Signals/Cross-capacitance.md

-TODO
-Virtual matrix
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+The chemical potential of the quantum dots used in spin-qubit devices determines the amount of electrons that
+will be loaded in the quantum dot. The chemical potential is controlled by a plunger gate close to the quantum dot.
+However, due to the small distance between quantum dots the potential of a quantum dot is also affected
+by neighboring plunger and barrier gates. This is the capacitive cross-talk, or cross-capacitance, from a gate to
+other quantum dots. This cross-capacitance is classical in nature and can be corrected by pulse_lib using
+"virtual gates".
+
+A virtual gate is a pulse_lib channel that outputs a pulse sequence on multiple AWG channels
+such that only a single chemical potential or single tunnel coupling will be affected by the pulses.
+Virtual gates are defined by a virtual gate matrix.
+
+
+# Virtual Gate Matrix
+
+A virtual gate matrix is a matrix that relates the virtual gate voltages to the real voltages.
+
+$`\begin{pmatrix} vP1 \\ vP2 \\ vP3 \end{pmatrix} = M \begin{pmatrix} P1 \\ P2 \\ P3 \end{pmatrix}`$
+
+When there would me no cross effects, this matrix would look like:
+
+$`M = \begin{pmatrix} 1  & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}`$
+
+Which gives a one to one map of the virtual gate voltages to the real voltages.
+In reality, this will not be the case. There will be cross-capacitances.
+
+$`M = \begin{pmatrix} 1  & C_{12} & C_{13} \\ C_{21} & 1 & C_{23} \\ C_{31} & C_{32} & 1 \end{pmatrix}`$
+
+The values on the diagonal are kept equal to 1. The lever arm of $`vP1`$ will then be equal to the lever
+arm of $`P1`$.
+A correct virtual matrix will remove the slant from the lines in the charge stability diagram without stretching it.
+
+See *Loading a quantum-dot based "Qubyte" register*, C. Volk (2019), for a detailed description of cross-capacitance and virtual gates.
+
+
+# Virtual gate matrix in pulse_lib
+
+Pulse_lib inverts cross-capacitance matrix $`M`$ to calculate the voltages on the output channels.
+
+	:math:`\begin{pmatrix} P1 \\ P2 \\ P3 \end{pmatrix} = M^{-1} \begin{pmatrix} vP1 \\ vP2 \\ vP3 \end{pmatrix}`
+
+The cross-capacitance matrix (real-to-virtual) can be passed to pulse_lib, but also the inverted
+matrix (virtual-to-real) can be passed. The cross-capacitance matrix has to be a square matrix, because it
+must be invertible. The virtual-to-real matrix doesn't have to be square.
+
+Example:
+```Python
+
+pulse.add_virtual_matrix(
+        name='virtual-gates',
+        real_gate_names=['B0', 'P1', 'B1', 'P2', 'B2'],
+        virtual_gate_names=['vB0', 'vP1', 'vB1', 'vP2', 'vB2'],
+        matrix=[
+            [1.00, 0.25, 0.05, 0.00, 0.00],
+            [0.14, 1.00, 0.12, 0.02, 0.00],
+            [0.05, 0.16, 1.00, 0.11, 0.01],
+            [0.02, 0.09, 0.18, 1.00, 0.16],
+            [0.00, 0.01, 0.013 0.21, 1.00]
+            ],
+        real2virtual=True)
+```
+See also core-tools (TODO link!)
+
+
+## Combining virtual gates to a combined parameter
+
+The use of the virtual matrix is not restricted to the definition of virtual gates to compensate
+for the cross-capacitance. It can also be used to define a new channel for voltage pulses that is
+a linear combination of virtual gates, e.g. to define a detuning parameter.
+
+Example:
+
+A detuning parameter e12 and a energy parameter U12 is defined using a virtual-to-real matrix.
+
+```Python
+
+pulse.add_virtual_matrix(
+        name='detuning12',
+        real_gate_names=['vP1', 'vP2'],
+        virtual_gate_names=['e12', 'U12'],
+        matrix=[
+            [+0.5, +1.0],
+            [-0.5, +1.0],
+            ],
+        real2virtual=False)
+```
+
+Likewise, virtual matrices can be stacked on top of each other.
+