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

• URL: http://arxiv.org/abs/2607.27342v1
• Date: Wed, 29 Jul 2026 18:01:14 GMT
• Title: Restrictions on non-Clifford fault tolerance and ruling out beyond-SQL quantum metrology
• Abstract summary: A potential route around known no-go results is to encode sensors in a quantum code where the physical signal acts inducely as a logical gate.<n>We show that single-qubit rotations by a small angle $$ can only be fault-tolerantly reconstructed from low-weight normalizer measurements.<n>We also prove a broader metrological no-go that avoids the assumptions of the quantum Cramér-Rao bound: constant-strength signal-aligned noise rules out any advantage over computation in AC or DC sensing.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $r\leq \lfloor \log_2 n/D\rfloor$, 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 $θ$ 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.

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