors that are being directly operated on are kept in memory.

peak_size_compressed(chi=None, order='surface_order', compress_late=None, accel='auto', log=None)

Compute the peak size of combined intermediate tensors when a compressed contraction is performed with maximum bond size chi, ordered by order. This is the practical space complexity if one is not swapping intermediates in and out of memory.

contraction_width_compressed(chi=None, order='surface_order', compress_late=None, log=2)

Compute log2 of the maximum sized tensor produced when a compressed contraction is performed with maximum bond size chi, ordered by order.

_update_tracked(node)

contract_nodes_pair(x, y, legs=None, cost=None, size=None, parent=None, check=False)

Contract node x with node y in the tree to create a new parent node, which is returned.

Parameters:

• x (node_type) – The first node to contract.

• y (node_type) – The second node to contract.

• legs (dict[str, int], optional) – The effective ‘legs’ of the new node if already known. If not given, this is computed from the inputs of x and y.

• cost (int, optional) – The cost of the contraction if already known. If not given, this is computed from the inputs of x and y.

• size (int, optional) – The size of the new node if already known. If not given, this is computed from the inputs of x and y.

• check (bool, optional) – Whether to check the inputs are valid.

Returns:

parent – The new parent node of x and y.

Return type:

node_type

contract_nodes(nodes, optimize='auto', grandparent=None, check=False, extra_opts=None)

Contract an arbitrary number of nodes in the tree to build up a subtree. The root of this subtree (a new intermediate) is returned.

is_complete()

Check every node has two children, unless it is a leaf.

get_default_order()

_traverse_dfs()

Traverse the tree in a depth first, non-recursive, order.

_traverse_ordered(order)

Traverse the tree in the order that minimizes order(node), but still constrained to produce children before parents.

traverse(order=None)

Generate, in order, all the node merges in this tree. Non-recursive! This ensures children are always visited before their parent.

Parameters:

order (None, "dfs", or callable, optional) – How to order the contractions within the tree. If a callable is given (which should take a node as its argument), try to contract nodes that minimize this function first.

Returns:

The bottom up ordered sequence of tree merges, each a tuple of (parent, left_child, right_child).

Return type:

generator[tuple[node]]

See also

descend

descend(mode='dfs')

Generate, from root to leaves, all the node merges in this tree. Non-recursive! This ensures parents are visited before their children.

Parameters:

mode ({'dfs', bfs}, optional) – How expand from a parent.

Returns:

The top down ordered sequence of tree merges, each a tuple of (parent, left_child, right_child).

Return type:

generator[tuple[node]

See also

traverse

get_subtree(node, size, search='bfs', seed=None)

Get a subtree spanning down from node which will have size leaves (themselves not necessarily leaves of the actual tree).

Parameters:

• node (node_type) – The node of the tree to start with.

• size (int) – How many subtree leaves to aim for.

• search ({'bfs', 'dfs', 'random'}, optional) –

  How to build the tree:

  • ’bfs’: breadth first expansion

  • ’dfs’: depth first expansion (largest nodes first)

  • ’random’: random expansion

• seed (None, int or random.Random, optional) – Random number generator seed, if search is ‘random’.

Returns:

• sub_leaves (tuple[node_type]) – Nodes which are subtree leaves.

• branches (tuple[node_type]) – Nodes which are between the subtree leaves and root.

remove_ind(ind, project=None, inplace=False)

Remove (i.e. by default slice) index ind from this contraction tree, taking care to update all relevant information about each node.

remove_ind_

restore_ind(ind, inplace=False)

Restore (unslice or un-project) index ind to this contraction tree, taking care to update all relevant information about each node.

Parameters:

• ind (str) – The index to restore.

• inplace (bool, optional) – Whether to perform the restoration inplace or not.

Return type:

ContractionTree

restore_ind_

unslice_rand(seed=None, inplace=False)

Unslice (restore) a random index from this contraction tree.

Parameters:

• seed (None, int or random.Random, optional) – Random number generator seed.

• inplace (bool, optional) – Whether to perform the unslicing inplace or not.

Return type:

ContractionTree

unslice_rand_

unslice_all(inplace=False)

Unslice (restore) all sliced indices from this contraction tree.

Parameters:

inplace (bool, optional) – Whether to perform the unslicing inplace or not.

Return type:

ContractionTree

unslice_all_

calc_subtree_candidates(pwr=2, what='flops')

_subtree_remove_and_optimize(sub_root, sub_leaves, sub_branches, already_optimized, node_cost, minimize, opt, pbar)

_subtree_reconfigure_descend(subtree_size, subtree_search, maxiter, seed, minimize, opt, already_optimized, node_cost, pbar)

_subtree_reconfigure_rand_select(subtree_size, subtree_search, weight_what, weight_pwr, select, maxiter, seed, minimize, opt, already_optimized, node_cost, pbar)

subtree_reconfigure(subtree_size=8, subtree_search='bfs', weight_what='flops', weight_pwr=2, select='max', maxiter='auto', maxiter_auto_cap=1024, seed=None, minimize=None, optimize=None, inplace=False, progbar=False)

Reconfigure subtrees of this tree with locally optimal paths.

Parameters:

• subtree_size (int, optional) – The size of subtree to consider. Cost is exponential in this.

• subtree_search ({'bfs', 'dfs', 'random'}, optional) –

  How to build the subtrees:

  • ’bfs’: breadth-first-search creating balanced subtrees

  • ’dfs’: depth-first-search creating imbalanced subtrees

  • ’random’: random subtree building

• weight_what ({'flops', 'size'}, optional) – When assessing nodes to build and optimize subtrees from whether to score them by the (local) contraction cost, or tensor size.

• weight_pwr (int, optional) – When assessing nodes to build and optimize subtrees from, how to scale their score into a probability: score**(1 / weight_pwr). The larger this is the more explorative the algorithm is when select='random'.

• select ({'descend', 'max', 'min', 'random'}, optional) –

  What order to select node subtrees to optimize:

  • ’descend’: start from the root and then descend into children. In this case the weights and weight_pwr are ignored since this is a deterministic order.

  • ’max’: choose the highest score first

  • ’min’: choose the lowest score first

  • ’random’: choose randomly weighted on score - see weight_pwr.

• maxiter (int, optional) – How many subtree optimizations to perform, the algorithm can terminate before this if all subtrees have been optimized. If ‘auto’, defaults to min(tree.N, maxiter_auto_cap)

• maxiter_auto_cap (int, optional) – The maximum cap to apply to the default value of maxiter when maxiter='auto'.

• seed (int, optional) – A random seed (seeds python system random module).

• minimize ({'flops', 'size'}, optional) – Whether to minimize with respect to contraction flops or size.

• inplace (bool, optional) – Whether to perform the reconfiguration inplace or not.

• progbar (bool, optional) – Whether to show live progress of the reconfiguration.

Return type:

ContractionTree

subtree_reconfigure_

subtree_reconfigure_forest(num_trees=8, num_restarts=10, restart_fraction=0.5, subtree_maxiter=100, subtree_size=10, subtree_search=('random', 'bfs'), subtree_select=('random',), subtree_weight_what=('flops', 'size'), subtree_weight_pwr=(2,), parallel='auto', parallel_maxiter_steps=4, minimize=None, seed=None, progbar=False, inplace=False)

‘Forested’ version of subtree_reconfigure which is more explorative and can be parallelized. It stochastically generates a ‘forest’ reconfigured trees, then only keeps some fraction of these to generate the next forest.

Parameters:

• num_trees (int, optional) – The number of trees to reconfigure at each stage.

• num_restarts (int, optional) – The number of times to halt, prune and then restart the tree reconfigurations.

• restart_fraction (float, optional) – The fraction of trees to keep at each stage and generate the next forest from.

• subtree_maxiter (int, optional) – Number of subtree reconfigurations per step. num_restarts * subtree_maxiter is the max number of total subtree reconfigurations for the final tree produced.

• subtree_size (int, optional) – The size of subtrees to search for and reconfigure.

• subtree_search (tuple[{'random', 'bfs', 'dfs'}], optional) – Tuple of options for the search kwarg of ContractionTree.subtree_reconfigure() to randomly sample.

• subtree_select (tuple[{'random', 'max', 'min'}], optional) – Tuple of options for the select kwarg of ContractionTree.subtree_reconfigure() to randomly sample.

• subtree_weight_what (tuple[{'flops', 'size'}], optional) – Tuple of options for the weight_what kwarg of ContractionTree.subtree_reconfigure() to randomly sample.

• subtree_weight_pwr (tuple[int], optional) – Tuple of options for the weight_pwr kwarg of ContractionTree.subtree_reconfigure() to randomly sample.

• parallel ('auto', False, True, int, or distributed.Client) – Whether to parallelize the search.

• parallel_maxiter_steps (int, optional) – If parallelizing, how many steps to break each reconfiguration into in order to evenly saturate many processes.

• minimize ({'flops', 'size', ..., Objective}, optional) – Whether to minimize the total flops or maximum size of the contraction tree.

• seed (None, int or random.Random, optional) – A random seed to use.

• progbar (bool, optional) – Whether to show live progress.

• inplace (bool, optional) – Whether to perform the subtree reconfiguration inplace.

Return type:

ContractionTree

subtree_reconfigure_forest_

simulated_anneal

simulated_anneal_

parallel_temper

parallel_temper_

slice(target_size=None, target_overhead=None, target_slices=None, temperature=0.01, minimize=None, allow_outer=True, max_repeats=16, reslice=False, seed=None, inplace=False)

Slice this tree (turn some indices into indices which are explicitly summed over rather than being part of contractions). The indices are stored in tree.sliced_inds, and the contraction width updated to take account of the slicing. Calling tree.contract(arrays) moreover which automatically perform the slicing and summation.

Parameters:

• target_size (int, optional) – The target number of entries in the largest tensor of the sliced contraction. The search algorithm will terminate after this is reached.

• target_slices (int, optional) – The target or minimum number of ‘slices’ to consider - individual contractions after slicing indices. The search algorithm will terminate after this is breached. This is on top of the current number of slices.

• target_overhead (float, optional) – The target increase in total number of floating point operations. For example, a value of 2.0 will terminate the search just before the cost of computing all the slices individually breaches twice that of computing the original contraction all at once.

• temperature (float, optional) – How much to randomize the repeated search.

• minimize ({'flops', 'size', ..., Objective}, optional) – Which metric to score the overhead increase against.

• allow_outer (bool, optional) – Whether to allow slicing of outer indices.

• max_repeats (int, optional) – How many times to repeat the search with a slight randomization.

• reslice (bool, optional) – Whether to reslice the tree, i.e. first remove all currently sliced indices and start the search again. Generally any ‘good’ sliced indices will be easily found again.

• seed (None, int or random.Random, optional) – A random seed or generator to use for the search.

• inplace (bool, optional) – Whether the remove the indices from this tree inplace or not.

Return type:

ContractionTree

See also

SliceFinder, ContractionTree.slice_and_reconfigure

slice_

slice_and_reconfigure(target_size, step_size=2, temperature=0.01, minimize=None, allow_outer=True, max_repeats=16, reslice=False, reconf_opts=None, progbar=False, inplace=False)

Interleave slicing (removing indices into an exterior sum) with subtree reconfiguration to minimize the overhead induced by this slicing.

Parameters:

• target_size (int) – Slice the tree until the maximum intermediate size is this or smaller.

• step_size (int, optional) – The minimum size reduction to try and achieve before switching to a round of subtree reconfiguration.

• temperature (float, optional) – The temperature to supply to SliceFinder for searching for indices.

• minimize ({'flops', 'size', ..., Objective}, optional) – The metric to minimize when slicing and reconfiguring subtrees.

• max_repeats (int, optional) – The number of slicing attempts to perform per search.

• progbar (bool, optional) – Whether to show live progress.

• inplace (bool, optional) – Whether to perform the slicing and reconfiguration inplace.

• reconf_opts (None or dict, optional) – Supplied to ContractionTree.subtree_reconfigure() or ContractionTree.subtree_reconfigure_forest(), depending on ‘forested’ key value.

slice_and_reconfigure_

slice_and_reconfigure_forest(target_size, step_size=2, num_trees=8, restart_fraction=0.5, temperature=0.02, max_repeats=32, reslice=False, minimize=None, allow_outer=True, parallel='auto', progbar=False, inplace=False, reconf_opts=None)

‘Forested’ version of ContractionTree.slice_and_reconfigure(). This maintains a ‘forest’ of trees with different slicing and subtree reconfiguration attempts, pruning the worst at each step and generating a new forest from the best.

Parameters:

• target_size (int) – Slice the tree until the maximum intermediate size is this or smaller.

• step_size (int, optional) – The minimum size reduction to try and achieve before switching to a round of subtree reconfiguration.

• num_restarts (int, optional) – The number of times to halt, prune and then restart the tree reconfigurations.

• restart_fraction (float, optional) – The fraction of trees to keep at each stage and generate the next forest from.

• temperature (float, optional) – The temperature at which to randomize the sliced index search.

• max_repeats (int, optional) – The number of slicing attempts to perform per search.

• parallel ('auto', False, True, int, or distributed.Client) – Whether to parallelize the search.

• progbar (bool, optional) – Whether to show live progress.

• inplace (bool, optional) – Whether to perform the slicing and reconfiguration inplace.

• reconf_opts (None or dict, optional) – Supplied to ContractionTree.slice_and_reconfigure().

Return type:

ContractionTree

slice_and_reconfigure_forest_

compressed_reconfigure(minimize=None, order_only=False, max_nodes='auto', max_time=None, local_score=None, exploration_power=0, best_score=None, progbar=False, inplace=False)

Reconfigure this tree according to peak_size_compressed.

Parameters:

• chi (int) – The maximum bond dimension to consider.

• order_only (bool, optional) – Whether to only consider the ordering of the current tree contractions, or all possible contractions, starting with the current.

• max_nodes (int, optional) – Set the maximum number of contraction steps to consider.

• max_time (float, optional) – Set the maximum time to spend on the search.

• local_score (callable, optional) –

  A function that assigns a score to a potential contraction, with a lower score giving more priority to explore that contraction earlier. It should have signature:

  local_score(step, new_score, dsize, new_size)
  

  where step is the number of steps so far, new_score is the score of the contraction so far, dsize is the change in memory by the current step, and new_size is the new memory size after contraction.

• exploration_power (float, optional) – If not 0.0, the inverse power to which the step is raised in the default local score function. Higher values favor exploring more promising branches early on - at the cost of increased memory. Ignored if local_score is supplied.

• best_score (float, optional) – Manually specify an upper bound for best score found so far.

• progbar (bool, optional) – If True, display a progress bar.

• inplace (bool, optional) – Whether to perform the reconfiguration inplace on this tree.

Return type:

ContractionTree

compressed_reconfigure_

windowed_reconfigure(minimize=None, order_only=False, window_size=20, max_iterations=100, max_window_tries=1000, score_temperature=0.0, queue_temperature=1.0, scorer=None, queue_scorer=None, seed=None, inplace=False, progbar=False, **kwargs)

windowed_reconfigure_

flat_tree(order=None)

Create a nested tuple representation of the contraction tree like:

((0, (1, 2)), ((3, 4), ((5, (6, 7)), (8, 9))))

Such that the contraction will progress like:

((0, (1, 2)), ((3, 4), ((5, (6, 7)), (8, 9))))
((0, 12), (34, ((5, 67), 89)))
(012, (34, (567, 89)))
(012, (34, 56789))
(012, 3456789)
0123456789

Where each integer represents a leaf (i.e. single element node).

get_leaves_ordered()

Return the list of leaves as ordered by the contraction tree.

Return type:

tuple[node_type]

get_path(order=None)

Generate a standard path (with linear recycled ids) from the contraction tree.

Parameters:

order (None, "dfs", or callable, optional) – How to order the contractions within the tree. If a callable is given (which should take a node as its argument), try to contract nodes that minimize this function first.

Returns:

path

Return type:

tuple[tuple[int, int]]

path

get_numpy_path(order=None)

Generate a path compatible with the optimize kwarg of numpy.einsum.

get_ssa_path(order=None)

Generate a single static assignment path from the contraction tree.

Parameters:

order (None, "dfs", or callable, optional) – How to order the contractions within the tree. If a callable is given (which should take a node as its argument), try to contract nodes that minimize this function first.

Returns:

ssa_path

Return type:

tuple[tuple[int, int]]

ssa_path

surface_order(node)

set_surface_order_from_path(ssa_path)

get_path_surface()

path_surface

get_ssa_path_surface()

ssa_path_surface

get_spans()

Get all (which could mean none) potential embeddings of this contraction tree into a spanning tree of the original graph.

Return type:

tuple[dict[frozenset[int], frozenset[int]]]

compute_centralities(combine='mean')

Compute a centrality for every node in this contraction tree.

get_hypergraph(accel=False)

Get a hypergraph representing the uncontracted network (i.e. the leaves).

reset_contraction_indices()

Reset all information regarding a) the explicit contraction indices ordering and b) cached contraction expressions. This should probably be called any time structural changes are made to the tree, e.g. reconfiguration.

sort_contraction_indices(priority='flops', make_output_contig=True, make_contracted_contig=True, reset=True)

Set explicit orders for the contraction indices of this self to optimize for one of two things: contiguity in contracted (‘k’) indices, or contiguity of left and right output (‘m’ and ‘n’) indices.

Parameters:

• priority ({'flops', 'size', 'root', 'leaves'}, optional) – Which order to process the intermediate nodes in. Later nodes re-sort previous nodes so are more likely to keep their ordering. E.g. for ‘flops’ the mostly costly contracton will be process last and thus will be guaranteed to have its indices exactly sorted.

• make_output_contig (bool, optional) – When processing a pairwise contraction, sort the parent contraction indices so that the order of indices is the order they appear from left to right in the two child (input) tensors.

• make_contracted_contig (bool, optional) – When processing a pairwise contraction, sort the child (input) tensor indices so that all contracted indices appear contiguously.

• reset (bool, optional) – Reset all indices to the default order before sorting.

print_contractions(sort=None, show_brackets=True)

Print each pairwise contraction, with colorized indices (if colorama is installed), and other information. The color codes are:

• blue: index appears on left and is kept

• green: index appears on right and is kept

• red: contracted index: appears on both sides and is removed

• pink: batch index: appears on both sides and is kept

Any trivial indices that appear only on one term and not in the output are removed and shown by the preprocessing steps.

Parameters:

• sort ({'flops', 'size'}, optional) – Sort the contractions by either the number of floating point operations or the size of the intermediate tensor. By default the contraction are show in the order they are performed.

• show_brackets (bool, optional) – Whether to show the brackets around contiguous sections of the same type of indices.

get_contractor(order=None, prefer_einsum=False, strip_exponent=False, check_zero=False, implementation=None, autojit=False, progbar=False)

Get a reusable function which performs the contraction corresponding to this tree, cached.

Parameters:

• tree (ContractionTree) – The contraction tree.

• order (str or callable, optional) – Supplied to ContractionTree.traverse(), the order in which to perform the pairwise contractions given by the tree.

• prefer_einsum (bool, optional) – Prefer to use einsum for pairwise contractions, even if tensordot can perform the contraction.

• strip_exponent (bool, optional) – If True, the function will eagerly strip the exponent (in log10) from intermediate tensors to control numerical problems from leaving the range of the datatype. This method then returns the scaled ‘mantissa’ output array and the exponent separately.

• check_zero (bool, optional) – If True, when strip_exponent=True, explicitly check for zero-valued intermediates that would otherwise produce nan, instead terminating early if encountered and returning (0.0, 0.0).

• implementation (str or tuple[callable, callable], optional) –

  What library to use to actually perform the contractions. Options are:

  • None: let cotengra choose.

  • ”autoray”: dispatch with autoray, using the tensordot and einsum implementation of the backend.

  • ”cotengra”: use the tensordot and einsum implementation of cotengra, which is based on batch matrix multiplication. This is faster for some backends like numpy, and also enables libraries which don’t yet provide tensordot and einsum to be used.

  • ”cuquantum”: use the cuquantum library to perform the whole contraction (not just individual contractions).

  • tuple[callable, callable]: manually supply the tensordot and einsum implementations to use.

• autojit (bool, optional) – If True, use autoray.autojit() to compile the contraction function.

• progbar (bool, optional) – Whether to show progress through the contraction by default.

Returns:

fn – The contraction function, with signature fn(*arrays).

Return type:

callable

contract_core(arrays, order=None, prefer_einsum=False, strip_exponent=False, check_zero=False, backend=None, implementation=None, autojit='auto', progbar=False)

Contract arrays with this tree. The order of the axes and output is assumed to be that of tree.inputs and tree.output, but with sliced indices removed. This functon contracts the core tree and thus if indices have been sliced the arrays supplied need to be sliced as well.

Parameters:

• arrays (sequence of array) – The arrays to contract.

• order (str or callable, optional) – Supplied to ContractionTree.traverse().

• prefer_einsum (bool, optional) – Prefer to use einsum for pairwise contractions, even if tensordot can perform the contraction.

• backend (str, optional) – What library to use for einsum and transpose, will be automatically inferred from the arrays if not given.

• autojit ("auto" or bool, optional) – Whether to use autoray.autojit to jit compile the expression. If “auto”, then let cotengra choose.

• progbar (bool, optional) – Show progress through the contraction.

slice_key(i, strides=None)

Get the combination of sliced index values for overall slice i.

Parameters:

i (int) – The overall slice index.

Returns:

key – The value each sliced index takes for slice i.

Return type:

dict[str, int]

slice_arrays(arrays, i)

Take arrays and slice the relevant inputs according to tree.sliced_inds and the dynary representation of i.

contract_slice(arrays, i, **kwargs)

Get slices i of arrays and then contract them.

gather_slices(slices, backend=None, progbar=False)

Gather all the output contracted slices into a single full result. If none of the sliced indices appear in the output, then this is a simple sum - otherwise the slices need to be partially summed and partially stacked.

gen_output_chunks(arrays, with_key=False, progbar=False, **contract_opts)

Generate each output chunk of the contraction - i.e. take care of summing internally sliced indices only first. This assumes that the sliced_inds are sorted by whether they appear in the output or not (the default order). Useful for performing some kind of reduction over the final tensor object like fn(x).sum() without constructing the entire thing.

Parameters:

• arrays (sequence of array) – The arrays to contract.

• with_key (bool, optional) – Whether to yield the output index configuration key along with the chunk.

• progbar (bool, optional) – Show progress through the contraction chunks.

Yields:

• chunk (array) – A chunk of the contracted result.

• key (dict[str, int]) – The value each sliced output index takes for this chunk.

contract(arrays, order=None, prefer_einsum=False, strip_exponent=False, check_zero=False, backend=None, implementation=None, autojit='auto', progbar=False)

Contract arrays with this tree. This function takes unsliced arrays and handles the slicing, contractions and gathering. The order of the axes and output is assumed to match that of tree.inputs and tree.output.

Parameters:

• arrays (sequence of array) – The arrays to contract.

• order (str or callable, optional) – Supplied to ContractionTree.traverse().

• prefer_einsum (bool, optional) – Prefer to use einsum for pairwise contractions, even if tensordot can perform the contraction.

• strip_exponent (bool, optional) – If True, eagerly strip the exponent (in log10) from intermediate tensors to control numerical problems from leaving the range of the datatype. This method then returns the scaled ‘mantissa’ output array and the exponent separately.

• check_zero (bool, optional) – If True, when strip_exponent=True, explicitly check for zero-valued intermediates that would otherwise produce nan, instead terminating early if encountered and returning (0.0, 0.0).

• backend (str, optional) – What library to use for tensordot, einsum and transpose, it will be automatically inferred from the input arrays if not given.

• autojit (bool, optional) – Whether to use the ‘autojit’ feature of autoray to compile the contraction expression.

• progbar (bool, optional) – Whether to show a progress bar.

Returns:

• output (array) – The contracted output, it will be scaled if strip_exponent==True.

• exponent (float) – The exponent of the output in base 10, returned only if strip_exponent==True.

See also

contract_core, contract_slice, slice_arrays, gather_slices

contract_mpi(arrays, comm=None, root=None, **kwargs)

Contract the slices of this tree and sum them in parallel - assuming we are already running under MPI.

Parameters:

• arrays (sequence of array) – The input (unsliced arrays)

• comm (None or mpi4py communicator) – Defaults to mpi4py.MPI.COMM_WORLD if not given.

• root (None or int, optional) – If root=None, an Allreduce will be performed such that every process has the resulting tensor, else if an integer e.g. root=0, the result will be exclusively gathered to that process using Reduce, with every other process returning None.

• kwargs – Supplied to contract_slice().

benchmark(dtype='float64', max_time=60, min_reps=3, max_reps=100, warmup=True, **contract_opts)

Benchmark the contraction of this tree.

Parameters:

• dtype ({"float32", "float64", "complex64", "complex128"}) – The datatype to use.

• max_time (float, optional) – The maximum time to spend benchmarking in seconds.

• min_reps (int, optional) – The minimum number of repetitions to perform, regardless of time.

• max_reps (int, optional) – The maximum number of repetitions to perform, regardless of time.

• warmup (bool or int, optional) – Whether to perform a warmup run before the benchmark. If an int, the number of warmup runs to perform.

• contract_opts – Supplied to contract_slice().

Returns:

A dictionary of benchmarking results. The keys are:

• ”time_per_slice”float

  The average time to contract a single slice.

• ”est_time_total”float

  The estimated total time to contract all slices.

• ”est_gigaflops”float

  The estimated gigaflops of the contraction.

Return type:

dict

See also

contract_slice

plot_ring

plot_tent

plot_span

plot_flat

plot_circuit

plot_rubberband

plot_contractions

plot_contractions_alt

plot_hypergraph(**kwargs)

describe(info='normal', join=' ')

Return a string describing the contraction tree.

__repr__()

__str__()

cotengra.core._reconfigure_tree(tree, *args, **kwargs)

cotengra.core._slice_and_reconfigure_tree(tree, *args, **kwargs)

cotengra.core._get_tree_info(tree)

cotengra.core._describe_tree(tree, info='normal')

class cotengra.core.ContractionTreeCompressed(inputs, output, size_dict, track_childless=False, track_flops=False, track_write=False, track_size=False, objective=None, nodeops='auto')

Bases: ContractionTree

A contraction tree for compressed contractions. Currently the only difference is that this defaults to the ‘surface’ traversal ordering.

set_state_from(other)

Set the internal state of this tree to that of other.

classmethod from_path(inputs, output, size_dict, *, path=None, ssa_path=None, autocomplete='auto', check=False, **kwargs)

Create a (completed) ContractionTreeCompressed from the usual inputs plus a standard contraction path or ‘ssa_path’ - you need to supply one. This also set the default ‘surface’ traversal ordering to be the initial path.

get_default_order()

get_default_objective()

Get the objective function for this tree.

get_default_chi()

get_default_compress_late()

total_flops

Sum the flops contribution from every node in the tree.

Parameters:

dtype ({'float', 'complex', None}, optional) – Scale the answer depending on the assumed data type.

total_write

Sum the total amount of memory that will be created and operated on.

combo_cost

total_cost

max_size

The size of the largest intermediate tensor.

peak_size

Get the peak concurrent size of tensors needed - this depends on the traversal order, i.e. the exact contraction path, not just the contraction tree.

contraction_cost

Get the total number of scalar operations ~ time complexity.

contraction_width

Get log2 of the size of the largest tensor.

total_flops_exact

total_write_exact

combo_cost_exact

total_cost_exact

max_size_exact

peak_size_exact

abstractmethod get_contractor(*_, **__)

Get a reusable function which performs the contraction corresponding to this tree, cached.

Parameters:

• tree (ContractionTree) – The contraction tree.

• order (str or callable, optional) – Supplied to ContractionTree.traverse(), the order in which to perform the pairwise contractions given by the tree.

• prefer_einsum (bool, optional) – Prefer to use einsum for pairwise contractions, even if tensordot can perform the contraction.

• strip_exponent (bool, optional) – If True, the function will eagerly strip the exponent (in log10) from intermediate tensors to control numerical problems from leaving the range of the datatype. This method then returns the scaled ‘mantissa’ output array and the exponent separately.

• check_zero (bool, optional) – If True, when strip_exponent=True, explicitly check for zero-valued intermediates that would otherwise produce nan, instead terminating early if encountered and returning (0.0, 0.0).

• implementation (str or tuple[callable, callable], optional) –

  What library to use to actually perform the contractions. Options are:

  • None: let cotengra choose.

  • ”autoray”: dispatch with autoray, using the tensordot and einsum implementation of the backend.

  • ”cotengra”: use the tensordot and einsum implementation of cotengra, which is based on batch matrix multiplication. This is faster for some backends like numpy, and also enables libraries which don’t yet provide tensordot and einsum to be used.

  • ”cuquantum”: use the cuquantum library to perform the whole contraction (not just individual contractions).

  • tuple[callable, callable]: manually supply the tensordot and einsum implementations to use.

• autojit (bool, optional) – If True, use autoray.autojit() to compile the contraction function.

• progbar (bool, optional) – Whether to show progress through the contraction by default.

Returns:

fn – The contraction function, with signature fn(*arrays).

Return type:

callable

simulated_anneal(minimize=None, tfinal=0.0001, tstart=0.01, tsteps=50, numiter=50, seed=None, inplace=False, progbar=False, **kwargs)

Perform simulated annealing refinement of this compressed contraction tree.

simulated_anneal_

class cotengra.core.PartitionTreeBuilder(partition_fn)

Function wrapper that takes a function that partitions graphs and uses it to build a contraction tree. partition_fn should have signature:

def partition_fn(inputs, output, size_dict,

weight_nodes, weight_edges, **kwargs):

… return membership

Where weight_nodes and weight_edges decsribe how to weight the nodes and edges of the graph respectively and membership should be a list of integers of length len(inputs) labelling which partition each input node should be put it.

partition_fn

build_divide(inputs, output, size_dict, random_strength=0.01, cutoff=10, parts=2, parts_decay=0.5, sub_optimize='greedy', super_optimize='random-greedy-128', check=False, seed=None, **partition_opts)

build_agglom(inputs, output, size_dict, random_strength=0.01, groupsize=4, sub_optimize='greedy', check=False, seed=None, **partition_opts)

trial_fn(inputs, output, size_dict, **partition_opts)

trial_fn_agglom(inputs, output, size_dict, **partition_opts)

cotengra.core.jitter(x, strength, rng)

cotengra.core.jitter_dict(d, strength, seed=None)

cotengra.core.separate(xs, blocks)

Partition xs into n different list based on the corresponding labels in blocks.