One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations

One-dimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a local, shift-invariant unitary evolution. By local we mean that no instantaneous long-range communication can occur. In order to define these over a Hilbert space we must restrict to a base of finite, yet unbounded configurations. We show that QCA always admit a two-layered block representation, and hence the inverse QCA is again a QCA. This is a striking result since the property does not hold for classical one-dimensional cellular automata as defined over such finite configurations. As an example we discuss a bijective cellular automata which becomes non-local as a QCA, in a rare case of reversible computation which does not admit a straightforward quantization. We argue that a whole class of bijective cellular automata should no longer be considered to be reversible in a physical sense. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [12]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions. One-dimensional cellular automata (CA) consist in a line of cells, each of which may take one in a finite number of possible states. These evolve in discrete time steps according to a local, shift-invariant function. When defined over infinite configurations, the inverse of a bijective CA is then itself a CA, and this structural reversibility leads to a natural block decomposition of the CA. None of this holds over finite, yet possibly unbounded, configurations. Because CA are a physics-like model of computation it seems very natural to study their quantum extensions. The flourishing research in quantum information and quantum computer science provides us with appropriate context for doing so, both in terms of the potential implementation and the theoretical framework. Right from the very birth of the field with Feynman’s 1986 paper, it was hoped that QCA may prove an important path to realistic implementations of quantum computers [7] – mainly because they eliminate the need for an external, classical control and hence the principal source of decoherence. Other possible aims include providing models of distributed quantum computation, providing bridges between computer science notions and modern theoretical physics, or anything like understanding the dynamics of some quantum physical system in discrete spacetime, i.e. from an idealized viewpoint. Studying QCA rather than quantum Turing machines for instance means we bother about the spatial structure or the spatial parallelism of things [2], whether for the purpose of describing a quantum protocol, modelling a quantum physical phenomena [11]. One-dimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a local, shift-invariant unitary evolution. By local we mean that information cannot be transmitted faster than a fixed number of cells per time step. Because the standard mathematical setting for quantum mechanics is the theory of Hilbert spaces, we must exhibit and work with a countable basis for our vectorial space. This is the reason why we restrict to finite, unbounded configurations. An elegant alternative to this restriction is to abandon Hilbert spaces altogether and use the more abstract mathematical setting of C∗-algebras [3] – but here we want our proofs to be self-contained and accessible to the Computer Science community. Our main result is that QCA can always be expressed as two layers of an infinitely repeating unitary gate even over such finite configurations. The existence of such a two-layered block representation implies of course that the inverse QCA is again a QCA. Our proof method is mainly a drastic simplification of that of the same theorem over infinite configurations, adapted to finite unbounded configurations. Moreover in its present form the theorem over infinite configurations is stated for n-dimensions [12], which we prove is incorrect by presenting a two-dimensional QCA which does not admit a two-layered block representation. It is a rather striking fact however that QCA admit the two-layered block representation in spite of their being defined over finite, unbounded configurations. For most purposes this saves us from complicated unitary tests such as [5, 6, 1]. But more importantly notice how this is clearly not akin to the classical case, where a CA may be bijective over such finite configurations, and yet not structurally reversible. In order to clarify this situation we consider a perfectly valid, bijective CA but whose inverse function is not a CA. It then turns out that its quantum version is no longer valid, as it allows superluminal signalling. Hence whilst we are used to think that any reversible computation admits a trivial quantization, this turns out not to be the case in the realm of cellular automata. Curiously the non-locality of quantum states (entanglement) induces more structure upon the cellular automata – so that its evolution may remain local as an operation (no-superluminal signalling). Based upon these remarks we prove that an important, well-studied class of bijective CA may be dismissed as not physically reversible. Outline. We provide a simple axiomatic presentation of QCA (Section 1). We reorganize a number of known mathematical results around the notion of subsystems in quantum theory (Section 2). Thanks to this small theory we prove the reversibility/block structure theorem in an elementary manner (Section 3). In the discussion we show why the theorem does not hold as such in further dimensions; we exhibit superluminal signalling in the XOR quantum automata, and end with a general theorem discarding all injective, non surjective CA over infinite configurations as unphysical (Section 4).