cotengrust provides fast rust implementations of contraction ordering
primitives for tensor networks or einsum expressions. The two main functions
are:
optimize_optimal(inputs, output, size_dict, **kwargs)optimize_greedy(inputs, output, size_dict, **kwargs)
The optimal algorithm is an optimized version of the opt_einsum 'dp'
path - itself an implementation of https://arxiv.org/abs/1304.6112.
There is also a variant of the greedy algorithm, which runs ntrials of greedy,
randomized paths and computes and reports the flops cost (log10) simultaneously:
optimize_random_greedy_track_flops(inputs, output, size_dict, **kwargs)
cotengrust is available for most platforms from
PyPI:
pip install cotengrustor if you want to develop locally (which requires pyo3 and maturin):
git clone https://github.com/jcmgray/cotengrust.git
cd cotengrust
maturin develop --release(the release flag is very important for assessing performance!).
If cotengrust is installed, then by default cotengra will use it for its
greedy, random-greedy, and optimal subroutines, notably subtree
reconfiguration. You can also call the routines directly:
import cotengra as ctg
import cotengrust as ctgr
# specify an 8x8 square lattice contraction
inputs, output, shapes, size_dict = ctg.utils.lattice_equation([8, 8])
# find the optimal 'combo' contraction path
%%time
path = ctgr.optimize_optimal(inputs, output, size_dict, minimize='combo')
# CPU times: user 13.7 s, sys: 83.4 ms, total: 13.7 s
# Wall time: 13.7 s
# construct a contraction tree for further introspection
tree = ctg.ContractionTree.from_path(
inputs, output, size_dict, path=path
)
tree.plot_rubberband()
The following benchmarks illustrate performance and may be a useful comparison point for other implementations.
First, the runtime of the optimal algorithm on random 3-regular graphs,
with all bond sizes set to 2, for different mimimize targets:
Taken over 20 instances, lines show mean and bands show standard error on mean. Note how much easier it is to find optimal paths for the maximum intermediate size or cost only (vs. total for all contractions). While the runtime generally scales exponentially, for some specific geometries it might reduce to polynomial.
For very large graphs, the random_greedy optimizer is appropriate, and there is a tradeoff between how
long one lets it run (ntrials) and the best cost it achieves. Here we plot these for various
ntrials
(labelled on each marker):
Again, data is taken over 20 runs, with lines and bands showing mean and standard error on the mean.
In most cases 32-64 trials is sufficient to achieve close to convergence, but for larger or harder
graphs you may need more. The empirical scaling of the random-greedy algorithm is very roughly
The depth 20 sycamore quantum circuit amplitude is a standard benchmark nowadays, it is generally
a harder graph than the 2d lattice. Still, the random-greedy approach can do quite well due to its
sampling of both temperature and costmod:
Again, each point is a ntrials setting, and the lines and bands show the mean and error on the mean
respectively, across 20 repeats. The dashed line shows the roughly best known line from other more
advanced methods.
The optimize functions follow the api of the python implementations in cotengra.pathfinders.path_basic.py.
def optimize_optimal(
inputs,
output,
size_dict,
minimize='flops',
cost_cap=2,
search_outer=False,
simplify=True,
use_ssa=False,
):
"""Find an optimal contraction ordering.
Parameters
----------
inputs : Sequence[Sequence[str]]
The indices of each input tensor.
output : Sequence[str]
The indices of the output tensor.
size_dict : dict[str, int]
The size of each index.
minimize : str, optional
The cost function to minimize. The options are:
- "flops": minimize with respect to total operation count only
(also known as contraction cost)
- "size": minimize with respect to maximum intermediate size only
(also known as contraction width)
- 'max': minimize the single most expensive contraction, i.e. the
asymptotic (in index size) scaling of the contraction
- 'write' : minimize the sum of all tensor sizes, i.e. memory written
- 'combo' or 'combo={factor}` : minimize the sum of
FLOPS + factor * WRITE, with a default factor of 64.
- 'limit' or 'limit={factor}` : minimize the sum of
MAX(FLOPS, alpha * WRITE) for each individual contraction, with a
default factor of 64.
'combo' is generally a good default in term of practical hardware
performance, where both memory bandwidth and compute are limited.
cost_cap : float, optional
The maximum cost of a contraction to initially consider. This acts like
a sieve and is doubled at each iteration until the optimal path can
be found, but supplying an accurate guess can speed up the algorithm.
search_outer : bool, optional
If True, consider outer product contractions. This is much slower but
theoretically might be required to find the true optimal 'flops'
ordering. In practical settings (i.e. with minimize='combo'), outer
products should not be required.
simplify : bool, optional
Whether to perform simplifications before optimizing. These are:
- ignore any indices that appear in all terms
- combine any repeated indices within a single term
- reduce any non-output indices that only appear on a single term
- combine any scalar terms
- combine any tensors with matching indices (hadamard products)
Such simpifications may be required in the general case for the proper
functioning of the core optimization, but may be skipped if the input
indices are already in a simplified form.
use_ssa : bool, optional
Whether to return the contraction path in 'single static assignment'
(SSA) format (i.e. as if each intermediate is appended to the list of
inputs, without removals). This can be quicker and easier to work with
than the 'linear recycled' format that `numpy` and `opt_einsum` use.
Returns
-------
path : list[list[int]]
The contraction path, given as a sequence of pairs of node indices. It
may also have single term contractions if `simplify=True`.
"""
...
def optimize_greedy(
inputs,
output,
size_dict,
costmod=1.0,
temperature=0.0,
simplify=True,
use_ssa=False,
):
"""Find a contraction path using a (randomizable) greedy algorithm.
Parameters
----------
inputs : Sequence[Sequence[str]]
The indices of each input tensor.
output : Sequence[str]
The indices of the output tensor.
size_dict : dict[str, int]
A dictionary mapping indices to their dimension.
costmod : float, optional
When assessing local greedy scores how much to weight the size of the
tensors removed compared to the size of the tensor added::
score = size_ab / costmod - (size_a + size_b) * costmod
This can be a useful hyper-parameter to tune.
temperature : float, optional
When asessing local greedy scores, how much to randomly perturb the
score. This is implemented as::
score -> sign(score) * log(|score|) - temperature * gumbel()
which implements boltzmann sampling.
simplify : bool, optional
Whether to perform simplifications before optimizing. These are:
- ignore any indices that appear in all terms
- combine any repeated indices within a single term
- reduce any non-output indices that only appear on a single term
- combine any scalar terms
- combine any tensors with matching indices (hadamard products)
Such simpifications may be required in the general case for the proper
functioning of the core optimization, but may be skipped if the input
indices are already in a simplified form.
use_ssa : bool, optional
Whether to return the contraction path in 'single static assignment'
(SSA) format (i.e. as if each intermediate is appended to the list of
inputs, without removals). This can be quicker and easier to work with
than the 'linear recycled' format that `numpy` and `opt_einsum` use.
Returns
-------
path : list[list[int]]
The contraction path, given as a sequence of pairs of node indices. It
may also have single term contractions if `simplify=True`.
"""
def optimize_simplify(
inputs,
output,
size_dict,
use_ssa=False,
):
"""Find the (partial) contracton path for simplifiactions only.
Parameters
----------
inputs : Sequence[Sequence[str]]
The indices of each input tensor.
output : Sequence[str]
The indices of the output tensor.
size_dict : dict[str, int]
A dictionary mapping indices to their dimension.
use_ssa : bool, optional
Whether to return the contraction path in 'single static assignment'
(SSA) format (i.e. as if each intermediate is appended to the list of
inputs, without removals). This can be quicker and easier to work with
than the 'linear recycled' format that `numpy` and `opt_einsum` use.
Returns
-------
path : list[list[int]]
The contraction path, given as a sequence of pairs of node indices. It
may also have single term contractions.
"""
...
def optimize_random_greedy_track_flops(
inputs,
output,
size_dict,
ntrials=1,
costmod=(0.1, 4.0),
temperature=(0.001, 1.0),
seed=None,
simplify=True,
use_ssa=False,
):
"""Perform a batch of random greedy optimizations, simulteneously tracking
the best contraction path in terms of flops, so as to avoid constructing a
separate contraction tree.
Parameters
----------
inputs : tuple[tuple[str]]
The indices of each input tensor.
output : tuple[str]
The indices of the output tensor.
size_dict : dict[str, int]
A dictionary mapping indices to their dimension.
ntrials : int, optional
The number of random greedy trials to perform. The default is 1.
costmod : (float, float), optional
When assessing local greedy scores how much to weight the size of the
tensors removed compared to the size of the tensor added::
score = size_ab / costmod - (size_a + size_b) * costmod
It is sampled uniformly from the given range.
temperature : (float, float), optional
When asessing local greedy scores, how much to randomly perturb the
score. This is implemented as::
score -> sign(score) * log(|score|) - temperature * gumbel()
which implements boltzmann sampling. It is sampled log-uniformly from
the given range.
seed : int, optional
The seed for the random number generator.
simplify : bool, optional
Whether to perform simplifications before optimizing. These are:
- ignore any indices that appear in all terms
- combine any repeated indices within a single term
- reduce any non-output indices that only appear on a single term
- combine any scalar terms
- combine any tensors with matching indices (hadamard products)
Such simpifications may be required in the general case for the proper
functioning of the core optimization, but may be skipped if the input
indices are already in a simplified form.
use_ssa : bool, optional
Whether to return the contraction path in 'single static assignment'
(SSA) format (i.e. as if each intermediate is appended to the list of
inputs, without removals). This can be quicker and easier to work with
than the 'linear recycled' format that `numpy` and `opt_einsum` use.
Returns
-------
path : list[list[int]]
The best contraction path, given as a sequence of pairs of node
indices.
flops : float
The flops (/ contraction cost / number of multiplications), of the best
contraction path, given log10.
"""
...
def ssa_to_linear(ssa_path, n=None):
"""Convert a SSA path to linear format."""
...
def find_subgraphs(inputs, output, size_dict,):
"""Find all disconnected subgraphs of a specified contraction."""
...