[Feature]: Pairwise regularization on unitary trajectories for gate interpolation

[andgoldschmidt]

Feature Description

Unitary gates can be parameterized for applications like VQAs. For example, $U(\theta, \phi)$ might be a gate within a circuit that is being optimized by some VQA. During optimization, updates $\Delta \theta$ and $\Delta \phi$ occur. It would be nice to have a lookup table of the pulse for any target $U(\theta, \phi)$.

Start with a few targets $\{U(\theta_j, \phi_j)\}^J_{j=1}$. The best way to find controls that interpolate between these gates is to make the unitary trajectories between neighboring goals as close as possible. This can be accomplished by a pairwise unitary trajectory regularizer. In one dimension, a chain of pairs can accomplish this goal. In higher dimensions, we may need pairwise weights according to some connectivity graph.

The way to solve this is probably to allow for quantum systems and trajectories to combined via a non-interacting direct sum. That is, we need $\oplus$ between quantum systems, named trajectories.

Feature to implement

• Direct sum of quantum systems
• Direct sum of named trajectories (@aarontrowbridge I can submit a separate PR in NamedTrajectories)
• Implement UnitaryPairwiseQuadraticRegularizer to keep paths close
• Implement UnitaryDirectSumProblem to construct direct sum problems from previous problems.
• Write tests
• Utilities for coordinating points on surfaces in higher dimensions (triangle for 2D, simplex for 3D, etc.).

Importance

3

What does this feature affect?

• quantum system construction
• problem setup
• problem solution
• problem performance
• solution analysis
• plotting
• documentation
• tests
• other (please specify below)

Other information

No response

[andgoldschmidt]

Some changes to the original solution, this now does a much better job of handling everything via NamedTrajectories.

The problem template structure from 0.2 is also being used to localize the scope of the new code to the direct sum problem template.