[Bug]: `_check_flow_general_properties` rejects empty graphs

[thierry-martinez]

_check_flow_general_properties rejects empty graphs with PartialOrderError(PartialOrderErrorReason.Empty), even though the formal definitions of flows do not exclude empty graphs. As a result, flow‑extraction procedures return an empty flow object for an empty graph, but that object then fails the well‑formedness check.

def test_empty_graph() -> None:
    og = OpenGraph(graph=nx.Graph(), input_nodes=[], output_nodes=[], measurements={})
    pf = og.extract_causal_flow()
    pf.check_well_formed() # raises PartialOrderError(PartialOrderErrorReason.Empty)

Edit: I copy-pasted the wrong exception. In this example, pf.check_well_formed() raises graphix.flow.exceptions.PartialOrderLayerError: The first layer of the partial order must contain all the output nodes of the open graph and cannot be empty. First layer: frozenset().

[matulni]

Thanks for the catch!

I think there're two issues here.

1. The partial order of flows (or XZCorrections) on empty open graphs should not be (set(),) but rather tuple().

2. PauliFlow.check_well_formed should allow for an empty partial order iff the OpenGraph does not have any nodes.

This is the behavior for XZCorrections:

p = Pattern()
xz = p.extract_xzcorrections()
assert xz.partial_order_layers == tuple()
xz.check_well_formed()

However, if we extract the xz-corrections from the flow, we obtain an ill-formed partial order, and the well-formedness check fails:

og = OpenGraph(graph=nx.Graph(), input_nodes=[], output_nodes=[], measurements={})
pf = og.extract_pauli_flow()
xz = pf.to_corrections()
assert xz.partial_order_layers == (set(), )
xz.check_well_formed() # Fails

Fixing 1) requires a small modification on the flow extraction routines, because the follow the convention that empty open graphs return a partial order with partial orde (set(),) .

From the documentation:

"A correct flow can only exist on an open graph with output nodes, so layers[0] always contains a finite set of nodes."

This is true, except for the edge case you found of empty open graphs. If I remember correctly, the emptiness check was a short-cut to verify this property together with the convention of the flow extraction routines.

If you agree with the suggested fix, I'll work on a PR to address this.

[ECZ-Quantum-Labs]

Hi @thierry-martinez and @matulni, I'd like to work on fixing this issue. Based on your discussion, I'll implement the fix to:

1. Change the partial order of flows on empty graphs from (set(),) to tuple()
2. Update PauliFlow.check_well_formed to allow empty partial order when the OpenGraph has no nodes

I can deliver a tested PR within 24-48 hours. Please assign this issue to me. Thank you!

[matulni]

Hi @ECZ-Quantum-Labs thanks for your interested in the project, sure feel free to submit a PR!

[thierry-martinez]

As @matulni have said, feel free to submit a PR, @ECZ-Quantum-Labs ! To clear up any possible confusion, note that this issue is not part of UnitaryHack 2026, and there is no bounty on it.