Adding a logical √T, or square-root-of-T, gate to a simulated quantum gate set was associated with lower modeled costs for fine-grained single-qubit synthesis, the process of assembling one-qubit operations from available gates. Among feasible queries in a benchmark of randomly selected single-qubit targets, the gate set that included the extra primitive had an average chosen cost about 25% to 30% lower than the set without it. In one representative target, the modeled normalized cost was 2.45 rather than 18.40, corresponding to an 86.7% reduction.
The comparison came from a computational simulation study of fault-tolerant preparation and injection of a logical √T magic state, a resource state used as an additional gate primitive. The study used Monte Carlo sampling for the tetrahedral-code preparation and full density-matrix simulation for the morphed-code preparation and the Steane-code T-state comparator. The document is an arXiv version 1 preprint dated 26 Aug 2026.
The cheaper compilation used the costlier primitive
The result does not mean that √T injection was modeled as a cheaper standalone resource. For a post-selected logical primitive, expected cost was defined as the space-time volume required to obtain one accepted implementation. On that basis, the model assigned normalized costs of about 0.02 to transversal Clifford operations, 1 to T injection and 2.45 to √T injection.
The same compilation model assigned depolarizing-noise coefficients of 21p², 951p² and 1,734p² to transversal Clifford, T injection and √T injection, respectively, where p is the physical error-rate parameter used in the simulations. The figures describe a trade-off in available synthesis choices: √T had a higher assigned cost, but the gate set containing it had lower benchmark costs.
The modeled √T resource was prepared in two code settings: the [[15, 1, 3]] tetrahedral code and the smaller [[10, 1, 2]] morphed variant. A Steane-code preparation of the |T⟩ state served as the comparator. At low noise, all three protocols showed quadratic logical-error suppression, with fitted scalings of about 56p² for tetrahedral √T, 29p² for morphed √T and 25p² for Steane T.
Across the simulated physical-error range, the Steane preparation had the lowest logical error rate and the highest acceptance. The tetrahedral √T preparation had the highest logical error rate and the lowest acceptance, while the morphed version fell between them on acceptance and was slightly worse than Steane on logical error rate. The modeled preparation-versus-compilation trade-off was that √T was more expensive to inject, while the gate set including it had lower synthesis costs in the benchmark.
The simulated injection error varied by input state
A six-state injection test found quadratic scaling of logical error for every tested input state. Computational-basis inputs had lower logical error rates than equatorial inputs, so the tested error rates varied with input state.
Process tomography—a reconstruction of how a channel acts from test-state measurements—found that at p = 10−4 the representative residual injection channel was close to trace-preserving and unital. Its off-diagonal Pauli-block terms were small relative to the diagonal contractions, supporting a Pauli-channel approximation for the remaining noise.
For the injection shown in the study, the fitted residual Pauli components scaled as 230p² for X, 57p² for Y and 578p² for Z. The residual was therefore Z-dominated in the model, while the conservative depolarizing upper bound was about 1,734p².
More targets fit within the modeled cost limit
The broader comparison used GS1 without √T and GS2 with √T. Synthesis queries retrieved the 5,000 nearest stored database entries using Euclidean distance in Pauli-vector space. The supplied extraction reports 104 Haar-random single-qubit unitaries and says that the same target set was used for both gate sets.
Among feasible queries in the plotted benchmark, the average selected cost for GS2 was about 25% to 30% lower than for GS1. The comparison was stable at p = 10−5 and p = 10−4. Because the searches used a finite stored database, the percentage is conditional on the feasible queries and database used.
The difference was larger when the question was whether a query could fit within a specified limit. Near εtot = 10−1.5 and p = 10−3, 2.6% of queries were feasible with GS1 under the normalized cost cap C ≤ 17, compared with about 86% with GS2. That coverage result is specific to the finite database and error-budget regime used in the simulation.
What the comparison does—and does not—settle
Taken together, the figures describe a compilation trade-off within a stated model: √T injection carried a higher normalized cost than T injection, yet making √T available was associated with lower average selected cost in the single-qubit benchmark. The simulations used the same circuit-level depolarizing noise model throughout, so the reported percentages and coverage figures belong to that modeled setting.
The study’s question was narrowly framed around fine-grained single-qubit synthesis and the logical preparation and injection of √T. Its cost conclusions came from finite nearest-neighbor searches and the stated error-budget rules, so they describe that benchmark setup.
The paper states that all data and code used for the presented results and figures are available through Zenodo, listed as reference 52.
Paper data and sources
Original title: Fault-tolerant $|\sqrt{ \mathrm{T} }\rangle$ state preparation and injection for more efficient fine-grained quantum circuit synthesis
Authors: Berat Yenilen, Markus Müller, Manuel Rispler
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text