OMEinsumContractionOrders

OMEinsumContractionOrders is a Julia package that provides an optimize_code function for finding optimal contraction orders for tensor networks. It is designed to work with multiple tensor network packages, such as: OMEinsum.jl package and ITensorNetworks.jl.

Installation

To install OMEinsumContractionOrders, please follow these steps:

1. Open Julia's interactive session (known as REPL) by typing julia in your terminal.
2. Press the ] key in the REPL to enter the package mode.
3. Type add OMEinsumContractionOrders to install the stable release of the package.
4. (Optional) If you want to use the KaHyParBipartite optimizer, which is based on the KaHyPar library, type add KaHyPar. Note that this step is optional because some users may have issues with the binary dependencies of KaHyPar (please check issues: this and this).

Example 1: Use it directly

The contraction order optimizer is implemented in the optimize_code function. It takes three arguments: code, size, and optimizer. The code argument is the einsum notation to be optimized. The size argument is the size of the variables in the einsum notation. The optimizer argument is the optimizer to be used. The optimize_code function returns the optimized contraction order. One can use contraction_complexity function to get the time, space and rewrite complexity of returned contraction order.

julia> using OMEinsumContractionOrders, Graphs, KaHyPar

julia> function random_regular_eincode(n, k; optimize=nothing)
        g = Graphs.random_regular_graph(n, k)
        ixs = [[minmax(e.src,e.dst)...] for e in Graphs.edges(g)]
        return EinCode([ixs..., [[i] for i in Graphs.vertices(g)]...], Int[])
    end

julia> code = random_regular_eincode(200, 3);

julia> optcode_tree = optimize_code(code, uniformsize(code, 2),
    TreeSA(sc_target=28, βs=0.1:0.1:10, ntrials=2, niters=100, sc_weight=3.0));

julia> optcode_tree_with_slice = optimize_code(code, uniformsize(code, 2),
    TreeSA(sc_target=28, βs=0.1:0.1:10, ntrials=2, niters=100, sc_weight=3.0, nslices=5));

julia> optcode_kahypar = optimize_code(code, uniformsize(code, 2), 
    KaHyParBipartite(sc_target=30, max_group_size=50));

julia> optcode_sa = optimize_code(code, uniformsize(code, 2),
    SABipartite(sc_target=30, max_group_size=50));

julia> contraction_complexity(code, uniformsize(code, 2))
Time complexity: 2^200.0
Space complexity: 2^0.0
Read-write complexity: 2^10.644757592516257

julia> contraction_complexity(optcode_kahypar, uniformsize(code, 2))
Time complexity: 2^39.5938886486877
Space complexity: 2^28.0
Read-write complexity: 2^30.39890775966298

julia> contraction_complexity(optcode_sa, uniformsize(code, 2))
Time complexity: 2^41.17129641027078
Space complexity: 2^29.0
Read-write complexity: 2^31.493976190321106

julia> contraction_complexity(optcode_tree, uniformsize(code, 2))
Time complexity: 2^35.06468305863757
Space complexity: 2^28.0
Read-write complexity: 2^30.351552349259258

julia> contraction_complexity(optcode_tree_with_slice, uniformsize(code, 2))
Time complexity: 2^33.70760100663681
Space complexity: 2^24.0
Read-write complexity: 2^32.17575935629581

Example 2: Use it in OMEinsum

OMEinsumContractionOrders is shipped with OMEinsum package. You can use it to optimize the contraction order of an OMEinsum expression.

julia> using OMEinsum

julia> code = ein"ij, jk, kl, il->"
ij, jk, kl, il -> 

julia> optimized_code = optimize_code(code, uniformsize(code, 2), TreeSA())
SlicedEinsum{Char, NestedEinsum{DynamicEinCode{Char}}}(Char[], ki, ki -> 
├─ jk, ij -> ki
│  ├─ jk
│  └─ ij
└─ kl, il -> ki
   ├─ kl
   └─ il
)

Extensions

LuxorTensorPlot

LuxorTensorPlot is an extension of the OMEinsumContractionOrders package that provides a visualization of the contraction order. It is designed to work with the OMEinsumContractionOrders package. To use LuxorTensorPlot, please follow these steps:

pkg> add OMEinsumContractionOrders, LuxorGraphPlot

julia> using OMEinsumContractionOrders, LuxorGraphPlot

and then the extension will be loaded automatically.

The extension provides the following to function, viz_eins and viz_contraction, where the former will plot the tensor network as a graph, and the latter will generate a video or gif of the contraction process. Here is an example:

julia> using OMEinsumContractionOrders, LuxorGraphPlot

julia> eincode = OMEinsumContractionOrders.EinCode([['a', 'b'], ['a', 'c', 'd'], ['b', 'c', 'e', 'f'], ['e'], ['d', 'f']], ['a'])
ab, acd, bcef, e, df -> a

julia> viz_eins(eincode, filename = "eins.png")

julia> nested_eins = optimize_code(eincode, uniformsize(eincode, 2), GreedyMethod())
ab, ab -> a
├─ ab
└─ acf, bcf -> ab
   ├─ acd, df -> acf
   │  ├─ acd
   │  └─ df
   └─ bcef, e -> bcf
      ├─ bcef
      └─ e

julia> viz_contraction(nested_code)
[ Info: Generating frames, 7 frames in total
[ Info: Creating video at: /var/folders/3y/xl2h1bxj4ql27p01nl5hrrnc0000gn/T/jl_SiSvrH/contraction.mp4
"/var/folders/3y/xl2h1bxj4ql27p01nl5hrrnc0000gn/T/jl_SiSvrH/contraction.mp4"

The resulting image and video will be saved in the current working directory, and the image is shown below:

The large white nodes represent the tensors, and the small colored nodes represent the indices, red for closed indices and green for open indices.

References

If you find this package useful in your research, please cite the relevant papers in CITATION.bib.

Multi-GPU computation

Please check this Gist:

https://gist.github.com/GiggleLiu/d5b66c9883f0c5df41a440589983ab99

Authors

OMEinsumContractionOrders was developed by Jin-Guo Liu and Pan Zhang.