Incorrect Behavior of Transformer `eject_phased_paulis` with Mid-Circuit Measurements

[p51lee]

Description of the issue

The eject_phased_paulis transformer exhibits incorrect behavior when interacting with X or Y gates preceding measurements. Specifically, the transformer erroneously merges these gates into measurements, as demonstrated by the following transformation:

Original circuit:

0: ───X───M('c_0')───M('c_1')───

Transformed circuit:

0: ───!M('c_0')───M('c_1')───

This issue notably affects the measurement outcome of c_1, potentially altering the execution results of circuits incorporating mid-circuit measurements. A temporary solution might involve verifying the position of the measurement within the circuit, ensuring it only merges operations when measurements are at the end of the circuit.

How to reproduce the issue

import cirq

circuit = cirq.Circuit(
    cirq.X(cirq.LineQubit(0)),
    cirq.measure(cirq.LineQubit(0), key='c_0'),
    cirq.measure(cirq.LineQubit(0), key='c_1'),
)
circuit_trans = cirq.eject_phased_paulis(circuit)

print(circuit)
print(circuit_trans)

Execution result:

0: ───X───M('c_0')───M('c_1')───
0: ───!M('c_0')───M('c_1')───

Cirq version 1.3.0