Restrictions on non-Clifford fault tolerance and ruling out beyond-SQL quantum metrology

Abstract

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 $d\ge 3$ supporting a transversal logical action in level $D$ of the Clifford hierarchy, every stabilizer generating set must contain a check of weight at least $2^D$. Moreover, any $r$-level concatenated realization satisfies $r\le \lfloor {\log }_{2}n/D\rfloor$, 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 $\hat{I}¸$ can only induce a nontrivial logical action on an $n$-qubit code if its checks include irreducible stabilizers of weight $\hat{I}\text{c◯}(1/(n|\hat{I}¸{|}^2))$. 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 $|\hat{I}¸|=o(n^{-1/2})$, 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.

6+20 pages, 1 figure

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