Random quantum circuits continue to inspire a wide range of applications in quantum information science, while remaining analytically tractable through probabilistic methods. Motivated by the need for deterministic circuits with similar applications, we construct classes of nonrandom unitary Clifford circuits by imposing translation invariance in both time and space. Further imposing dual- unitarity, our circuits effectively become crystalline lattices whose vertices are SWAP or iSWAP cores and whose edges are decorated with single-qubit gates. Working on the square and kagome lattice, one can further impose invariance under (subgroups of) the crystal's point group. We also break unitarity by adding spacetime-translation-invariant measurements and find a class of circuits with fractal dynamics. We use the formalism of Clifford quantum cellular automata to describe operator spreading, entanglement generation, and recurrence times in the unitary versions of these circuits. A full classification on the square lattice reveals, of particular interest, a "non-fractal good scrambling class" with dense operator spreading that generates codes with linear contiguous code distance and high performance under erasure errors.
, Huse, D.
and Gullans, M.
Crystalline Quantum Circuits, PRX Quantum, [online], https://doi.org/10.1103/PRXQuantum.4.030313, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=935726
(Accessed September 25, 2023)