Capacitively Coupled Charge Qubits - Revision history

AdamFrees: /* Numerical Simulations */

2014-12-02T15:56:59Z

Numerical Simulations

← Older revision		Revision as of 15:56, 2 December 2014
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	Animations of these transformations can be found on the right.		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>

AdamFrees: /* Numerical Simulations */

2014-11-17T01:25:45Z

Numerical Simulations

← Older revision		Revision as of 01:25, 17 November 2014
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	===Numerical Simulations===		===Numerical Simulations===
	[[Image:x1 gate.gif\|thumb\|200px]]		[[Image:x1 gate.gif\|thumb\|200px\|Animation of the density matrix during the application of the X1 gate]]
	[[Image:x2 gate.gif\|thumb\|200px]]		[[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>		<gallery>
	File:X1 00.png\|Beginning of X1 Gate		File:X1 00.png\|Beginning of X1 Gate
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	File:X2 99.png\|End of X2 Gate		File:X2 99.png\|End of X2 Gate
	</gallery>		</gallery>

			Animations of these transformations can be found on the right.

AdamFrees: /* Numerical Simulations */

2014-11-17T01:15:46Z

Numerical Simulations

← Older revision		Revision as of 01:15, 17 November 2014
Line 363:		Line 363:

	===Numerical Simulations===		===Numerical Simulations===
	[[~~File~~:x1 gate.gif\|thumb\|~~400px~~]]		[[Image:x1 gate.gif\|thumb\|200px]]
	[[~~File~~:x2 gate.gif\|thumb\|~~400px~~]]		[[Image:x2 gate.gif\|thumb\|200px]]
	<gallery>		<gallery>
	File:X1 00.png\|Beginning of X1 Gate		File:X1 00.png\|Beginning of X1 Gate

AdamFrees: /* Rotations */

2014-11-17T01:12:41Z

Rotations

← Older revision		Revision as of 01:12, 17 November 2014
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	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.		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===
			[[File:x1 gate.gif\|thumb\|400px]]
			[[File:x2 gate.gif\|thumb\|400px]]
			<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>

AdamFrees at 03:16, 11 November 2014

2014-11-11T03:16:04Z

← Older revision		Revision as of 03:16, 11 November 2014
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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.		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==		==General Formulation==
	[[Image:Detuning graph.png\|thumb\|Energy levels of the 2 qubit system as a function of both detunings.\|300px]]		[[Image:Detuning graph.png\|thumb\|Energy levels of the 2 qubit system as a function of both detunings.\|300px]]

AdamFrees: /* Notes on Logical Operating Points */

2014-11-03T23:05:02Z

Notes on Logical Operating Points

← Older revision		Revision as of 23:05, 3 November 2014
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	===Notes on Logical Operating Points===		===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.

AdamFrees: /* Constructing Logical Gates */

2014-11-03T21:46:14Z

Constructing Logical Gates

← Older revision		Revision as of 21:46, 3 November 2014
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	* <math>Z_1</math> gate (<math>Z</math> on qubit 1)		* <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)		* <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)		* <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)		* <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)		* <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)		* <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>		* <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===		===Notes on Logical Operating Points===

AdamFrees: /* Rotations */

2014-11-03T21:22:58Z

Rotations

@@ Line 107: / Line 107: @@
 ==Rotations==
 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.
@@ Line 165: / Line 164: @@
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>
 |}
@@ Line 220: / Line 224: @@
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>
 |}
 {| border="1" cellpadding="2"
@@ Line 322: / Line 330: @@
 \end{matrix}\right)</math>
 |}

AdamFrees: /* Rotations */

2014-11-03T17:27:31Z

Rotations

@@ Line 219: / Line 219: @@
 -B\frac{g}{\lambda_1}& 0 & 0 &B\frac{\Delta_1+\Delta_2}{\lambda_1}
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>
 |}

AdamFrees: /* Rotations */

2014-11-03T16:48:46Z

Rotations

@@ Line 123: / Line 123: @@
 & 0 & 0 & -B
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math> || <math>\frac{1}{2}\left( \begin{matrix}
-& A_1\times\text{Sign}(\Delta_1-\Delta_2) & A_2 & 0 \\
+& A_1\times\text{Sign}(\Delta_1-\Delta_2) & -A_2 & 0 \\
-A_1\times\text{Sign}(\Delta_1-\Delta_2) & 0 & 0 & A_2\times\text{Sign}(\Delta_1-\Delta_2) \\
+A_1\times\text{Sign}(\Delta_1-\Delta_2) & 0 & 0 & -A_2\times\text{Sign}(\Delta_1-\Delta_2) \\
-A_2 & 0 & 0 & -A_1 \\
+-A_2 & 0 & 0 & -A_1 \\
-& A_2\times\text{Sign}(\Delta_1-\Delta_2) & -A_1 & 0
+& -A_2\times\text{Sign}(\Delta_1-\Delta_2) & -A_1 & 0
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>
 |-
@@ Line 135: / Line 135: @@
 & 0 & 0 & -B
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math> || <math>\frac{1}{2}\left( \begin{matrix}
-& C_1 & C_2 & 0 \\
+& -C_1\times\text{Sign}(\Delta_1-\Delta_2) & -C_2 & 0 \\
-C_2 & 0 & 0 & C_4 \\
+-C_2 & 0 & 0 & -C_1 \\
-& C_3 & C_4 & 0
+& C_2\times\text{Sign}(\Delta_1-\Delta_2) & -C_1 & 0
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>
 |-
@@ Line 162: / Line 162: @@
 & B\frac{\Delta_1-\Delta_2}{\lambda_2} & B\frac{g}{\lambda_2}\times\text{Sign}(\Delta_1-\Delta_2)  & 0 \\
 & B\frac{g}{\lambda_2}\times\text{Sign}(\Delta_1-\Delta_2)  & -B\frac{\Delta_1-\Delta_2}{\lambda_2} & 0 \\
 -B\frac{g}{\lambda_1}& 0 & 0 &B\frac{\Delta_1+\Delta_2}{\lambda_1}
 \end{matrix}\right)\cos{(\omega_{AC}t)}</math>