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Capacitively Coupled Charge Qubits: Difference between revisions
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Capacitively Coupled Charge Qubits (view source)
Revision as of 15:56, 2 December 2014
, 2 December 2014→Numerical Simulations
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This project started in the Fall of 2014, focusing on the operation of two or more charge qubits which are capacitively coupled. Current PDF Summary can be found [[File:Capacitively Coupled Qubits.pdf]]
==General Formulation==
[[Image:Detuning graph.png|thumb|Energy levels of the 2 qubit system as a function of both detunings.|300px]]
* <math>Z_1</math> gate (<math>Z</math> on qubit 1)
*# Wait for a time <math>\tau = \frac{h}{2(\lambda_1+\lambda_2)}</math> (always on)
* <math>Z_2</math> gate (<math>Z</math> on qubit 2)
*# Wait for a time <math>\tau = \frac{h}{2(\lambda_1-\lambda_2)}</math> (always on)
* <math>X_1</math> gate (<math>X</math> on qubit 1)
*# Pulse <math>\epsilon_1</math> at a frequency of <math>\omega_{AC} = (\lambda_1+\lambda_2)/\hbar</math> for a time <math>\tau = \frac{2h}{A_2}</math>
* <math>X_2</math> gate (<math>X</math> on qubit 2)
*# Pulse <math>\epsilon_2</math> at a frequency of <math>\omega_{AC} = (\lambda_1-\lambda_2)/\hbar</math> for a time <math>\tau = \frac{2h}{C_1}</math>
* <math>\text{CNOT}_1</math> gate (<math>\text{CNOT}</math> with qubit 1 as control)
*# Pulse <math>\epsilon_1</math> at a frequency of <math>\omega_{AC} = (\lambda_1-\lambda_2)/\hbar</math> for a time <math>\tau = \frac{h}{A_1}</math>
*# Pulse <math>\epsilon_2</math> at a frequency of <math>\omega_{AC} = (\lambda_1-\lambda_2)/\hbar</math> for a time <math>\tau = \frac{h}{C_1}</math>
* <math>\text{CNOT}_2</math> gate (<math>\text{CNOT}</math> with qubit 2 as control)
*# Pulse <math>\epsilon_2</math> at a frequency of <math>\omega_{AC} = (\lambda_1+\lambda_2)/\hbar</math> for a time <math>\tau = \frac{h}{C_2}</math>
*# Pulse <math>\epsilon_1</math> at a frequency of <math>\omega_{AC} = (\lambda_1+\lambda_2)/\hbar</math> for a time <math>\tau = \frac{3h}{A_2}</math>
* <math>\text{SWAP}</math>
*# Pulse either <math>\Delta_1</math> or <math>\Delta_2</math> at a frequency of <math>\omega_{AC} = 2\lambda_2/\hbar</math> for a time <math>\tau = \frac{2h\lambda_2}{Bg}</math>
===Notes on Logical Operating Points===
Here we discuss the values of <math>A_1</math>, <math>A_2</math>, <math>C_1</math>, and <math>C_2</math>. Analytical expressions can and have been found for these quantities, but they yield little intuition about their nature.
[[File:Rotation coeffs.png|border|600px]]
Above is a graph of the 4 values as a function of <math>\Delta_1</math>. If we wish to consider the regime in which <math>\Delta_1>\Delta_2</math> as above, we should consider the part of the graph to the right of the black dotted line.
A few qualitative statements can be made about the logical gates based off of this graph. First, it is clear that the <math>X_2</math> gate will always be faster than the <math>X_1</math> gate. A feature of concern is the behavior of <math>C_2</math>, namely that it tends towards zero, as this increases the time for the <math>\text{CNOT}_2</math> gate.
===Numerical Simulations===
[[Image:x1 gate.gif|thumb|200px|Animation of the density matrix during the application of the X1 gate]]
[[Image:x2 gate.gif|thumb|200px|Animation of the density matrix during the application of the X2 gate]]
Using the QuTiP package, we can simulate the Master Equation for the system in question. First, we can verify the gates that we proposed under ideal conditions. Below are pictures of the density matrix as it undergoes pulses that we have predicted to correspond to X1 and X2 gates. Note that the behavior of the density matrix matches what we would expect.
<gallery>
File:X1 00.png|Beginning of X1 Gate
File:X1 99.png|End of X1 Gate
File:X2 00.png|Beginning of X2 Gate
File:X2 99.png|End of X2 Gate
</gallery>
Animations of these transformations can be found on the right.
Issues with CNOT gate:
<gallery>
File:State population oscillations 00 state.png|00 state
File:State population oscillations 10 state.png|10 state
File:State population oscillations super.png|Superpostition between 00 and 01 state
</gallery>
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