Weighted Acquisition produces skewed IQ blobs.
First noticed by measurements run in the DiCarlo Lab; performing a NumericalWeightedIntegrationComplex
acquisition with optimal weights produces skewed IQ blobs. This issue is an attempt to explain why this occurs. Note that there are no complex numbers in the description, only real numbers.
Before the physical IQ mixer, we assume a signal coming into the port of the instrument to be:
\rm{RF} = A_r(t)\cos{(\omega_{rf}t + \phi_r(t))}
The signal is mixed with two signals from the LO:
\rm{LO1} = A_l\cos{(\omega_{lo}t + \phi_l)}
\rm{LO2} = A_l\sin{(\omega_{lo}t + \phi_l)}
The two outputs of the mixer are:
I_r(t) = \rm{RF} \times \rm{LO1} = \frac{A_r(t)A_l}{2}\cos{(\omega_{\rm{IF}}t + \phi_r(t) - \phi_l)}
Q_r(t) = \rm{RF} \times \rm{LO2} = -\frac{A_r(t)A_l}{2}\sin{(\omega_{\rm{IF}}t + \phi_r(t) - \phi_l)}
These two signals are then processed by the NCO (Digitally downconverted) using the following equations:
I(t) = I_r(t)\cos(\omega_{\rm{NCO}}t) - Q_r(t)\sin(\omega_{\rm{NCO}}t)
Q(t) = I_r(t)\sin(\omega_{\rm{NCO}}t) + Q_r(t)\cos(\omega_{\rm{NCO}}t)
If the NCO and IF are the same frequency, then:
I(t) = \frac{A_r(t)A_l}{2}(t)\cos{(\phi_r(t) - \phi_l)}
Q(t) = -\frac{A_r(t)A_l}{2}(t)\sin{(\phi_r(t) - \phi_l)}
Weighted integration in Qblox hardware is as follows:
I_{\rm{int}} = \sum_{i=0}^N I(i)W_a(i)
Q_{\rm{int}} = \sum_{i=0}^N Q(i)W_b(i)
To show the appearance of skewed blobs, we first measure the ground and excited state traces of a qubit:
Without weighted integration, and adding a gaussian noise to every trace, here are what the blobs look like:
The data is scaled, translated and rotated such that the mean of the ground state measurements is at I=0, Q=0, and the mean of the excited state measurements is at I=1, Q=0. The SNR reported is defined as the reciprocal of the standard deviation of I_int of the ground state measurements.
The optimal weights are calculated using the difference between the average ground and excited state traces:
W_a(t) = I_1(t) - I_0(t)
W_b(t) = Q_1(t) - Q_0(t)
Using the weighted integration scheme described above, here are the new integrated values:
We could instead adopt a complex multiplication scheme, defined as:
I_{\rm{int}} = \sum_{i=0}^N I(i)W_a(i) + Q(i)W_b(i)
Q_{\rm{int}} = \sum_{i=0}^N Q(i)W_a(i) - I(i)W_b(i)
In this case, here are the integrated values, along with the SNR: