Unitary k-designs without Hamiltonian quenches

Quantum information protocols, such as those for black hole information recovery, often rely on Haar-random unitaries. However, these are exponentially difficult to implement. Recent research has explored using chaotic Hamiltonian dynamics to generate 'unitary designs'—ensembles that mimic Haar randomness with fewer resources. This paper investigates whether such designs can be generated using a single, fixed chaotic Hamiltonian, avoiding the need for complex Hamiltonian quenches or ensembles of different Hamiltonians.

The author introduces the 'Two-Pauli-Kick' (2PK) protocol. Instead of switching between different Hamiltonians, the system evolves under a single, fixed chaotic Hamiltonian $H$. The protocol involves three periods of evolution separated by two insertions of the same randomly chosen Pauli operator $P$. By sampling the evolution times from a uniform distribution, the resulting temporal ensemble forms an approximate unitary $k$-design. This approach significantly reduces the computational and experimental overhead compared to multi-Hamiltonian quench protocols.

• Efficiency: The 2PK protocol successfully converges to the Haar value for the $k$-th frame potential, confirming the emergence of approximate unitary $k$-designs.
• Universality: The protocol was verified across various chaotic systems, including Gaussian random matrix ensembles (GUE/GOE) and the Sachdev-Ye-Kitaev (SYK) model. The results indicate that the emergence of designs is a universal feature of chaotic dynamics rather than a property of specific interaction structures.
• Finite Temperature: Unlike previous quench-based methods, the 2PK protocol allows for a canonical definition of finite-temperature frame potentials. The author establishes a direct relationship between the frame potential and the partition function, showing that the protocol remains robust at finite temperatures.
• Holographic Interpretation: The author suggests that the protocol can be viewed through a holographic lens, where the Pauli kicks act as bulk shockwaves, providing a potential link to gravity duals.