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Unformatted text preview: arXiv:quant-ph/0210187v1 27 Oct 2002 A 2 rebit gate universal for quantum computing Terry Rudolph and Lov Grover Bell Labs, 600-700 Mountain Ave., Murray Hill, NJ 07974, U.S.A. (Dated: February 1, 2008) We show, within the circuit model, how any quantum computation can be efficiently performed using states with only real amplitudes (a result known within the Quantum Turing Machine model). This allows us to identify a 2-qubit (in fact 2- re bit) gate which is universal for quantum computing, although it cannot be used to perform arbitrary unitary transformations. Performing universal quantum computation is gener- ally equated with the ability to build up arbitrary uni- tary transformations acting on n qubits, out of a set of unitary transformations that act on a small number of qubits at a time. Deutsch  originally presented a single 3 qubit universal quantum gate from which all n qubit unitary transformations could be built. It was subse- quently shown that two qubit gates suffice . 1 In general, when evaluating a new proposal for imple- menting quantum computation, the standard procedure is to check whether one can perform (i) a controlled-NOT (CNOT) operation between two qubits, and (ii) arbitrary single qubit unitary transformations. If so, then univer- sal quantum computing is certainly possible. Recently some beautiful ideas for implementing quantum compu- tation by performing measurements on appropriate states have been presented . However the general principle of proving universality in accordance with the ability to ob- tain evolution corresponding to (i) and (ii) has still been followed. The ability to perform a CNOT and arbitrary single qubit gates allows one to evolve an n qubit state to any point in the Hilbert space, i.e. it allows the construction of arbitrary unitary transforms. However it has been shown within the Quantum Turing Machine model that universal quantum computing can be performed using only real amplitudes . The Quantum Turing Machine model is not particu- larly intuitive for either thinking about construction of a practical quantum computer, nor for design of quantum algorithms. The purpose of this note is to point out how any quantum computation can be simply translated into...
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