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arXiv:2610.16401

Magic-Free Certification of Logical Error Rates in Universal Fault-Tolerant Circuits

Gebrat Porta-San Miguel, Shahram Panahiyan, Dieter Jaksch

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

As quantum hardware enters the fault-tolerant regime, algorithms are beginning to be executed on encoded qubits, yet scalable certification of how well they run remains an open problem. Here we introduce magic-free certification, a certification protocol for universal fault-tolerant circuits that consumes no magic, the resource that generally places them beyond efficient classical simulation and is by far the most expensive element of a fault-tolerant computation. The protocol is code- and decoder-agnostic, runs interleaved with the target circuit on the same device, and has a sample complexity that grows linearly with the number of logical qubits and non-Clifford gates while remaining decoupled from the logical error rate itself. Given a predefined error tolerance $p_L^*$, the protocol determines at a chosen confidence level whether the device executes the target circuit at a logical error rate below $p_L^*$. The method relies on trap circuits, copies of the target with every magic state replaced by a stabilizer state, whose outcomes are deterministic and which, being indistinguishable from the target to stochastic Pauli noise, fail at the same rate. We illustrate the protocol in Steane-code simulations of a magic state injection gadget and a quantum Fourier transform.

Comments
21 pages, 3 figures

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Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity 7−1+3×10−57^{+3}_{-1}\times 10^{-5}7−1+3​×10−5 with a 14.8−1+1%14.8^{+1}_{-1}\%14.8−1+1​% discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity ≤2.3−9+9×10−4\leq 2.3^{+9}_{-9}\times 10^{-4}≤2.3−9+9​×10−4. Notably, this significantly outperforms the unencoded physical CH infidelity of 10−310^{-3}10−3. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of 6×10−106\times 10^{-10}6×10−10 (5×10−145\times 10^{-14}5×10−14) at two-qubit error rate of 10−310^{-3}10−3 (10−410^{-4}10−4) respectively.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
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