Computing With Ellipses
18 Apr 2019

Generic 1-bit hermitian operators

In two octants of the computational sphere

The circuit shown last time can be used to create any hermitian operator in two octants of the computational sphere: one in which the X, Y, and Z dot products are all positive, and one in which the X, Y, and Z dot products are all negative, since corresponding operators in these two octants are actually the same (since they are half rotations of the sphere). Shown below are 7 axes in these two octants. The middle axis of the group is formed when RX = RY = RZ, as shown previously. The other 6 axes are formed by using 2:1 and 1:2 ratios of dot products, which move the central axis either toward, or away from, each of the X, Y, or Z axes: The orientation of the operator axis can be adjusted by simply changing pairs of resistors in the final summation stage of the circuit. Resistor ratios are inverses of dot product ratios, since the dot products represent conductances. In the following image, RX = 2 * RY = 2 * RZ, so that the operator axis moves away from the X axis. The table shows the orientation of the axis for each resistor combination: ```RX      RY      RZ         Psi           Chi
-----   -----   -----      -----------   -----------
100 K   100 K   100 K      0.392699082   0.307739854
100 K   100 K   200 K      0.553574      0.364864
200 K   100 K   100 K      0.231824      0.364864
100 K   200 K   100 K      0.392699      0.169918
100 K    50 K   100 K      0.392699      0.477658
50 K   100 K   100 K      0.553574      0.210267
100 K   100 K    50 K      0.231824      0.210267
```

Below are tests of the last 6 resistor combinations on oriented figures (green = input, blue = output) and random points. The left images show the measured operator axes (yellow), and the right show the registration of calculated points with actual outputs. Only the average axis (white) is shown in the random case.

RX = 100 K, RY = 100 K, RZ = 200 K: The measured axis is within 0.097 degrees of the true axis, and the average point-by-point error is 0.332 degrees.

RX = 200 K, RY = 100 K, RZ = 100 K: The measured axis is within 0.065 degrees of the true axis, and the average point-by-point error is 0.287 degrees.

RX = 100 K, RY = 200 K, RZ = 100 K: The measured axis is within 0.109 degrees of the true axis, and the average point-by-point error is 0.343 degrees.

RX = 100 K, RY = 50 K, RZ = 100 K: The measured axis is within 0.120 degrees of the true axis, and the average point-by-point error is 0.309 degrees.

RX = 50 K, RY = 100 K, RZ = 100 K: The measured axis is within 0.165 degrees of the true axis, and the average point-by-point error is 0.354 degrees.

RX = 100 K, RY = 100 K, RZ = 50 K: The measured axis is within 0.088 degrees of the true axis, and the average point-by-point error is 0.284 degrees.

In summary, here are the locations of the 7 axes and their measured points (in yellow) in the first octant: 