Computing With Ellipses
17 Mar 2019

Automated state preparation (left), and -90 degree rotation of states (right): Generic 1-bit operator building-blocks

-i and +i phase filters

The X, Y, and Z operators shown previously can be combined in different proportions to create a generic hermitian operator having any axis through the computational sphere. To do this, however, we will need to use a version of the Y operator that distributes the phase shift equally between both channels, rather than the one demonstrated previously that lumps it all into one channel by means of an inverter, or multiplication by -1, which is actually a phase shift of +pi or -pi. Instead, the phase shift will be distributed equally to each channel, by -pi/2 and +pi/2. These phase shifts correspond to multiplication by -i and +i, respectively. Multiplication by +i rotates a signal 90 degrees counter-clockwise in the complex plane, while multiplication by -i rotates a signal 90 degrees clockwise in the plane.

These operations are implemented in the analog electronics by two phase filters. The -i filter is shown first: it is based on two RC filter-follower stages, which cause the attenuated output to lag. Re-amplification of the filtered signal is provided by a non-inverting op-amp biased near mid-supply. The result is the input signal delayed in phase by 1/4 period. Here is a diagram of this circuit: The fractional resistance values were arrived at, by combining resistors in series and parallel, in order to make the phase delay as close to -pi/2 as possible, and then to amplify the signal back to its original magnitude. This was first done roughly with an oscilloscope and reference signal:  and then more finely on the computer using the state preparation and detection electronics and software. The -i filter is applied to the second channel of 1-bit state data, so that the relationship between input and output is:

```C1 -->      C1, and
C2 --> -i * C2``` This rotates the computational sphere -90 degrees about the HV axis, which has the effect of turning the linearly polarized state D into the circularly polarized state R: This is similar to the Z operator, but only half-way, and in the other direction. Since this is not a half-rotation of the sphere, it is not a hermitian operator, but it is one of the building blocks of generic hermitian operators. This filter was tested with a set of input signals distributed as points depicting two oriented figures on the sphere, and with another set consisting of random points on the sphere. One computer program compares the transformed points to their mathematical images. In this case, the overall average error is less than one half of a degree, which is about the same as the error in the untransformed data. Another program uses an iterative procedure to estimate the best rotation from the acquired data. It is interesting to note that this process is accurate enough to detect the slight (known from above) phase tuning error that causes a very small overrotation.

In the first two images below, green points are inputs, blue points are the transformed outputs. In the third image, the inputs have been transformed mathematically to compare to the outputs (plot rotated for a better view). The fourth image shows the best fit rotation axis in white, which is a positive rotation about the V axis (using the right hand rule). The last two images show results for random points. The green and blue dots are about one degree wide:  The +i filter is also partially based on two RC filter-follower stages. However, re-amplification of the filtered signal is now provided by an inverting op-amp biased near mid-supply. This creates an additional phase change of -pi (which is also +pi), resulting in a total phase change of -3pi/2, which is the same thing as +pi/2, because the signal is periodic. The result is the input signal advanced in phase by 1/4 period. Here is a diagram and photo of this circuit:  The +i filter is applied to the second channel of 1-bit state data, so that the relationship between input and output is:

```C1 -->      C1, and
C2 --> +i * C2```

This rotates the computational sphere +90 degrees about the HV axis, which has the effect of turning the linearly polarized state D into the other circularly polarized state G: In the images below, green points are inputs, blue points are transformed outputs. In the third image, the inputs have been transformed mathematically to compare to the outputs. The fourth image shows the best fit rotation axis in white, which is now a positive rotation about the H axis (using the right hand rule). The last two images show results for random points. Again, the average errors for all tests are less than one half degree, and the phase delay is more accurate than before:  