State k-designs from Hamiltonian evolution

Can deterministic time evolution under a fixed, physically realistic Hamiltonian generate ensembles of states that mimic the statistical properties of Haar-random states (state k-designs)? While temporal ensembles generated by a single Hamiltonian are typically constrained by energy conservation, this paper investigates whether carefully selecting the initial state ensemble can overcome these limitations to produce high-order designs.

The authors analyze the frame potential of ensembles generated by evolving initial states under a Hamiltonian. They derive a recursion relation for Gaussian Unitary Ensemble (GUE) Hamiltonians, showing that if the initial ensemble is a state 1-design, the evolved ensemble approaches a state k-design in the thermodynamic limit. They extend this to local, nonintegrable mixed-field Ising models, identifying that product states in the Y-basis are particularly effective due to their thermalization properties. To address finite-time convergence, they introduce an M-step quench protocol that uses alternating Hamiltonians to suppress errors.

• Recursive Design Generation: For GUE Hamiltonians, the frame potential of the evolved ensemble is directly related to the frame potential of the initial ensemble, allowing for the systematic construction of k-designs.
• Local Hamiltonian Success: For the mixed-field Ising model, initializing with Y-basis product states generates approximate state k-designs with high precision, as these states sample the bulk of the spectrum where eigenstate thermalization occurs.
• Error Suppression: The leading finite-time correction scales as O(1/T). By employing an M-step quench protocol, the authors demonstrate that this error can be suppressed to O(1/T^M), verified through both analytical derivation and numerical simulation.
• Unified Framework: The derived recursion relations provide a unified explanation for previous constructions of unitary k-designs, showing that sequential quenches effectively build up the necessary randomness.

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