Stretch Qats and DoQs towards two distinct quantum states

[muttley2k]

The question whether we can overcome the ~82% one-shot accuracy that we get when we use Northern/Southern hemispheres, measure PauliZ and our classifier doesn't do anything.

To achieve this, somehow the Qat and DoQ regions should be stretched towards two distinct quantum states, e.g. using 1 qubit, imagine stretching the surface of the Bloch sphere towards its North and South poles at the same time. This clearly won't happen if the classifier is just a 1-qubit unitary, because that would just rotate the points on the sphere, but wouldn't stretch anything.

Two strategies which may help:

• Use data re-uploading to bring in non-linearity. (During the hackathon the way I tried did not really work out.)
• Feed the 1-qubit sensor output into a multi-qubit classifier. (I have doubts whether this can work even in principle, as a multi-qubit unitary is still linear. But maybe if we classify by doing a measurement that has more than 2 possible outcomes.)

[Zed-Is-Dead]

Reflecting on the issue statement above, I tried to figure out how one could stretch a given region of the Bloch sphere towards N or S pole. I conjecture that this can't be done by discrete (unitary) operations; but how can we implement a continuous transformation that would drive each state of the given region towards N/S pole? Similar to a loxodrome curve (used in navigation): https://en.wikipedia.org/wiki/Rhumb_line Maybe this is too complicated or not relevant?

[muttley2k]

If you can manage to pull along the rhumb line, you're gonna be famous :-) I think we either need data re-upload or post selection (after measurement) to introduce non-linearity. I recall in Schuld's book there is a method of post selection described to introduce non-linearity.

[Zed-Is-Dead]

I will have a look - I think there is also a reference to re-uploading in Maria's last paper

[muttley2k]

Regarding the second option, the multi-qubit classifier (without data re-uploading). Wondering whether using a measurement that has K possible outcomes would mean we have effectively a poor man's nearest neighbor classifier if K is big enough.

Imagine a 1-qubit classifier where in the end we do a POVM measurement which can have 100 outcomes, corresponding to 100 vectors equally distributed all around the Bloch sphere (surely, we'll need many ancilla qubits to do that I assume, if PennyLane only allows projective measurements).

It's definitely true for K=2 that when we use expected value, it's basically a nearest neighbor classifier which checks whether the input state is closer to the North or to the South Pole.

The nice thing is that with n qubits we can have 2^n different measurement results, which suggests that we don't need to calculate expected values, 1 shot is enough when measuring, as 2^n measurement outcome vectors can be very dense on a Bloch sphere. Just thinking loud...