C075 - As long as the two pi/2 parity-analysis pulses have phases offset by pi/2, the sum of the...
Verdict: verified
Location: Supp. Mat. S5
Type / expected artifact: math / math
Claim: As long as the two pi/2 parity-analysis pulses have phases offset by pi/2, the sum of the two parity measurements equals claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain verified -> verified.
Source location(s): source/supp_content.tex:144 (Supp. Mat. S5).
Conclusion
Summed parity of two pi/2 pulses offset by pi/2 equals XX+YY exactly, independent of absolute phase phi (verified symbolically for generic phi; cross terms cancel). Matches paper.
Verification details
Derivation excerpt: The protocol measures the ground-state parity $\langle Z\otimes Z\rangle$ after applying a global $\pi/2$ analysis pulse to both ions. A $\pi/2$ pulse with phase $\varphi$ rotates each qubit about the axis $\hat n(\varphi)=(\cos\varphi, \sin\varphi,0)$ in the $x$-$y$ plane: $$R(\varphi)=\exp\!\big(-i\tfrac{\pi}{4}(\cos\varphi\,\sigma_x+\sin\varphi\,\sigma_y)\big) =\cos\tfrac{\pi}{4}\,\mathbb 1-i\sin\tfrac{\pi}{4}(\cos\varphi\,\sigma_x+\sin\varphi\,\sigma_y).$$ The measured observable for analysis phase $\varphi$ is $$\Pi(\varphi)=\big(R(\varphi)\!\otimes\!R(\varphi)\big)^\dagger (Z\otimes Z)\big(R(\varphi)\!\otimes\!R(\varphi)\big).$$
Executable rerun: sympy_check.py exited 0 in 1.976s; log verification/C075/attempts/R002/sympy_check.log.
Output excerpt:
summed parity observable (matrix):
?0 0 0 0?
? ?
?0 0 2 0?
? ?
?0 2 0 0?
? ?
?0 0 0 0?
O == (XX+YY): True
O == -(XX+YY): False