Designing fusion network to generate resource graph state

In this example, we decompose a graph state into a set of GHZ and linear cluster resource states, such that fusion operations can be used on these ‘micro-resource’ states to obtain the desired graph state. This is an important compilation stage to perform MBQC on discrete-variable optical QPUs.

The decomposition algorithm is based on [1].

[1] Zilk et al., A compiler for universal photonic quantum computers, 2022 arXiv:2210.09251

import itertools

import graphix
from graphix.extraction import get_fusion_network_from_graph

/home/docs/checkouts/readthedocs.org/user_builds/graphix-fork/envs/latest/lib/python3.10/site-packages/cotengra/hyperoptimizers/hyper.py:33: UserWarning: Couldn't import `kahypar` - skipping from default hyper optimizer and using basic `labels` method instead.
  warnings.warn(

Here we say we want a graph state with 9 nodes and 12 edges. We can obtain resource graph for a measurement pattern by using nodes, edges = pattern.get_graph().

gs = graphix.GraphState()
nodes = [0, 1, 2, 3, 4, 5, 6, 7, 8]
edges = [(0, 1), (1, 2), (2, 3), (3, 0), (3, 4), (0, 5), (4, 5), (5, 6), (6, 7), (7, 0), (7, 8), (8, 1)]
gs.add_nodes_from(nodes)
gs.add_edges_from(edges)
gs.draw()

Decomposition with GHZ and linear cluster resource states with no limitation in their sizes.

get_fusion_network_from_graph(gs)

[GHZ[0, 7, 5, 3, 1], GHZ[1, 2, 8], LINEAR[2, 3, 4, 5, 6, 7, 8]]

If you want to know what nodes are fused in each resource states, you can use get_fusion_nodes() function. Currently, we consider only type-I fusion. See [2] for the definition of fusion.

[2] Daniel E. Browne and Terry Rudolph. Resource-efficient linear optical quantum computation. Physical Review Letters, 95(1):010501, 2005.

fused_graphs = get_fusion_network_from_graph(gs)
for idx1, idx2 in itertools.combinations(range(len(fused_graphs)), 2):
    print(
        f"fusion nodes between resource state {idx1} and "
        f"resource state {idx2}: {graphix.extraction.get_fusion_nodes(fused_graphs[idx1], fused_graphs[idx2])}"
    )

fusion nodes between resource state 0 and resource state 1: [1]
fusion nodes between resource state 0 and resource state 2: [7, 5, 3]
fusion nodes between resource state 1 and resource state 2: [2, 8]

You can also specify the maximum size of GHZ clusters and linear clusters available, for more realistic fusion scheduling.

get_fusion_network_from_graph(gs, max_ghz=4, max_lin=4)

[GHZ[0, 7, 5, 3], GHZ[1, 8, 2, 0], LINEAR[2, 3, 4, 5], LINEAR[5, 6, 7, 8]]