Quantum SDP additive-optimality guarantee

For a reached positive-$R$, positive-dimension round, condition on its prior history. By [F2], except on its allocated shift event the single measured shift produces a fixed $G$ with minimum eigenvalue in $[1,2]$. Its scalar identity shift changes no normalized Gibbs density. The finite Fourier coefficient construction and all evolution/decoded-value errors meet the allocated successful-block error. The numerator uses $F_A\mathrm{/}{B}_A^{+}$ and the baseline uses $I\mathrm{/}(2B_A^{+})$, so [F3] applies to the same frozen Gibbs block. Its actual probabilities exceed $1\mathrm{/}(1024n)$ and the returned trace label is $\theta$-good except on the declared event. Retaining every internal branch as orthogonal history and padding the finite cap supplies the coherent indexed circuit and its true inverse; no normalized-state postselection inverse is assumed.

The scalar estimate is measured once. For either minimum the declared deterministic call cap is known before choosing $\nu \le (\beta \mathrm{/}(16Q_{min}){)}^2$. Step 1.1 gives a bad probability at most $\nu$ on every preserved index. [F4]'s per-index good-component rotation is used only in analysis; its controlled/inverse operator-norm hybrid proves the required selected-label nondomination for the actual finite circuit. Both angular score orders assign sentinel zero inside the region. Thus the two selected labels either yield the exact rational nonnegative two-sparse weight list, with objective at most $\alpha$, norm at most $r$ and true trace residual at least $-2(r+1)\theta$, or return EMPTY, implying $\alpha <\mathrm{O}\mathrm{P}\mathrm{T}$ by the bounded-optimal-vector contrapositive. No uniform probability lower bound for an individual good label or simultaneous deterministic estimate table is assumed.

In each threshold call the algorithm allocates $\beta ={\delta }_{call}\mathrm{/}(10T)$ to the shift, scalar estimate and each of the two complete minima, with all implementation hybrids included in the stated budgets. Summing conditional bad probabilities over at most $T$ reached rounds gives at most $4T\beta \le {\delta }_{call}$. This is a predictable adaptive union bound, not an independence assumption. On the complementary event every returned loss has norm at most one and the true round trace bound of step 2.1. Hence [F5] applies to the ideal densities, whose finite implementations were compared with them in step 1.1: EMPTY may return HIGHER, whereas $T$ weights give a LOWER certificate $\bar{y}=(e\mathrm{/}R)e_1+T^{-1}{\sum }_t{y}_t$ with matrix slack at least $(e\mathrm{/}(2R))I$ and objective at most $\alpha +e$. Exact rational weights and their average retain that margin without a rounding inference.

By [F1], the initial threshold $R+e$ has a LOWER response on the good event. By [F6], at most $L+1$ good threshold calls suffice; their stored certificate has objective within $\epsilon$ of the optimum. Conditional bad probability $\delta \mathrm{/}(L+1)$ at each reached call bounds the probability of a first bad call by $\delta$. Operations on a history that already contains a bad event do not require an unbounded additional union bound. Thus with probability at least $1-\delta$ the finite algorithm returns the claimed feasible vector. An unsupported failure has no exclusion implication; the supplied proved complete implementations ensure such a failure does not occur on this good event. Taking $\delta =1\mathrm{/}3$ proves the stated probability.

Each of the $T$ weights is two-sparse and the identity shift adds at most one index, so the final certificate has support at most $2T+1$ before merging. Its rational finite list and objective are computed classically at their actual cost. Dimension zero, $R=0$, and $\epsilon \ge 2R$ use the direct boundary branches proved in [F6], preceding all positive-$R$ formulas. These complete the claimed scope.