Single-qubit research demo on a link between p-bits, Riemannian geometry, and the quantum Bloch sphere

[toncho11]

Demo 1)

This is a single-qubit research demo. It is a proof of concept. The objective is to make a link between p-bit probabilistic computing, Riemannian geometry, and the quantum Bloch sphere. It explores this combination for a potential future quantum simulator.

A pure single-qubit state is treated first as a point on the Riemannian manifold CP^1, represented by the Bloch sphere S^2 with the Fubini-Study metric. Three independent p-bits statistically realize the Bloch coordinates, gates act as CP^1 isometries (distance-preserving transformations), and one additional p-bit samples projective measurements.

So the the qubit lives on CP^1, p-bits realize that state statistically, gates move it geometrically, and measurements are sampled probabilistically.

GATE_SEQUENCE represents a small quantum circuit test.

ManifoldPBitQubit # single-qubit simulator object
├── CP1BlochManifold # geometry
└── PBitCP1Backend # realization using p-bits

Demo 2)

Calibrate and execute a quantum circuit with a p-bit/Riemannian runtime backend. The main question explored by this demo is: can a noisy stochastic p-bit quantum runtime be calibrated using quantum-state Riemannian geometry so that it still produces the expected quantum-state result?

It also adds a few Python classes for clarity.

[gcattan]

Ok, I think the idea is very interesting.
If I understand correctly, your qubit is represented by 3 p-bit (+1 for the measurement), encoding each one for one of the Bloch sphere axis.

Your test at the end, is to show that if you apply different gate to the qubit, your retrieve the same stochastic distribution as for 1 single qubit.

I am right?

I guess next step would be to check 2-qubit circuit and entanglement.

Sounds promising.

[toncho11]

I updated demo 1 comments follwing your comment.
And I added demo 2. Demo 2 adds noise and calibration.