Computing With Ellipses
2 Apr 2019

Generic 1-bit hermitian operators

As sums of the canonical operators X, Y, and Z

An arbitrary hermitian operator can be represented as a weighted sum of the three canonical operators, such that A = xX + yY + zZ, where x, y, and z are real numbers between +1 and -1. These coefficients are the dot products of the axis of the desired operator A and the axes of each of the X, Y, and Z operators. In the examples below, A is white, and X, Y, and Z are red, green, and blue. The projections of A along X, Y, and Z are also shown: The math is shown below. The notation '(a, b)' refers to a 1-bit state given by signals 'a' and 'b' on channels 1 and 2 respectively. Each of the a's and b's are complex numbers:

```X(a, b) = (b, a);     Y(a, b) = (-i * b, +i * a);     Z(a, b) = (a, -b)

A(a, b) = x * X(a, b) + y * Y(a, b) + z * Z(a, b);    {x, y, z} in [-1, 1]

= x * (b, a) + y * (-i * b, +i * a) + z * (a, -b)```

For separate channels, this looks like:

```     a' = x * b + y * (-i * b) + z * a

b' = x * a + y * (+i * a) + z * (-b)
```

Conceptually, this can be effected by the following circuit for non-negative values of x, y, and z. These coefficients become conductances, which are the reciprocals of resistance. The -i and +i components are from last time, and the -1 component is an inverter which was demonstrated previously as well: For x < 0, y < 0, or z < 0, these are replaced by their absolute values, and an inverter (-1) is added to (or removed from) the appropriate term for each channel, or -i becomes +i and vice-versa. In all, there are 4 similar circuits which cover the entire sphere of axes.

The simplest operator to calculate using this circuit is for x = y = z. These can be any value, as the relative proportions are what matters, so that the operator A is composed of equal amounts of the operators X, Y, and Z. The axis of A is therefore equally between the axes for X, Y, and Z, and has coordinates psi = pi / 8 and chi = atan(1 / sqrt(2)) / 2 on the computational sphere: The final circuit for this operator looks like the following. RX, RY, and RZ are all 100 K: On the breadboard, CH1 and CH2 switch sides at the output. This is due to the legacy layout of the data acquisition electronics, and has nothing to do with the X operator: Here is the complete apparatus. State preparation is at left (red USB cord), and state detection is at right (black USB cord): Here are tests of this circuit 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:  In this test, the measured axis is within 0.062 degrees of the true axis, and the average point-by-point error is 0.314 degrees (colored dots are 1 degree wide).