Classical Simulability of Quantum Circuits with Shallow Magic Depth
Bibliographic record
Abstract
Quantum magic is a necessary resource for quantum computers to be not efficiently simulable by classical computers. Previous results have linked the of quantum magic, characterized by the number of T gates or the stabilizer rank, to classical simulability. However, the effect of the of quantum magic on the hardness of simulating a quantum circuit remains open. In this work, we investigate the classical simulability of quantum circuits with alternating Clifford and T layers across three tasks: amplitude estimation, sampling, and evaluating Pauli observables. In the case in which all T gates are distributed in a single layer, performing amplitude estimation and sampling to multiplicative error are already classically intractable under reasonable assumptions, but Pauli observables are easy to evaluate. Surprisingly, with the addition of just one T -gate layer or merely replacing all T gates with T 1 / 2 , the Pauli evaluation task reveals a sharp complexity transition from being in P to being GapP-complete. Nevertheless, when the precision requirement is relaxed to 1 / poly ( n ) additive error, we are able to give a polynomial-time classical algorithm to compute amplitudes, Pauli observables, and sampling from log ( n ) -sized marginal distributions for any magic-depth-1 circuit that is decomposable into a product of diagonal gates. This rules out certain forms of quantum advantage in these circuits. Our research provides new techniques to simulate highly magical circuits while shedding light on their complexity and their significant dependence on the magic depth.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".