[IN REVIEW] Check if an analog AnalogGate implements an Ising-like Hamiltonian

[benjimaclellan]

What's the goal:

Analog-mode quantum programs define the evolution of the quantum system in terms of continuous-time evolution. In Open Quantum Design's stack, we define such programs with the analog interface, where a circuit is composed of AnalogGate's which implement a continuous-time Hamiltonian operator.
In this bounty, the goal is to check if a provided AnalogGate object is Ising-like.

Some details:

Documentation on the analog layer is here. Relevant classes in the analog layer include:

• AtomicGate, which has a hamiltonian attribute.
• The operators in src/oqd_core/interface/analog/operators.py, which are composed to create an abstract data tree of a continuous-time Hamiltonian.
• The base compiler passes in oqd-compiler-infrastructure. More documentation of how the compiler infrastructure works is found here, and an example here.

The analysis on an AnalogGate object should check if the hamiltonian is Ising-like and satisfies the following properties:

• Composed of only qubit registers (i.e., no bosonic mode operators in the Hamiltonian expression).
• Composed of weight-2 Pauli strings, i.e., each additive term of the Hamiltonian is composed of only PauliI, PauliX, PauliY, and PauliZ terms, with only two terms in the Kronecker product being non identity.
• Has only time-independent coefficients.

The analysis should also extract the coupling matrix for each pair of Pauli operators, $XX$ vs $YY$ vs $ZZ$ vs $XZ$ vs $XY$, etc.
This is useful and important as certain class of Ising-like Hamiltonians can be engineered in trapped-ion systems by tuning the control laser parameters. Our goal, then, is to compile from a top-level AtomicCircuit program defined by a user into the real OQD hardware to simulate physical systems of interest.

One output of the analysis pass is the interaction matrix for all two-qubit Pauli operators:

def analysis(gate: AnalogGate):
    # compiler logic to extract Ising couplings
   return {
    'XX': np.array([ ... ]),
    'ZZ': np.array([ ... ]),
    ....
}

Possible extensions

• Allow the selection of which Ising-like terms are allowed, i.e., $XX$ vs. $XY$ vs. $ZZ$, etc. Some of these interaction Hamiltonians can be engineered with trapped ion systems more natively than others can.

[idriss-hamadi]

Hello, can you please check my PR