Unitary and Nonunitary Quantum Cellular Automata with Rydberg Arrays

We propose a physical realization of quantum cellular automata (QCA) using arrays of ultracold atoms excited to Rydberg states. The key ingredient is the use of programmable multifrequency couplings which generalize the Rydberg blockade and facilitation effects to a broader set of nonadditive, unitary and nonunitary (dissipative) conditional interactions. Focusing on a 1D array we define a set of elementary QCA rules that generate complex and varied quantum dynamical behavior. Finally, we demonstrate theoretically that Rydberg QCA is ideally suited for variational quantum optimization protocols and quantum state engineering by finding parameters that generate highly entangled states as the steady state of the quantum dynamics.

Figure 1

Physical platform for quantum cellular automata based on arrays of Rydberg atoms. (a) Proposed setup showing a 1D array of atoms held in optical microtraps with period $a$ and nearest-neighbor Rydberg-Rydberg interaction strength $V$. (b) Each atom can be described as a three-state system: $|g⟩$ (open symbols), $|r⟩$ (solid symbols), and an additional short lived state $|e⟩$. The $|g⟩\leftrightarrow |r⟩$ and $|r⟩\leftrightarrow |e⟩$ transitions on site $j$ are coupled by multifrequency fields with detunings $kV$ and coupling strengths ${\theta }_j^k$ and ${\varphi }_j^k$, respectively. This system can be reduced to an effective two-state system with programmable $K$-body interactions (shown on the right for $K=3$, see text for details), where the couplings ${\theta }_j^k$ and ${\tilde{\varphi }}_j^k$ realize unitary (reversible) and nonunitary (dissipative) conditional interactions dependent on the number of excited neighbors $k$.

Figure 2

Numerical simulations of quantum dynamics for discrete and continuous time evolution according to different QCA rules starting from the state $|000010000⟩$. Purely unitary rules are indicated with a green border. (a) Discrete time evolution of the magnetization $⟨{\widehat{Z}}_j⟩$ of $N=9$ sites with block partitioning $ABABABABA$ according to Eq. (4) for $t=20$ update steps. The different panels correspond to a subset of rules indexed by the parameters $[{\theta }^0,{\theta }^1,{\theta }^2,{\tilde{\varphi }}^0,{\tilde{\varphi }}^1,{\tilde{\varphi }}^2]$. (b) Corresponding continuous time evolution without block partitioning.

Figure 3

Variational quantum optimization of the Rydberg QCA stationary state towards highly entangled states for a chain of $N=6$ sites. (a) Hybrid quantum-classical feedback loop used to steer the QCA dynamics to desired quantum states. (b) Convergence of particle swarm optimization with a population size of 10 (gray lines) towards states with large covariance coefficient $⟨C⟩$. The solid blue line highlights the best individual. The optimal parameters are shown in the inset. (c) Graphical representation of the density matrix of the optimized state (imaginary parts are $<{10}^{-3}$). (d) Time evolution of the fidelity between the resulting QCA state using the optimal variational parameters and the ${\mathrm{GHZ}}^N$ state. The system evolves to a highly entangled state within approximately 70 time units. The dashed orange (dashed-dotted green) line shows the same evolution including a Rydberg state decay rate of $\gamma /2\pi =0.8\mathrm{kHz}$ (2.4 kHz) (see the text for other parameters).