Intrinsically Universal One-dimensional Quantum Cellular Automata in Two Flavours

Abstract

We give a one-dimensional quantum cellular automaton (QCA) capable of simulating all others. By this we mean that the initial configuration and the local transition rule of any onedimensional QCA can be encoded within the initial configuration of the universal QCA. Several steps of the universal QCA will then correspond to one step of the simulated QCA. The simulation preserves the topology in the sense that each cell of the simulated QCA is encoded as a group of adjacent cells in the universal QCA. The encoding is linear and hence does not carry any of the cost of the computation. We do this in two flavours: a weak one which requires an infinite but periodic initial configuration and a strong one which needs only a finite initial configuration.

Partitioned quantum cellular automata are intrinsically universal

“…In the bounded-size configurations case, circuit universality coincides with intrinsic universality, as noted by Van Dam [37]. Intrinsically universal QCA in the one-dimensional case have also been resolved [2]. Finally, a subsequent work, which crucially builds upon the result of this paper, exhibits an n-dimensional intrinsically universal QCA [3].…”

“…that they can all simulate each other in a space-preserving manner. The definition of intrinsic simulation has already been translated in the quantum context [2], however as it stands this is not sufficient to obtain the desired result. In this paper the definition of intrinsic simulation in the quantum context is discussed and developed, before the equivalence between all the various above-mentioned definitions of QCA is tackled.…”

The quantum game of life

“…In the bounded-size configurations case, circuit universality coincides with intrinsic universality, as noted by Van Dam [49]. QCA intrinsic universality in the onedimensional case has been resolved [5], and also in the general n > 1-dimensional case [7]. Both results rely on recent work [6], where it was shown that a simple subclass of QCA, namely Partitioned QCA (PQCA), are intrinsically universal.…”

“…• Dimension three is particularly interesting, not only because of the relevance to physics, but because intrinsic universality QCA constructions lend themselves to a striking simplification. In dimension one the simplest known intrinsically universal QCA has cell dimension 36 [5]. In dimension two it has cell dimension 4 [7], and seems reducible to 3 via a costly and inelegant variation of the scheme [7], but probably no further.…”

Intrinsically universal n-dimensional quantum cellular automata

“…that they can all simulate each other in a space-preserving manner. The definition of intrinsic simulation has already been translated in the quantum context [37], however as it stands this is not sufficient to obtain the desired result. In this paper the definition of intrinsic simulation in the quantum context is discussed and developed, before the equivalence between all the various above-mentioned definitions of QCA is tackled.…”