Generating pseudo-random unitaries with a Floquet driven chaotic quantum system

We explore using an ergodic Floquet quantum system on a torus to generate pseudo-random unitary operators. We choose a regime of the perturbed Harper model with strong perturbations and perturbation frequency exceeding the libration frequency to ensure that the system has an ergodic region that covers phase space and lacks resonant substructure. We generate a sample of unitary operators in a finite dimensional space by computing Floquet propagators from a distribution of its control parameters. To compare the distribution of unitaries to that of a Haar-random distribution, we compute k-frame potentials from samples of numerically generated unitaries. We find that uniform distributions of 4 control parameters can generate an approximate 3-design. Distributions of fewer control parameters are required to create an approximate 3-design if the Floquet system parameters drift.

💡 Research Summary

This paper investigates the generation of pseudo‑random unitary operators using a chaotic Floquet quantum system defined on a torus. The authors focus on a perturbed Harper model, selecting a regime where the perturbation strength is comparable to the underlying potential (μ≈μ′≈ε) and the perturbation frequency exceeds the characteristic libration frequency, yielding a ratio λ=ν/ω₀≈1 but slightly below unity. In this regime the classical phase space becomes globally ergodic with minimal resonant islands, which is essential for producing a quantum propagator that explores a large portion of the unitary group U(N).

The quantum dynamics are described by the time‑periodic Hamiltonian ˆh(τ)=ˆh₀+ˆh₁(τ), where ˆh₀ = a