TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Conditional Distance-Three Extended-Rectangle Threshold

Statement

Assume the finite recursively closed distance-three, $t = 1$ gadget family, complete syndrome decoder, explicit port discipline, local channel model and checked elementary correction, gate, pure-preparation and destructive-basis-measurement identities of Distance-Three exRec Correctness and Truncated Witness Reduction. Each level-zero rectangle is one elementary location, faulty independently with probability $η \in [0, 1]$ . Let $b \geq 2$ bound the number of child-location roots in each one-scale exRec template, including leading ECs, and put $A_{1} = (\binom{b}{2})$ . Use that lemma's reverse scan and expanded-support truncation convention.

Define the uniform probability envelope $p_{0} = η$ and $p_{r + 1} = A_{1} p_{r}^{2}$ . Then

p_{r} = A_{1}^{- 1} (A_{1} η)^{2^{r}} .

For any specified $k$ roots, the probability that they are jointly selected bad at level $r$ is at most $p_{r}^{k}$ . For $N$ top-level roots, the probability of any selected bad root is at most $N p_{r}$ , and every final classical success probability differs from ideal by at most $N p_{r}$ . If $η < 1 / A_{1}$ , then $p_{r} \to 0$ super-exponentially. For fixed gadgets and fixed subthreshold $η$ , achieving error at most $δ$ requires $O (\log \log (N / δ))$ levels and polylogarithmic multiplicative size and depth overhead. Elementary gadget correctness remains an explicit hypothesis; recursive exRec correctness and the witness reduction are supplied by the cited lemma.

Facts & Assumptions

Given: The checked local gadget and recursive port hypotheses in the statement, $b \geq 2$ , the independent fault process, $N \geq 1$ , and target $δ \in (0, 1]$ . Prove the probability and overhead bounds.

[F1]

Distance-Three exRec Correctness and Truncated Witness Reduction supplies correctness of good complete or truncated exRecs and the joint bound $(A_{1}^{2^{r} - 1} η^{2^{r}})^{k}$ from fixed disjoint elementary witness leaves.

[F2]

Independent Local Quantum Fault Model supplies the finite independent elementary fault process and fixed patternwise local replacement maps.

[F3]

The finite union bound bounds a finite union by the sum of its probabilities.

Proof

1.1givenF1F2algebra

The local hypotheses in the statement are precisely those of [F1], and [F2] supplies its independent elementary model. Thus [F1] bounds the joint selected-bad event for any specified $k$ roots by $(A_{1}^{2^{r} - 1} η^{2^{r}})^{k}$ . Since $A_{1}^{2^{r} - 1} η^{2^{r}} = A_{1}^{- 1} (A_{1} η)^{2^{r}}$ , this is $p_{r}^{k}$ . This argument uses only fixed elementary witness leaves; the selected bad-root events need not be independent.

2.1step 1.1algebra

The expression $p_{r} = A_{1}^{- 1} (A_{1} η)^{2^{r}}$ equals $η$ at $r = 0$ and satisfies $A_{1} p_{r}^{2} = A_{1}^{- 1} (A_{1} η)^{2^{r + 1}} = p_{r + 1}$ . Hence it is the stated recurrence envelope. In particular step 1.1 with $k = 1$ bounds each selected bad-root probability by $p_{r}$ .

3.1givenF1F3step 2.1algebra

By [F3], the probability $α$ of any selected bad top-level root is at most $N p_{r}$ . For every pattern in its complement, [F1] gives the ideal output instrument. Consequently the full output law is $(1 - α) P_{i d e a l} + α Q$ for some probability law $Q$ ; at $α = 0$ choose $Q$ arbitrarily. Each success probability differs from ideal by at most $α \leq N p_{r}$ . This remains true if the complement has probability zero, since then $α = 1$ .

4.1givenstep 2.1algebra∎

If $η = 0$ , $p_{r} = 0$ at every level. Otherwise for fixed $η < 1 / A_{1}$ put $θ = A_{1} η \in (0, 1)$ and $c = - \log θ > 0$ . The bound $N p_{r} \leq δ$ holds whenever $2^{r} \geq c^{- 1} \log (N / (A_{1} δ))$ , and level zero suffices when the right side is at most one. Hence the least qualifying level is $O (\log \log (N / δ))$ . The finite recursive gadget family has a constant size and depth expansion bound $κ \geq 1$ , so a circuit of original size $S$ and depth $D$ has size at most $S κ^{r}$ and depth at most $D κ^{r}$ , with a fixed boundary factor. Its multiplicative overheads are polylogarithmic, and $p_{r} = A_{1}^{- 1} e^{- c 2^{r}} \to 0$ super-exponentially.

Remarks

The local gadget conditions and port discipline are essential. The witness-tree proof handles fault-dependent truncation without independence of adaptive selected events. It applies to the stated pure-preparation and destructive-basis-measurement primitive family; no general mixed multioutput preparation or arbitrary POVM primitive is inferred.

Depends on

Used by

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Dependency tree · two levels

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Sources

Aliferis, Gottesman and Preskill, Quantum accuracy threshold for concatenated distance-3 codes (standard reference, not scraped)