Fig. 1: One-dimensional quantum cellular automata circuits.
From: Small-world complex network generation on a digital quantum processor
a Schematic for embedding one-dimensional chains into a subset of a two-dimensional Sycamore-class quantum processor. Gray crosses represent transmon qubits and blue rectangles represent couplers. Purple, green, yellow, and red paths are hypothetical example embeddings. b Generic structure of a one-dimensional quantum cellular automata (QCA) circuit where time flows to the right. An initialization step is applied to a chain of L qubits, typically to place them into a classical product state with some number of bit flips (\(\left|1\right\rangle\)s). A number of unitary QCA update cycles, t, are applied to all L qubits before a measurement is performed. c The specific structure of a Goldilocks QCA for one QCA cycle (red box), wherein the initial state is \(\left|0\ldots 010\ldots 0\right\rangle\), the local update unitary is a controlled Hadamard gate, and measurement is performed in the computational basis. d After moment alignment, spin-echo insertion, and compilation down to hardware-native gates a single QCA cycle (red box) results in 4 × (L − 1) number of \({\sqrt{{{\mbox{iSWAP}}}}}^{{{{\dagger}}} }\) gates and 8 × L number of individually-parameterized PhXZ(a, x, z) ≡ ZzZaXxZ−a gates. The number of single and two-qubit layers per QCA cycle does not change as a function of system size, only total gate volume does.
