2026/07/29 by Constantin Cedillo Vayson de Pradenne, Ishaan Kannan, Harald Putterman +1
Physics and Astronomy · #quant-ph
6+20 pages, 1 figure
arxiv created 2026/07/29 · arxiv updated 2026/07/31
Quantum metrology promises a quadratic speedup over the standard quantum limit (SQL), but signal-aligned noise is expected to preclude this advantage in realistic settings. A potential route around known no-go results is to encode the sensors in a quantum code where the physical signal acts transversally as a logical gate. Understanding restrictions on transversal non-Clifford gates is therefore central to both quantum metrology and fault-tolerant quantum computation. Here, we prove such restrictions and apply them to transversal sensing. For any stabilizer code of distance d≥ 3 supporting a transversal logical action in level D of the Clifford hierarchy, every stabilizer generating set must contain a check of weight at least 2D. Moreover, any r-level concatenated realization satisfies r≤ \lfloor log2 n/D\rfloor, forcing r=1 and ruling out concatenation when applied to beyond-SQL metrology. We then show that transversal single-qubit rotations by a small angle θ can only induce a nontrivial logical action on an n-qubit code if its checks include irreducible stabilizers of weight Ω(1/(n|θ|2)). Here, many single-qubit errors commute with every stabilizer or logical Pauli below this weight and are only detected by a high-weight check, so their syndromes cannot be fault-tolerantly reconstructed from low-weight normalizer measurements. Since beyond-SQL transversal sensing requires |θ| = o(n-1/2), the weight of checks required for syndrome extraction diverges with n. Finally, we prove a broader metrological no-go theorem that avoids the assumptions of the quantum Cramér-Rao bound: constant-strength signal-aligned noise rules out any asymptotic advantage over the SQL in AC or DC sensing, even with biased estimators, nonstabilizer or approximate encodings, quantum memory, intermediate measurements, or adaptive control.