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

TL;DR

This work investigates generating pseudo-random unitaries by driving a finite-dimensional quantum system with chaotic Floquet dynamics on a torus, focusing on the perturbed Harper model. By tuning strong perturbations and a specific frequency ratio, the authors produce ergodic phase-space coverage and analyze four propagators (two Floquet, Haar-random, and drifted) using Husimi distributions, quasi-energy spacings, transition probabilities, and inverse participation ratios to assess randomness. They construct quantum samplers by sampling Floquet-control parameters and quantify their Haar-likeness with k-frame potentials, identifying Floquet and drift-based samplers that closely approximate 3-designs with a small number of parameters. The results suggest Floquet-based samplers can rival Haar randomness with favorable scaling, offering a potentially fast, nonlocal alternative to quantum circuits for sampling unitaries, and point to future extensions with more general perturbations and higher dimensions.

Abstract

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.

Figures (16)

• Figure 1: a) We show an illustration of how orbits in a chaotic system are sensitive to initial conditions. b) If the Lyapunov exponent, measuring the divergence rate of orbits, is sensitive to the parameter $\mu$, then orbits begun at the same initial conditions but with slightly different parameters would diverge. c) A quantized version of the classical system shown in b) would have transition probabilities between localized states that are sensitive to the parameter $\mu$. The unitary propagator operator would be sensitive to the parameter $\mu$.
• Figure 2: The surface of the blue sphere represents the space of unitary operators in $N$ dimensions; U$(N)$. The green trajectory represents how a unitary operator (or propagator) $\hat{U}(\mu)$ depends on a parameter $\mu$. As $\mu$ increases, the propagator wanders. If the propagator is highly sensitive to the parameter $\mu$, then a range of $\mu$ can give unitaries that are widely distributed across U$(N)$.
• Figure 3: a) We show Poincaré maps for the classical Hamiltonian system given by equations \ref{['eqn:Hclass']}, \ref{['eqn:H0']}, \ref{['eqn:H1']} with ratio of perturbation frequency to libration frequency $\lambda = (a \epsilon)^{-\frac{1}{2}} = 2$. The panels from left to right have different values of perturbation strength ratio $\mu/\epsilon$. Initial conditions for the orbits are drawn from uniform distributions in phase space. Each orbit is a set of points of the same color and they are plotted in phase space $\phi,p$. The parameters of the Hamiltonian are written on the plot except for $\tau_0$ which is set to 0. b) Similar to a) but showing Poincaré maps for the case $\lambda =1/3$ and the same values of the ratio $\mu/\epsilon$ as in a). Comparison between a) and b) illustrates that when the perturbation frequency is near but somewhat lower than the characteristic libration frequency, there are fewer and smaller resonant islands. We have chosen strong perturbation parameters $\mu, \mu'$ so that the ergodic regions cover phase space.
• Figure 4: a) Poincaré maps of the associated classical models for the Floquet propagator $\hat{U}_{Ta}$ with properties listed in Table \ref{['tab:props']}. The ergodic region covers phase space. b) similar to a) but for propagator $\hat{U}_{Tb}$. There are both integrable and ergodic regions in phase space. The Poincaré maps are similar to those shown in Figure \ref{['fig:SS']} but for different classical Hamiltonians.
• Figure 5: a) Husimi distributions of the eigenvectors of the Floquet propagator $\hat{U}_{Ta}$ with parameters listed in Table \ref{['tab:props']} and dimension $N=49$. Each panel shows the Husimi distribution for a single eigenvector in phase space. The classical version of this operator is ergodic; see the Poincaré map in Figure \ref{['fig:SSab']}a. Most of the eigenfunctions are distributed across phase space. b) The Husimi distributions for $\hat{U}_{Tb}$ which is hybrid in the sense that there are both chaotic and integrable regions in phase space. There are Husimi distributions resembling non-chaotic localized orbits in the Poincaré map of Figure \ref{['fig:SSab']}b. c) The Husimi distributions are generated from the randomly generated unitary operator $\hat{U}_\text{Haar}$ and are widely distributed in phase space. d) The Husimi distributions are generated from $\hat{U}_\text{Drift}$ which is the drifting system with parameters listed in Table \ref{['tab:props']}. Husimi distributions are also widely distributed in phase space.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Corollary 2
• proof