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Capacitively Coupled Charge Qubits: Difference between revisions
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Capacitively Coupled Charge Qubits (view source)
Revision as of 21:22, 3 November 2014
, 3 November 2014→Rotations
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==Rotations==
===Shifting to the Rotating Frame===
After bringing the system adiabatically to the sweet spot <math>\epsilon_1=\epsilon_2=0</math>, we can apply an AC pulse to some of our parameters to induce a rotation within the system.
\end{matrix}\right)\cos{(\omega_{AC}t)}</math>
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where <math>\lambda_1 = \sqrt{(\Delta_1+\Delta_2)^2+g^2}</math>, <math>\lambda_2 = \sqrt{(\Delta_1-\Delta_2)^2+g^2}</math>, and <math>A_1</math>, <math>A_2</math>, <math>C_1</math>, and <math>C_2</math> are all complicated functions of the parameters which are discussed below.
Based off of the rotation matrices in the energy basis, it's clear that the situation in which <math>\Delta_1>\Delta_2</math> fundamentally differs from the case in which <math>\Delta_1<\Delta_2</math>. Moreover, it can be seen that the case in which <math>\Delta_1=\Delta_2</math> yields diverging results (as some of the matrix elements feature a step function at this point). So in practice, it would be good to avoid this point. Writing the rotations explicitly for the two regimes:
\end{matrix}\right)\cos{(\omega_{AC}t)}</math>
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We can further analyze these matrices by considering the effective rotations in the rotating frame for different applied frequencies, using the rotating wave approximation:
{| border="1" cellpadding="2"
\end{matrix}\right)</math>
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===Constructing Logical Gates===
We will assume in this section that we are in the regime <math>\Delta_1>\Delta_2</math>. The same gates can be determined in the other regime as well.
* <math>Z_1</math> gate (<math>Z</math> on qubit 1)
* <math>Z_2</math> gate (<math>Z</math> on qubit 2)
* <math>X_1</math> gate (<math>X</math> on qubit 1)
* <math>X_2</math> gate (<math>X</math> on qubit 2)
* <math>\text{CNOT}_1</math> gate (<math>\text{CNOT}</math> with qubit 1 as control)
* <math>\text{CNOT}_2</math> gate (<math>\text{CNOT}</math> with qubit 2 as control)
* <math>\text{SWAP}</math>
===Notes on Logical Operating Points===
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