Uniform finite-arity local-energy measurement synthesis

The effective encoding and fixed arity in [F5] give a fixed-dimensional matrix with computable entries. By Hermiticity and the norm bound, every eigenvalue $\lambda$ of $K\mathrm{/}B$ lies in $[-1,1]$; hence the eigenvalues of $E$ lie in $[0,1]$. The finite-dimensional spectral theorem therefore gives positive square roots $S=\sqrt{E}$ and $R=\sqrt{I-E}$. The block matrix $U_0=U=\left(\begin{array}{cc}S& R\\ R& -S\end{array}\right)$ is unitary: $S,R$ are commuting functions of $E$, $S^2+R^2=I$, and the off-diagonal products cancel. For input flag $0$, its flag-$0$ block is $S$. Follow the dilation by $X$ on the flag, so the accepted branch is labeled outcome $1$; define the marked-accept dilation $U_{acc}$ to be this full unitary. Its accepted effect is $S^{\ast }S=E$.

Choose degree $d\ge 1\mathrm{/}(4{\delta }^4)$ and use [F1] to form Bernstein polynomials $S_d=B_d(E)$ and $R_d=B_d(I-E)$. Functional calculus on the eigenbasis gives $\mathrm{\parallel }{S}_d-S\mathrm{\parallel },\mathrm{\parallel }{R}_d-R\mathrm{\parallel }\le \delta$. Thus the block matrix $M=\left(\begin{array}{cc}{S}_d& R_d\\ R_d& -S_d\end{array}\right)$ satisfies $\mathrm{\parallel }M-U\mathrm{\parallel }\le 2\delta$ by the triangle inequality for its two block columns. Taking $\delta \le 1\mathrm{/}16$ and applying [F2] gives an exactly unitary polar correction $Q$ with $\mathrm{\parallel }Q-U\mathrm{\parallel }\le 4\delta$.

Polynomial-time Cauchy names for the ideal dilation require a precision-polynomial construction. For $A=E$ or $I-E$, put $X_0=I$, $X_{j+1}=(X_j+A{X}_j^{-1})\mathrm{/}2$. On each eigenvalue $a\in [0,1]$, the scalar iterate satisfies $x_j\ge \max \{\sqrt{a},2^{-j}\}$ and $0\le x_{j+1}-\sqrt{a}=(x_j-\sqrt{a}{)}^2\mathrm{/}(2x_j)\le (x_j-\sqrt{a})\mathrm{/}2$. Thus $\mathrm{\parallel }{X}_j-\sqrt{A}\mathrm{\parallel }\le 2^{-j}$ and $\mathrm{\parallel }{X}_j^{-1}\mathrm{\parallel }\le 2^j$, even when $a=0$. For error $2^{-p}$, take $m=p+3$, approximate $A$ once to operator error $h=2^{-10(m+1{)}^2}$, and round every iterate to dyadic entries with total operator rounding error at most $h$. If the iterate error is $e_j\le 2^{-j-1}$, the inverse identity bounds inverse error by $2^{2j+1}{e}_j$ and approximate inverse norm by $2^{j+1}$. Therefore $e_{j+1}\le 2^{2j+2}{e}_j+2^{j+1}h+h$. With $e_0=0$, this gives $e_j\le 2^{2(m+1{)}^2}h$ for $j\le m$, smaller than $2^{-m}$ and $2^{-j-1}$, so the bounds hold throughout. In fixed local dimension $D$, rational inversion by adjugate and determinant uses a fixed number of arithmetic operations; the approximate iterates have least singular value at least $2^{-j-1}$ and determinant modulus at least $2^{-D(j+1)}$. Rounding after each iteration keeps all stored entries at $O_D(m^2)$ bits. The $m$ iterations and the input approximation consequently cost polynomial time in the encoding length and $p$. They supply polynomial-time Cauchy names for $S,R$ and $U_{acc}$.

Select $\delta \le \eta \mathrm{/}16$, and compile $Q$ to a circuit with a phase representative $V^{\mathrm{\prime }}$ satisfying $\mathrm{\parallel }{V}^{\mathrm{\prime }}-Q\mathrm{\parallel }\le \eta \mathrm{/}4$ using [F3]. The accepted flag outcome effect of the physical circuit is $\tilde{E}=V_{00}^{\ast }{V}_{00}$, where $V_{00}$ is the flag-$0$ compression before the final $X$ relabeling; the global scalar relating the physical unitary to $V^{\mathrm{\prime }}$ cancels in this effect. Compressions are contractions, so $\mathrm{\parallel }\tilde{E}-E\mathrm{\parallel }\le \mathrm{\parallel }{V}^{\mathrm{\prime }\ast }{P}_0{V}^{\mathrm{\prime }}-U^{\ast }{P}_{0}U\mathrm{\parallel }\le 2\mathrm{\parallel }{V}^{\mathrm{\prime }}-U\mathrm{\parallel }\le 2(\eta \mathrm{/}4+4\delta )\le \eta \mathrm{.}$ The fixed dimension is at most $2^{k+1}$; Bernstein degree is $O({\eta }^{-4})$, and its coefficients and matrix products have polynomially many bits at the needed precision. The polar series and fixed-width synthesis also have polynomial generator cost by [F2]–[F3]. Step 2.2 supplies polynomial-time Cauchy entries for $U_{acc}=(X_{flag}\otimes I)U_0$, so the same fixed-dimensional synthesis applies to the block diagonal matrix $I\oplus U_{acc}$ after adding one selector bit. Thus no controlled operation is assumed without synthesis.