Quantum CNOT and CV gates

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: q Alternatively the state of a single qubit may be described in as a density matrix:

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

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

6. Z e-iφ Construction of the CNOT Gate: By adjusting the Controlled Phase gate one may build many uses two qubit gates: Φ=π Hadamard: This form allows the construction of many useful gates: <=> H H Z

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

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

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.