Quantum computation via translation-invariant operations on a chain of qubits

Symmetry reduces complexity. In physical systems realizing quantum computers, the highest degree of symmetry is therefore not the most desirable. A quantum computer needs to be sufficiently simple and robust to be controllable in an experiment yet complex enough to be universal. One may therefore ask the question, “how much symmetry does a quantum computer allow for?” In fact, a number of physical systems considered for the realization of a quantum computer such as optical lattices 1 or arrays of microlenses 2 are translation invariant, and the above question acquires a practical aspect. Quite surprisingly, it turns out that universal quantum computation can tolerate a fair amount of symmetry. Recently, a scheme of quantum computation using the rotationinvariant measurement of the total “spin” of two qubits as the only gate has been devised 3 . Furthermore, a translation-invariant computation scheme has been described 4 . The computational power of translation-invariant or nearly translation invariant quantum systems was revealed in Lloyd’s 5 and Watrous’ 6 work on quantum cellular automata QCA . In Ref. 6 it was shown that a onedimensional QCA can simulate any quantum Turing machine. Translation invariance is broken only by the initial state which encodes the program. The schemes in Refs. 5 and 7 allow one to simulate quantum circuits using a chain of qubits with a generic translation-invariant interaction. They require different species of qubits in a periodic arrangement and local addressability at one end-point of the chain. In the scheme in Ref. 8 such individual addressing is only required in the initialization. The method proposed in Ref. 4 is completely translation invariant in space. It requires homogeneous oneand two-local operations on five-level systems. Here I describe a scheme for universal quantum computation via translation-invariant operations on a chain of qubits. No individual addressability is required. The scheme uses an Ising-type interaction and spatially uniform onequbit gates. The qubits are all of the same species. Cold atoms in optical lattices 1 , where the requirement of local control adds to the overall technological challenge, are a candidate for the realization of the presented scheme. II. CONSTRUCTIVE ELEMENTS, UNIVERSALITY, AND SCALABILITY