RFC: quantum assertions/debugging tools

[Roger-luo]

Following the "off-topic" conversation from #218 (comment) and #218 (comment) (copying @cduck 's comment here to show the references)

As a side note, this reminds me that there is quantum programming language research that develops "quantum > assertions" like assert qubit_a entangled with qubit_b, assert qubit_a shares no entanglement with qubit_c, > assert qubit_d==|+>, or assert qubits_abcd are in a uniform superposition, etc. These assertions can be easily checked on the state vector. And you can approximately check these during execution at the cost of destroying the quantum state, but still useful for validation.
Here's one paper: https://arxiv.org/pdf/2303.01487
Also: https://docs.quantinuum.com/tket/user-guide/manual/manual_assertion.html

I think this is a good timing thinking about some academia results on this as we are trying to improve the UX of this package.

state recording for simulators

Idea proposed by @jon-wurtz

@bloqade.kernel
def main():
    qubits = bloqade.make_qubits()
    kernel_that_does_things_to(qubits)
    bloqade.state.record("label 1",qubits)
    if measure(qubits[0])==0:
        kernel2_that_does_things_to(qubits)
    else:
        kernel3_that_does_things_to(qubits)
    bloqade.state.record("label 2",qubits)

bloqade.emulator.statevector( main )
-> {"label 1": [(probability_A, measurement_outcomeA, statevectorA)], "label 2": [(probability_B, measurement_outcomeB, statevectorB), (probability_C, measurement_outcomeC, statevectorC), ...] }

I think this could be combined as part of the logging with debug dialect's debug.info which records a message and given object (QuEraComputing/kirin#342).

quantum assertions

I think Gushu Li has some work around this a few years ago, maybe a good feature to get some help from. We can add similar projector assertion here as well.

cc: @jon-wurtz @cduck

[cduck]

Another academic reference: https://dl.acm.org/doi/pdf/10.1145/3307650.3322213