1. Quantum CNOT and CV gates Jacob D. Biamonte

2. 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.

3. 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:

4. 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:

5. 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? ?

6. 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:

7. 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

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

9. 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.