Extend the tree-tensor-network method to Clifford circuits

Determine how to apply the BlockRowCol-based circuit-rewriting technique to Clifford circuits on perfect binary tree tensor networks without relying solely on Clifford normal forms.

Background

The paper applies BlockRowCol recursively to CNOT circuits represented on perfect binary tree tensor networks. By grouping qubits into blocks and eliminating high-level interactions, the authors derive an asymptotic improvement over Patel–Markov–Hayes synthesis for reducing the cost of simulating CNOT circuits with tensor networks.

The corresponding extension to general Clifford circuits is not provided. The authors state that the same technique is unclear for Clifford circuits, apart from using existing Clifford normal forms, leaving open whether an analogous block-elimination strategy can exploit the tensor-network structure more effectively.

References

Here, we will consider CNOT circuits; unfortunately it is unclear how to apply the same technique for Clifford circuits, except via Clifford normal forms .

— Clifford Circuit Synthesis for Distributed Quantum Architectures with Arbitrary Network Topology  (2608.13543 - Laakkonen, 13 Aug 2026) in Section 8.4, “Application to Tree Tensor Networks”

Finally, we close with an open question: in Theorem \ref{thm:blockrowcolapproxopt}, we were able to show that BlockRowCol is approximately optimal for $k = 2$ because we could define the concept of the rank of a submatrix, which is modified in a controlled way when a ZX-type generalized CNOT gate is applied. Is there an analogous concept of rank for subsections of a stabilizer tableau, so that (generic) generalized CNOT gates look like `rank-one updates'? This does not seem readily apparent, even considering the representation of tableaux as symplectic matrices.

— Clifford Circuit Synthesis for Distributed Quantum Architectures with Arbitrary Network Topology  (2608.13543 - Laakkonen, 13 Aug 2026) in Section 8.5, “Future Work”