Entanglement geometry separates circuit cutting, classical hardness, and trainability

Circuit cutting is a technique used to distribute large quantum computations across smaller devices by replacing nonlocal gates with local operations and classical post-processing. While this enables the execution of larger circuits, it introduces a fundamental trade-off: the properties required for quantum advantage—specifically classical hardness and trainability—must be balanced against the sampling overhead of the cuts. This paper investigates whether a variational quantum circuit can be simultaneously cheaply cuttable, classically hard, and trainable.

The authors analyze the structural constraints imposed by entanglement geometry on circuit architectures. They compare standard Matrix Product State (MPS) and Tree Tensor Network (TTN) circuits, which possess bounded entanglement by design, against a custom two-block circuit family. This two-block architecture allows for independent control over the entanglement at the seam (the cut boundary) and the entanglement within each block. The researchers use numerical simulations up to $n=100$ to verify that they can maintain low cutting overhead while increasing internal entanglement, and they evaluate the compatibility of these configurations with trainability and classical simulation.

The study establishes three key results:

1. Negative Baseline: MPS and TTN circuits with constant bond dimensions are cheaply cuttable but are also efficiently classically simulable, ruling out quantum advantage.
2. The Two-Block Wedge: By decoupling seam entanglement from internal entanglement, the authors construct circuits that remain cheaply cuttable while requiring super-polynomial global bond dimensions. However, they prove that the depth required for classical hardness ($d = \omega(\log n)$) and the depth required for trainability ($d = O(\log n)$) are mutually exclusive.