PauliMeasurementGate fails to construct when given an empty observable

[Strilanc]

The following should not raise an exception. It should produce a circuit which, when sampled, samples a=0 and b=1.

import cirq
circuit = cirq.Circuit(
    cirq.PauliMeasurementGate(cirq.DensePauliString(""), key="a").on(),
    cirq.PauliMeasurementGate(-cirq.DensePauliString(""), key="b").on(),
)

Failing to measure the empty pauli string is sort of like failing to search for matches to the empty string. It's a corner case but it should still work.

[tanujkhattar]

Since cirq.PauliMeasurementGate currently decomposes into a cirq.MeasurementGate, this would require us to also extend support for empty qubits to cirq.MeasurementGate so that cirq.MeasurementGate(key="a") samples a=0 and cirq.MeasurementGate(key="b", invert_mask=1) measures b=1. Although, for now invert_mask expects a tuple with 1 entry per qubit. In this case, the number of qubits are 0 so that breaks down.

Another approach would be to not change the interface of cirq.MeasurementGate and instead have a special decomposition / override the _act_on_ protocol on PauliMeasurementGate here to achieve the desired effect, but from a quick glance it looks like SimulationState.measure also expects a non-zero qubits parameter to be measured.

We can also consider leveraging the newly added capabilities of allocating a new ancilla qubit within _decompose_with_context_ to perform the measurement using a clean ancilla and that would allow us to reuse all the existing infrastructure of making measurements on non-zero qubits; at the cost of a performance penalty.

Further investigation is needed to decide what's the best way to achieve this without breaking backwards compatibility and hopefully with minimal API changes.