Designs without disorder: unitary k-designs from a single-site bulk defect

Can a simple, non-random quantum system generate unitary k-designs—the gold standard for simulating random quantum processes—without requiring extensive disorder or complex circuit architectures? Specifically, the authors investigate whether a single local defect (a bulk impurity) is enough to break the integrability of an XXZ spin chain and enable the formation of unitary designs.

The authors employ a two-Pauli-kick (2PK) protocol. Unlike standard methods that rely on random gates or ensembles of Hamiltonians, this approach uses a single, fixed Hamiltonian and a fixed Pauli string. Randomness is introduced solely by sampling evolution times from a temporal ensemble. The authors derive an analytical selection rule based on the domain-wall structure of the Pauli string to identify which kicks effectively break integrability and produce designs. They validate this by comparing the frame potential of the resulting ensemble against the Haar measure, which represents the ideal random distribution.

The study demonstrates that a single bulk defect is sufficient to transform an integrable XXZ chain into a chaotic system capable of producing approximate unitary k-designs. The effectiveness of the protocol depends on the spatial structure of the Pauli kick; specifically, the authors derive a selection rule showing that kicks with an optimal domain-wall count—determined by the system's coupling parameters—can successfully generate designs. In contrast, boundary defects do not break integrability sufficiently, and the protocol fails in those cases. The mechanism remains robust even for weak bulk defects as the system size increases.

This work provides a highly efficient, low-overhead method for generating unitary designs in many-body systems. By showing that a single local defect is sufficient, the authors simplify the requirements for experimental implementations in platforms like ultracold atoms or Rydberg-atom arrays. This approach bridges the gap between integrable models and chaotic dynamics, offering a practical tool for quantum benchmarking and shadow tomography without needing complex, time-dependent control.