AG-CoT: Verified Algorithmic Traces for LLM Program Synthesis on Clifford Circuits

Lu Wei; Yufeng Wang; Chenfeng Cao; Lu Pang; Haibin Ling

cs.LGSep 27, 2026

AG-CoT: Verified Algorithmic Traces for LLM Program Synthesis on Clifford Circuits

Authors: Lu Wei, Yufeng Wang, Chenfeng Cao, Lu Pang, Haibin Ling

Organizations: Data Science Department, Stony Brook University · Department of Computer Science, Stony Brook University · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin · Westlake University

Abstract

Scientific code generation can produce executable programs that fail to compute the intended scientific object. We study this problem in language-model synthesis of Clifford circuits, which prepare the stabilizer states used in quantum error correction and admit exact classical verification. In our target-conditioned framework, each target is given as compact signed stabilizer generators, and an exact verifier checks the generated OpenQASM circuits. We supervise models with Aaronson-Gottesman chain-of-thought (AG-CoT) traces checked by the verifier, and continue training on model generations that the verifier accepts. Across two independently trained model families (3B and 7B), AG-CoT supervision multiplies greedy-decode state-equivalence accuracy by four to six times over circuit-only baselines, and verifier-filtered continuation training adds a further consistent gain atop both. A complementary 32B study shows that supervised models achieve near-perfect syntax and Clifford validity while the strongest direct model reaches 6.14% state equivalence per target, rising to over 10% under verifier-guided selection with multiple candidates. These results show that algorithmic trace supervision gives a large, statistically significant gain in both model families and that verifier-filtered continuation adds a further repeated gain. The persistent gap between Clifford validity and state equivalence confirms that exact verification is necessary: a circuit can be syntactically and physically valid yet prepare the wrong quantum state.

Figures & tables

Panel of Figure 1 — Figure 1: Running verifier example. Both candidates parse and remain Clifford-valid, but Candidate B is rejected because it prepares $+Z$ rather than the requested $+X$ target.

Figure 2: Target-conditioned circuit synthesis with algorithmic trace supervision and verifier-filtered continuation. Panel a constructs a verified teacher circuit from signed target generators; b pairs circuit-only and AG-CoT supervision with the same prompt and final circuit. The two-qubit example is schematic, with the blue reduction operations inverted to obtain the preparation circuit. Panel c forms a fixed dataset of accepted AG-CoT responses for supervised continuation (RFT). Acceptance checks the emitted circuit and task requirements. Panel d shows correct single outputs on the same 3,661 test targets for each trained system in Table 1 . The three Qwen RFT versions remain separate.

Figure 3: A target-conditioned AG-CoT training example. The reformatted input and trace excerpt retain all four reduction operations and the corresponding auxiliary and stabilizer rows for qubit 0. Inverting the reduction gives the complete program shown on the right. Circuit labels identify the reduction steps in reverse order. The adjacent final Hadamard gates remain as emitted by the teacher.

System	Correct/total	Suc(%)
Qwen2.5-3B-Instruct
Base model	0 / 3,661	0.00
Direct SFT	33 / 3,661	0.90
AG-CoT SFT	201 / 3,661	5.49
RFT ver. 1	213 / 3,661	5.82
RFT ver. 2	215 / 3,661	5.87

Table 1: Single-output training comparison on the same 3,661 target prompts. Each row uses one greedy completion per target and no program repair.

Panel of Figure 4 — Figure 4: Qwen2.5-3B-Instruct task success by target size, following requirements in Sec. 5 . Each rate includes all targets at that size and a greedy output per target (Appendix Table 13 ).

Comparison	only	only	$p$	$p$
	Earlier	Later	Exact	Holm
Direct SFT to AG-CoT SFT	2	170	4.97 $\times$ 10 -48	–
AG-CoT SFT to RFT ver. 1	7	19	0.029	0.058
AG-CoT SFT to RFT ver. 2	7	21	0.013	0.038
AG-CoT SFT to RFT ver. 3	9	18	0.122	0.122

Table 2: Paired Qwen2.5-3B-Instruct task success comparisons on the 3,661 targets. Earlier only and later only are discordant successes in the order shown. Exact

p

uses the two sided McNemar test. Holm adjusted

p

covers the three comparisons from AG-CoT SFT to RFT.

System	Syntax	Clifford	Exact@1 lab.	StateEq@1	95% CI
	Surface validity		Target match
Zero-shot	39.9	39.9	0.10	0.33	[0.15, 0.52]
Basic AG-CoT	99.9	99.9	2.62	5.37	[4.67, 6.06]
Step AG-CoT	99.9	99.9	2.31	4.83	[4.15, 5.50]
GRPO-v2	99.9	99.9	2.21	5.26	[4.57, 5.98]
RFT-v1	99.6	99.6	3.13	6.14	[5.37, 6.91]

Table 3: 32B direct results on 3,894 prompts. Rates are percentages. Exact@1 lab. checks saved-label equality, while StateEq@1 checks state equivalence over the same targets. The final column gives the reported 95% confidence interval for StateEq@1.

Model	Criterion	1	4	8	16	32	64
RFT-v1	Label	3.18	3.69	4.00	4.37	4.78	5.19
Basic AG-CoT	Label	2.13	2.93	3.44	3.78	4.01	4.42
Basic AG-CoT	StateEq	4.65	6.40	7.33	8.29	9.24	10.07

Table 4: Pass@

N

(%) from 64 candidates per target. Values for

N<64

are estimated from this pool, and the

N=64

column gives observed coverage. The Basic AG-CoT rows use the same 249,216 candidates, with StateEq-Pass@64 of 10.07% versus 4.42% under strict saved label equality.

System	Errors	Mean deficit	Median deficit	Deficit $=1$ Count (%)	Deficit $\leq 2$ Count (%)
Direct SFT	3,616	8.22	9	39 (1.08)	148 (4.09)
AG-CoT SFT	3,425	7.66	8	103 (3.01)	266 (7.77)
RFT ver. 1	3,398	7.61	8	121 (3.56)	278 (8.18)
RFT ver. 2	3,409	7.60	8	108 (3.17)	279 (8.18)
RFT ver. 3	3,416	7.57	8	115 (3.37)	274 (8.02)

Table 5: Constraint deficits among valid but state-incorrect outputs of adapted Qwen2.5-3B models in Table 1 . The deficit counts missing independent target constraints. Each model has its own error set. Means, medians and percentages use the corresponding error count, not a common set of failed targets. Outputs with deficit one also contribute to the final column.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Object	Verifier role
Evaluation source	QCircuitBench Clifford random-circuit instances	Provides targets with exact stabilizer-tableau verification.
Prompt target	Signed Pauli generators $T_{x}$	Specifies the target state without exposing a reference circuit.
Candidate output	OpenQASM 3.0 program $y$	Defines an executable state-preparation circuit from $\|0\rangle^{\otimes n}$ .
Strict 32B criterion	$P(y)$ , $K(y)$ , and $\mathcal{L}(T_{y})=\mathcal{L}(T_{x})$	Measures saved-label Exact@1.
State-equivalence criterion	$P(y)$ , $K(y)$ , and $\mathrm{Equiv}(T_{y},T_{x})=1$	Removes generator-basis sensitivity.
Controlled task criterion	StateEq plus gate, qubit, header, and completion requirements	Measures task success in Table 1 .

Appendix

Table 6: Verifier protocol for the Clifford state-preparation task.

Panel of Figure 5 — Figure 5: Target representation shift. Signed stabilizer generators replace long numeric state vectors with a compact target interface that still supports exact Clifford-state verification. Here $n$ is the qubit count, $M_{x}$ stores binary $X/Z$ support, and $r_{x}$ stores generator signs.

Representation	Model-facing object	Role in this study
State vector	$2^{n}$ complex amplitudes	General but long text.
Stabilizer-state tableau	Signed Pauli generators; $2n^{2}+n$ bits before formatting	Compact exact Clifford target.
OpenQASM output	Clifford circuit program	Executable candidate.

Appendix

Table 7: Representations used in the preprocessed Clifford state-preparation task.

Split or check	Count	Notes
Accepted AG-CoT examples	38,940	Before train/test split.
Generation failures	0	During AG-CoT construction.
Training examples	35,046	90% split.
Held-out test prompts	3,894	10% split.
Conversion check	500	Random tableau/circuit checks.
AG validation	1,000	Random Clifford tests.

Appendix

Table 8: Dataset construction and validation counts.

Qubits	2	3	4	5	6	7	8	9	10	11	12
Prompts	30	51	107	145	219	311	367	475	618	702	869

Appendix

Table 9: Held-out test prompt counts by qubit count. The counts sum to 3,894.

Panel of Figure 6 — Figure 6: Overview of the 32B construction and verification. The target $T_{x}=(M_{x},r_{x})$ stores binary Pauli support $M_{x}$ and generator signs $r_{x}$ . The trace $\rho_{x}$ records $m$ reduction steps through full reference Clifford tableaus $T_{0},\ldots,T_{m}$ to the identity $I$ , and $y_{x}$ is the emitted OpenQASM program. For a generated program $y$ , $P(y)$ and $K(y)$ indicate parsing and Clifford validity, $T_{y}$ is its prepared state tableau, and $\mathcal{L}$ extracts saved stabilizer labels. The rightmost panel reports 32B direct results and basic AG-CoT search with 64 candidates per target.

Panel of Figure 7 — Figure 7: AG-CoT data construction for the 32B study. The construction records the row-reduction trace and the synthesized OpenQASM circuit from the same process, then retains completions whose circuits pass the strict saved-label verifier.

Component	Required record	Reproducibility role
Held-out prompts	3,894 targets plus metadata	Replay held-out evaluation inputs.
Verifier/evaluator	Extraction, Clifford conversion, Exact@1 , StateEq@1 , Pass@ $N$	Recompute reported metrics.
AG-CoT construction	Row-reduction traces plus verifier acceptance	Rebuild SFT data.
Evaluation records	Commands, evaluated checkpoints, decoding settings, checksums	Identify the evaluated configuration.
Dependencies	Qiskit and model-stack versions	Recreate the software environment.

Appendix

Table 10: Components required to reproduce the 32B study.

Package	32B study	3B eval.
Python	3.11	3.12.3
PyTorch	2.9.1	2.5.1
Transformers	4.57.6	4.52.4
TRL	0.27.2	–
PEFT	0.19.1	–
vLLM	0.15.0	0.6.4.post1

Appendix

Table 11: Software versions for the 32B study and the 3B single-output evaluations. A dash marks a package version not specified for the 3B evaluations.

System	A6000	H200 NVL	Shared successes
AG-CoT SFT	201	202	201
RFT version 1	213	213	213

Appendix

Table 12: Exact task success across two inference environments on the same 3,661 targets. Shared successes count targets solved in both environments. The A6000 counts are those reported in Table 1 .

Qubits	Targets	Direct SFT	AG-CoT SFT
3	30	0	3
4	95	1	8
5	135	3	12
6	201	2	10
7	282	4	16
8	355	7	29

Appendix

Table 13: Task success counts by qubit count for the 3B comparison. Both models use the same targets and one greedy completion per target. The target count is the denominator for both success rates in each row.

Family	Checkpoint	Correct / total	Task success (%)
Qwen	Base	0 / 3,661	0.00
Qwen	Direct SFT	41 / 3,661	1.12
Mistral	Base	0 / 3,661	0.00
Mistral	Direct SFT	78 / 3,661	2.13

Appendix

Table 14: Matched cross-family Direct-SFT replication on a common 3,661-row target-disjoint development partition. Each row uses one greedy FP16 completion with a 2,048-token generation cap and strict no-repair evaluation. These results are separate from the 8,000-token comparisons in Table 1 .

Qubits	Prompts	Pass@1 (%)	Pass@64 (%)
2	30	21.15	86.67
3	51	7.20	21.57
4	107	6.57	13.08
5	145	4.70	7.59
6	219	3.30	4.57
7	311	7.11	8.04

Appendix

Table 15: Search results by qubit count for RFT. Pass@1 averages the fraction of correct candidates among 64 samples per target, and Pass@64 gives observed target coverage.

Run	Samples	Metric	Value (%)
Zero-shot	1	Exact / StateEq	0.10 / 0.33
Basic AG-CoT	1	Exact / StateEq	2.62 / 5.37
Step AG-CoT	1	Exact / StateEq	2.31 / 4.83
GRPO-v2	1	Exact / StateEq	2.21 / 5.26
RFT-v1	1	Exact / StateEq	3.13 / 6.14
RFT-v1 Best-of-64	64	Pass@64	5.19

Appendix

Table 16: Evaluation budgets for the 32B results over 3,894 held-out prompts.

Run	Max tok.	Rate (%)
		Syntax	Exact@1	StateEq@1	Trunc.
QASM-only SFT	2048	24.91	5.50	7.68	83.00

Appendix

Table 17: Exploratory QASM-only format stress test under a 2,048-token generation cap.

Event	Errors	Observed	Expected	Std. dev.
Deficit at most two	961	112	8.84	2.45
Sign conflict	2,581	1,335	1,639.2	16.5

Appendix

Table 18: Random reference comparisons for 32B RFT-v1 errors. The program reference preserves gate types and order on five to eight qubit targets. The sign reference preserves each output’s unsigned overlap on all valid errors. The final column is the reference standard deviation, not a confidence interval.

(a) Mean constraint deficit
$n$	$N$	Cap (%)	Observed (SE)	Empty	Length	Uniform
2	19	0.0	1.47 (0.12)	1.84	1.58	1.75
3	46	0.0	2.09 (0.12)	2.24	2.51	2.65
4	95	0.0	3.11 (0.10)	3.29	3.51	3.61
5	131	0.0	3.79 (0.11)	4.08	4.39	4.59
6	207	0.0	4.68 (0.10)	4.84	5.37	5.58

Appendix

Table 19: Complete reference comparison for the 2,581 valid, state-incorrect 32B RFT-v1 outputs. Here

n

is the qubit count,

N

is the number of analyzed errors, and

r

counts missing independent signed target constraints. Empty denotes the zero-gate circuit, length denotes random operands with the observed gate sequence, and uniform denotes a uniformly random stabilizer state. Both random references exclude exact target matches. Cap is the percentage of all outputs at that qubit count excluded by the 2,000-character limit, not a percentage of

N

. Panel (a) reports mean deficits, with standard errors (SE) across observed outputs in parentheses. Panel (b) gives actual counts for observed and empty candidates and expected counts for the random references.

Deficit $r$	Nonzero $F$	$F=0$ count	$F>0$ count	Total
1	$2^{-1}$	70	72	142
2	$2^{-2}$	100	92	192
3	$2^{-3}$	106	72	178
4	$2^{-4}$	143	92	235
5	$2^{-5}$	116	88	204
6	$2^{-6}$	153	147	300

Appendix

Table 20: Complete constraint-deficit distribution of the 2,581 valid, state-incorrect 32B RFT-v1 outputs with saved QASM shorter than 2,000 characters, pooled over two to twelve qubits. Deficit

r

counts missing independent signed target constraints and

F

is squared overlap with the target state. The second column gives the fidelity value only for the nonzero-overlap branch. The two count columns partition each row, and none of these outputs prepares the target exactly.

System	Exact	Syntax fail	Label mismatch
Basic AG-CoT	102	5	3787
GRPO-v2	86	5	3803
RFT-v1	122	17	3755

Appendix

Table 21: 32B direct-verifier outcomes over 3,894 held-out prompts. Counts use the strict saved-label decision and separate exact matches from syntax failures, non-Clifford parses, and valid Clifford outputs with a saved-label mismatch.

Layer bucket	Correct / total	Exact@1 (%)	Test share (%)
1–2	35 / 47	74.5	1.2
3–5	27 / 64	42.2	1.6
6–10	17 / 132	12.9	3.4
11–14	5 / 102	4.9	2.6
15–20	1 / 142	0.7	3.6
21–50	0 / 664	0.0	17.1

Appendix

Table 22: GRPO-v2 direct evaluation by target circuit-layer bucket under strict saved-label Exact@1. Bucket percentages are rounded to one decimal place. The last bucket contains one correct output, whose rate of approximately 0.0365% rounds to 0.0%.

System	Exact@1 lab. (%)	StateEq@1 (%)	95% CI
	Direct target match
Zero-shot	0.10	0.33	[0.15, 0.52]
Basic AG-CoT	2.62	5.37	[4.67, 6.06]
Step AG-CoT	2.31	4.83	[4.15, 5.50]
GRPO-v2	2.21	5.26	[4.57, 5.98]
RFT-v1	3.13	6.14	[5.37, 6.91]

Appendix

Table 23: State-equivalence confidence intervals over 3,894 held-out prompts; lab. denotes the saved-label verifier convention.

Run	Exact successes	Exact@1 (%)	95% CI
Basic AG-CoT	102 / 3894	2.62	[2.13, 3.13]
GRPO-v2	86 / 3894	2.21	[1.77, 2.70]
RFT-v1	122 / 3894	3.13	[2.59, 3.70]

Appendix

Table 24: Bootstrap confidence intervals for the direct Exact@1 metric (%) computed over 10,000 resamples of the 3,894 held-out prompts.

Search run	Covered	Missed
RFT-v1 Best-of-64	202	3692

Appendix

Table 25: 32B strict saved-label search coverage over 3,894 held-out prompts.

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