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Quantum CNOT and CV gates Jacob D. Biamonte

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Direction Realize CNOT and CV Gates as NMR pulses J. Jones, R. Hansen and M. Mosca, Quantum Logic Gates and Nuclear Magnetic Resonance Pulse Sequences, J.Magn.Resonance 135, pages 353-360, (1998), quant-ph/9805070.

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The goal of single qubit gates are to rotate a vector on the Bloch Sphere Basic building blocks of all single qubit gates: Alternatively the state of a single qubit may be described in as a density matrix:

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Common Single Operation Gates (our building blocks): These can be used to create the Pauli Spin Matrices: Phase gate: Basic building blocks of all single qubit gates: YZXS e -iφ Controlled Phase gate: The phase shift and controlled Phase shift gates are also important:

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Common Gates that are often needed in quantum algorithms: Controlled Pauli Spin Matrices: Hadamard V Gate: Controlled single qubit gates: R(θ) σiσi Other Gates: ? H ? ? H V V How can we build these useful gates from elementary building blocks? ?

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Construction of the CNOT Gate: Hadamard: Z HH Ze -iφ By adjusting the Controlled Phase gate one may build many uses two qubit gates: Φ=πΦ=π This form allows the construction of many useful gates:

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Construction of the CV Gate: Hadamard: S HH Se -iφ By adjusting the Controlled Phase gate one may build many uses two qubit gates: Φ=π/2 This form allows the construction of many useful gates: V

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Toffoli Gate: V V+V+ V Now smaller gates can be used to build larger gates:

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Additional information: A. Barenco, C. Bennett, R. Cleve, D. DiVincenzo, N. Margolus, P. Shor. T. Sleator, J. Smolin, and H. Weinfurter, Elementary gates of quantum computation, Physical Review A, 52(5):3457-3467, (1995), quant- ph/9503016. J. Jones, R. Hansen and M. Mosca, Quantum Logic Gates and Nuclear Magnetic Resonance Pulse Sequences, J.Magn.Resonance 135, pages 353-360, (1998), quant-ph/9805070.

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