Derivation — C076 (drive A on |Psi+> excites one ion with prob p/2)

The bit-flip probability is an error process

The ideal drive-(A) propagator at a loop-closing time is $U_A=\exp(i\Phi S_{x,e}^2)$ (C012/C013). At the working point $\Phi=\pi/4$ this gives perfect $|\!\downarrow\downarrow\rangle\to|ee\rangle$ collective transfer (C003) and produces zero single-ion ($D_{5/2}$) excitation — the ideal MS drive never leaves exactly one ion in $|e\rangle$. (Confirmed in run.py: the ideal $U_A$ gives $P(\text{one ion in }e)=0$ for all $\Phi$ from both $|dd\rangle$ and $|\Psi^+\rangle$.)

The bit-flip probability $p$ in Supp. Mat. S5 is therefore an error quantity: a residual single-ion excitation $|\!\downarrow\rangle\to|e\rangle=D_{5/2}$ caused by imperfections (residual displacement $|\alpha|>0$, finite temperature, off-resonant terms). Let $r$ be the per-ion probability that one round of drive (A) excites a single $|\!\downarrow\rangle$ ion to $|e\rangle$. This is the lowest-order error process; double excitations are $O(r^2)$.

Counting argument

The drive (A) is the collective $\{|\!\downarrow\rangle,|e\rangle\}$ MS force. $|\!\uparrow\rangle$ is not coupled to $|e\rangle$ by this drive (it is a spectator).

From $|\!\downarrow\downarrow\rangle$ (definition of $p$). Both ions are in $|\!\downarrow\rangle$, so both are susceptible. The probability to find exactly one ion excited to $|e\rangle$ is $$p\equiv P(\text{one in }e\mid dd)=2r(1-r)\stackrel{r\ll1}{\approx}2r.$$ This is the paper's definition: "apply a single round of drive (A) and measure the probability to find exactly one ion in $D_{5/2}$ ... this probability equals the bit-flip probability $p$."

From $|\Psi^+\rangle=\tfrac1{\sqrt2}(|!\uparrow\downarrow\rangle +|!\downarrow\uparrow\rangle)$. In each branch exactly one ion is $|\!\downarrow\rangle$ (susceptible) and the other is $|\!\uparrow\rangle$ (spectator). So in either branch only the single $|\!\downarrow\rangle$ ion can be excited to $|e\rangle$, with probability $r$. The two branches excite different ions, giving orthogonal final states, so the coherence does not change the single-ion populations: $$P(\text{one in }e\mid\Psi^+)=r.$$

Result

$$\frac{P(\text{one in }e\mid\Psi^+)}{p}=\frac{r}{2r(1-r)}=\frac{1}{2(1-r)} \xrightarrow{r\ll1}\frac12,$$ so $$P(\text{one in }e\mid\Psi^+)=\frac{p}{2}\quad(\text{to leading order in }r).$$ Exactly the claim: applying a single round of drive (A) to $|\Psi^+\rangle$ excites one ion to $D_{5/2}$ with probability $p/2$. The factor $1/2$ is a pure counting factor — $|dd\rangle$ has two susceptible ($|\!\downarrow\rangle$) ions, $|\Psi^+\rangle$ has one per branch.

Verification

run.py evaluates $p=2r(1-r)$ and $P(\text{one}|\Psi^+)=r$ for a range of $r$; the relation $P(\text{one}|\Psi^+)=p/2$ holds with relative error equal to $r$ itself (the $O(r^2)$ double-excitation correction), e.g. $r=0.01\Rightarrow$ $p=0.0198$, $p/2=0.0099$ vs $P(\Psi^+)=0.0100$ — a $1\%$ discrepancy at the few-% bit-flip level. For the measured per-cycle bit-flip probabilities this correction is negligible. The statement is exact in the relevant small-error limit; the $O(r^2)$ deviation is a known limitation of treating $p/2$ as the single-ion excitation out of $|\Psi^+\rangle$ (the paper itself notes $p/2$ "underestimates the total excitation probability"). Reimplemented from the paper text + MS model (the self-contained model).