Quantum cellular automaton for universal quantum computation

A number of physical systems that are considered for the realization of a universal quantum computer, such as optical lattices 1 or arrays of microlenses 2 , possess a translation symmetry in the arrangement of qubits and their mutual interaction. Quantum cellular automata QCA represent a suitable framework to explore the computational power of such physical systems, because they respect this symmetry. A priori, translation invariance may be regarded as a severe limitation since it constrains the degree of control that can be exerted to the quantum system. However, it has been demonstrated that one-dimensional QCA can efficiently simulate any quantum Turing machine 3 . Further it has been shown that there exists a universal QCA which can simulate any other automaton with linear slow down 4 , and that every reversible QCA can be represented in a generalized Margolus partitioning scheme 5 . Proposals with an emphasis on experimental viability have outlined how generic physical systems can be used as quantum computation devices if equipped with a minimal amount of external control. Among the described mechanisms are global control via sequences of resonant light pulses 6 or modulation of a coupling constant 7,8 , and individual control over one of the elementary cells 9 . At this point, one may abandon all algorithm-specific control during the process of computation and ask “How intricate do quantum cellular automata have to be such that they can perform useful tasks in quantum information processing?.” A quick answer may be “Simple, by definition.” However, when QCAs are tuned for algorithmic application, it may occur that—while the simple composition is retained— the elementary cells and neighborhood schemes become complicated. An interesting facet of the answer to the above question has been provided in Ref. 10 , where a very simple QCA for quantum data transmission has been devised also see Ref. 5 . Motivated by a recent result 11 , where universal computation via autonomous evolution of a 10-local Hamiltonian is described, here we consider quantum computation in the cellular automaton scenario. We explicitly construct a computationally universal two-dimensional QCA whose transition rule is based on a four-qubit unitary.